diff --git a/changes-set.txt b/changes-set.txt index 0be996339c..c34483b6ee 100644 --- a/changes-set.txt +++ b/changes-set.txt @@ -71,7 +71,6 @@ proposed syl6eleq eleqtrdi compare to eleqtri or eleqtrd proposed syl6eleqr eleqtrrdi compare to eleqtrri or eleqtrrd proposed syl6ss sstrdi compare to sstri or sstrd proposed syl6sseq sseqtrdi compare to sseqtri or sseqtrd -proposed sseqtr4i sseqtrri proposed sseqtr4d sseqtrrd proposed syl6sseqr sseqtrrdi proposed syl6eqss eqsstrdi compare to eqsstri or eqsstrd @@ -81,6 +80,7 @@ make a github issue.) DONE: Date Old New Notes +14-Jan-24 sseqtr4i sseqtrri 13-Jan-24 mndlsmidm [same] moved from AV's mathbox to main set.mm 13-Jan-24 smndlsmidm [same] moved from AV's mathbox to main set.mm 13-Jan-24 cycsubm [same] moved from AV's mathbox to main set.mm diff --git a/discouraged b/discouraged index 5ffe0c6764..4d78eec0c9 100755 --- a/discouraged +++ b/discouraged @@ -398,7 +398,6 @@ "4syl" is used by "qtopcmap". "4syl" is used by "qtopf1". "4syl" is used by "restmetu". -"4syl" is used by "revccat". "4syl" is used by "revrev". "4syl" is used by "rpvmasum2". "4syl" is used by "rpvmasumlem". @@ -406,7 +405,6 @@ "4syl" is used by "sdclem2". "4syl" is used by "sdomsdomcard". "4syl" is used by "serf0". -"4syl" is used by "signstfvp". "4syl" is used by "signstres". "4syl" is used by "smoiso". "4syl" is used by "srng0". @@ -415,7 +413,6 @@ "4syl" is used by "stoweidlem11". "4syl" is used by "stoweidlem14". "4syl" is used by "subfacp1lem5". -"4syl" is used by "swrdccat2". "4syl" is used by "symgtrinv". "4syl" is used by "tmsxms". "4syl" is used by "tocyc01". @@ -13191,7 +13188,7 @@ New usage of "4atex2-0aOLDN" is discouraged (1 uses). New usage of "4atex2-0bOLDN" is discouraged (0 uses). New usage of "4atex2-0cOLDN" is discouraged (0 uses). New usage of "4ipval2" is discouraged (2 uses). -New usage of "4syl" is discouraged (193 uses). +New usage of "4syl" is discouraged (190 uses). New usage of "5oai" is discouraged (0 uses). New usage of "5oalem1" is discouraged (1 uses). New usage of "5oalem2" is discouraged (2 uses). @@ -13625,7 +13622,6 @@ New usage of "bj-csbsnlem" is discouraged (1 uses). New usage of "bj-currypeirce" is discouraged (0 uses). New usage of "bj-denot" is discouraged (0 uses). New usage of "bj-dtru" is discouraged (0 uses). -New usage of "bj-dvdemo1" is discouraged (0 uses). New usage of "bj-equsalhv" is discouraged (0 uses). New usage of "bj-eximALT" is discouraged (1 uses). New usage of "bj-gl4" is discouraged (0 uses). @@ -14059,6 +14055,7 @@ New usage of "cbncms" is discouraged (5 uses). New usage of "cbv2OLD" is discouraged (0 uses). New usage of "cbvabvOLD" is discouraged (0 uses). New usage of "cbval2OLD" is discouraged (0 uses). +New usage of "cbval2vOLD" is discouraged (0 uses). New usage of "cbvalvOLD" is discouraged (0 uses). New usage of "cbveuALT" is discouraged (0 uses). New usage of "cbvexsv" is discouraged (2 uses). @@ -15171,14 +15168,12 @@ New usage of "erngring-rN" is discouraged (0 uses). New usage of "erngset-rN" is discouraged (3 uses). New usage of "eu1OLD" is discouraged (0 uses). New usage of "euaeOLD" is discouraged (0 uses). -New usage of "euanvOLD" is discouraged (0 uses). New usage of "eubiOLD" is discouraged (0 uses). New usage of "eubidOLD" is discouraged (0 uses). New usage of "eubiiOLD" is discouraged (0 uses). New usage of "euequOLD" is discouraged (0 uses). New usage of "euimOLD" is discouraged (0 uses). New usage of "eujustALT" is discouraged (0 uses). -New usage of "euorvOLD" is discouraged (0 uses). New usage of "ex-decpmul" is discouraged (0 uses). New usage of "ex-gt" is discouraged (0 uses). New usage of "ex-gte" is discouraged (0 uses). @@ -16989,7 +16984,6 @@ New usage of "pmod2iN" is discouraged (0 uses). New usage of "pmodN" is discouraged (0 uses). New usage of "pmodl42N" is discouraged (1 uses). New usage of "pn0sr" is discouraged (4 uses). -New usage of "pncan3OLD" is discouraged (0 uses). New usage of "pnonsingN" is discouraged (3 uses). New usage of "pointpsubN" is discouraged (0 uses). New usage of "pointsetN" is discouraged (1 uses). @@ -17136,7 +17130,6 @@ New usage of "retbwax1" is discouraged (0 uses). New usage of "retbwax2" is discouraged (3 uses). New usage of "retbwax3" is discouraged (0 uses). New usage of "retbwax4" is discouraged (0 uses). -New usage of "reubidvaOLD" is discouraged (0 uses). New usage of "reueq1OLD" is discouraged (0 uses). New usage of "rexab2OLD" is discouraged (0 uses). New usage of "rexanidOLD" is discouraged (0 uses). @@ -17746,7 +17739,6 @@ New usage of "w-bnj17" is discouraged (103 uses). New usage of "w-bnj19" is discouraged (8 uses). New usage of "watfvalN" is discouraged (1 uses). New usage of "watvalN" is discouraged (1 uses). -New usage of "wfrlem4OLD" is discouraged (0 uses). New usage of "wl-embant" is discouraged (0 uses). New usage of "wl-impchain-a1-1" is discouraged (1 uses). New usage of "wl-impchain-a1-2" is discouraged (1 uses). @@ -18046,7 +18038,6 @@ Proof modification of "bj-cbv2hv" is discouraged (67 steps). Proof modification of "bj-cbv2v" is discouraged (47 steps). Proof modification of "bj-cbv3hv2" is discouraged (10 steps). Proof modification of "bj-cbval" is discouraged (42 steps). -Proof modification of "bj-cbval2v" is discouraged (85 steps). Proof modification of "bj-cbval2vv" is discouraged (20 steps). Proof modification of "bj-cbvaldv" is discouraged (20 steps). Proof modification of "bj-cbvaldvav" is discouraged (22 steps). @@ -18054,7 +18045,6 @@ Proof modification of "bj-cbvalim" is discouraged (64 steps). Proof modification of "bj-cbvalimi" is discouraged (34 steps). Proof modification of "bj-cbvalimt" is discouraged (112 steps). Proof modification of "bj-cbvex" is discouraged (42 steps). -Proof modification of "bj-cbvex2v" is discouraged (70 steps). Proof modification of "bj-cbvex2vv" is discouraged (20 steps). Proof modification of "bj-cbvex4vv" is discouraged (61 steps). Proof modification of "bj-cbvexdv" is discouraged (55 steps). @@ -18087,13 +18077,9 @@ Proof modification of "bj-dfif" is discouraged (39 steps). Proof modification of "bj-dfnnf3" is discouraged (34 steps). Proof modification of "bj-disj2r" is discouraged (88 steps). Proof modification of "bj-disjsn01" is discouraged (18 steps). -Proof modification of "bj-dral1v" is discouraged (36 steps). -Proof modification of "bj-drex1v" is discouraged (42 steps). -Proof modification of "bj-drnf1v" is discouraged (50 steps). Proof modification of "bj-drnf2v" is discouraged (10 steps). Proof modification of "bj-dtru" is discouraged (146 steps). Proof modification of "bj-dtrucor2v" is discouraged (31 steps). -Proof modification of "bj-dvdemo1" is discouraged (28 steps). Proof modification of "bj-dvelimdv" is discouraged (64 steps). Proof modification of "bj-dvelimdv1" is discouraged (63 steps). Proof modification of "bj-dvelimv" is discouraged (25 steps). @@ -18219,6 +18205,7 @@ Proof modification of "cayleyhamiltonALT" is discouraged (657 steps). Proof modification of "cbv2OLD" is discouraged (40 steps). Proof modification of "cbvabvOLD" is discouraged (12 steps). Proof modification of "cbval2OLD" is discouraged (85 steps). +Proof modification of "cbval2vOLD" is discouraged (85 steps). Proof modification of "cbvalvOLD" is discouraged (53 steps). Proof modification of "cbveuALT" is discouraged (48 steps). Proof modification of "cbvexsv" is discouraged (29 steps). @@ -18543,14 +18530,12 @@ Proof modification of "equsexvwOLD" is discouraged (36 steps). Proof modification of "equviniOLD" is discouraged (68 steps). Proof modification of "eu1OLD" is discouraged (86 steps). Proof modification of "euaeOLD" is discouraged (49 steps). -Proof modification of "euanvOLD" is discouraged (7 steps). Proof modification of "eubiOLD" is discouraged (15 steps). Proof modification of "eubidOLD" is discouraged (48 steps). Proof modification of "eubiiOLD" is discouraged (17 steps). Proof modification of "euequOLD" is discouraged (36 steps). Proof modification of "euimOLD" is discouraged (37 steps). Proof modification of "eujustALT" is discouraged (188 steps). -Proof modification of "euorvOLD" is discouraged (7 steps). Proof modification of "ex-decpmul" is discouraged (304 steps). Proof modification of "ex-natded5.13" is discouraged (67 steps). Proof modification of "ex-natded5.13-2" is discouraged (21 steps). @@ -19068,7 +19053,6 @@ Proof modification of "pm2.21ddALT" is discouraged (10 steps). Proof modification of "pm2.43bgbi" is discouraged (16 steps). Proof modification of "pm2.43cbi" is discouraged (34 steps). Proof modification of "pm2.61iOLD" is discouraged (13 steps). -Proof modification of "pncan3OLD" is discouraged (49 steps). Proof modification of "preleqALT" is discouraged (115 steps). Proof modification of "prmgaplcm" is discouraged (247 steps). Proof modification of "prmgapprmo" is discouraged (387 steps). @@ -19150,7 +19134,6 @@ Proof modification of "retbwax1" is discouraged (195 steps). Proof modification of "retbwax2" is discouraged (127 steps). Proof modification of "retbwax3" is discouraged (20 steps). Proof modification of "retbwax4" is discouraged (13 steps). -Proof modification of "reubidvaOLD" is discouraged (10 steps). Proof modification of "reueq1OLD" is discouraged (11 steps). Proof modification of "rexab2OLD" is discouraged (67 steps). Proof modification of "rexanidOLD" is discouraged (37 steps). @@ -19446,7 +19429,6 @@ Proof modification of "vtocl2dOLD" is discouraged (118 steps). Proof modification of "vtoclALT" is discouraged (11 steps). Proof modification of "vtoclgftOLD" is discouraged (142 steps). Proof modification of "vtxdusgr0edgnelALT" is discouraged (94 steps). -Proof modification of "wfrlem4OLD" is discouraged (543 steps). Proof modification of "wl-cases2-dnf" is discouraged (85 steps). Proof modification of "wl-dfclab" is discouraged (73 steps). Proof modification of "wl-embant" is discouraged (12 steps). diff --git a/iset.mm b/iset.mm index c44c429a9e..afae85e990 100644 --- a/iset.mm +++ b/iset.mm @@ -27333,11 +27333,11 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $} ${ - sseqtr4.1 $e |- A C_ B $. - sseqtr4.2 $e |- C = B $. + sseqtrri.1 $e |- A C_ B $. + sseqtrri.2 $e |- C = B $. $( Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.) $) - sseqtr4i $p |- A C_ C $= + sseqtrri $p |- A C_ C $= ( eqcomi sseqtri ) ABCDCBEFG $. $} @@ -27533,7 +27533,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.) $) eqimss2i $p |- B C_ A $= - ( ssid sseqtr4i ) BBABDCE $. + ( ssid sseqtrri ) BBABDCE $. $} $( Two classes are different if they don't include the same class. @@ -28918,7 +28918,7 @@ subset operation on the right hand side could be an equality (that is, 27-Jul-2018.) $) difdif2ss $p |- ( ( A \ B ) u. ( A i^i C ) ) C_ ( A \ ( B \ C ) ) $= ( cin cun cvv wss inssdif unss2 ax-mp difindiss sstri invdif eqcomi difeq2i - cdif sseqtr4i ) ABPZACDZEZABFCPZDZPZABCPZPTRAUAPZEZUCSUEGTUFGACHSUERIJABUAK + cdif sseqtrri ) ABPZACDZEZABFCPZDZPZABCPZPTRAUAPZEZUCSUEGTUFGACHSUERIJABUAK LUDUBAUBUDBCMNOQ $. $( De Morgan's law for union. Theorem 5.2(13) of [Stoll] p. 19. @@ -31615,44 +31615,44 @@ we define it to be empty in this case (see ~ opprc1 and ~ opprc2 ). For $( A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.) $) snsspr1 $p |- { A } C_ { A , B } $= - ( csn cun cpr ssun1 df-pr sseqtr4i ) ACZIBCZDABEIJFABGH $. + ( csn cun cpr ssun1 df-pr sseqtrri ) ACZIBCZDABEIJFABGH $. $( A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) $) snsspr2 $p |- { B } C_ { A , B } $= - ( csn cun cpr ssun2 df-pr sseqtr4i ) BCZACZIDABEIJFABGH $. + ( csn cun cpr ssun2 df-pr sseqtrri ) BCZACZIDABEIJFABGH $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp1 $p |- { A } C_ { A , B , C } $= - ( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtr4i ) ADZABEZCDZFZABCGMNPA + ( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtrri ) ADZABEZCDZFZABCGMNPA BHNOIJABCKL $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp2 $p |- { B } C_ { A , B , C } $= - ( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtr4i ) BDZABEZCDZFZABCGMNPA + ( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtrri ) BDZABEZCDZFZABCGMNPA BHNOIJABCKL $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp3 $p |- { C } C_ { A , B , C } $= - ( csn cpr cun ctp ssun2 df-tp sseqtr4i ) CDZABEZKFABCGKLHABCIJ $. + ( csn cpr cun ctp ssun2 df-tp sseqtrri ) CDZABEZKFABCGKLHABCIJ $. $( A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.) $) prsstp12 $p |- { A , B } C_ { A , B , C } $= - ( cpr csn cun ctp ssun1 df-tp sseqtr4i ) ABDZKCEZFABCGKLHABCIJ $. + ( cpr csn cun ctp ssun1 df-tp sseqtrri ) ABDZKCEZFABCGKLHABCIJ $. $( A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.) $) prsstp13 $p |- { A , C } C_ { A , B , C } $= - ( cpr ctp prsstp12 tpcomb sseqtr4i ) ACDACBEABCEACBFABCGH $. + ( cpr ctp prsstp12 tpcomb sseqtrri ) ACDACBEABCEACBFABCGH $. $( A pair is a subset of an unordered triple containing its members. (Contributed by Jim Kingdon, 11-Aug-2018.) $) prsstp23 $p |- { B , C } C_ { A , B , C } $= - ( cpr ctp prsstp12 tprot sseqtr4i ) BCDBCAEABCEBCAFABCGH $. + ( cpr ctp prsstp12 tprot sseqtrri ) BCDBCAEABCEBCAFABCGH $. ${ prss.1 $e |- A e. _V $. @@ -33374,7 +33374,7 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and 10-Dec-2004.) $) iunxdif2 $p |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C ) $= - ( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtr4i + ( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtrri wa jctil eqss sylibr ) EFHBCDIZJACKZBUDFLZACELZHZUGUFHZTUFUGMUEUIUHABCUDE FNUFBCFLZUGUDCHUFUJHCDOBUDCFPQABCEFGRSUAUFUGUBUC $. $} @@ -33750,7 +33750,7 @@ same disjoint variable group (meaning ` A ` cannot depend on ` x ` ) and the double power class of that class. (Contributed by BJ, 29-Apr-2021.) $) pwpwssunieq $p |- { x | U. x = A } C_ ~P ~P A $= - ( cv cuni wceq cab wss cpw eqimss ss2abi pwpwab sseqtr4i ) ACDZBEZAFMBGZA + ( cv cuni wceq cab wss cpw eqimss ss2abi pwpwab sseqtrri ) ACDZBEZAFMBGZA FBHHNOAMBIJABKL $. $} @@ -38109,7 +38109,7 @@ the empty set as an element and is not a successor (i.e. that is the union [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.) $) sssucid $p |- A C_ suc A $= - ( csn cun csuc ssun1 df-suc sseqtr4i ) AAABZCADAHEAFG $. + ( csn cun csuc ssun1 df-suc sseqtrri ) AAABZCADAHEAFG $. $( Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, @@ -41129,7 +41129,7 @@ is most interesting when the natural number is a successor (as seen in (Contributed by NM, 16-Jul-1995.) $) opabssxp $p |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B ) $= - ( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtr4i ) BFDGCFEGHZAHZBCIOB + ( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtrri ) BFDGCFEGHZAHZBCIOB CIDEJPOBCOAKLBCDEMN $. $} @@ -43393,7 +43393,7 @@ is most interesting when the natural number is a successor (as seen in $( The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) $) dmresi $p |- dom ( _I |` A ) = A $= - ( cid cdm wss cres wceq cvv ssv dmi sseqtr4i ssdmres mpbi ) ABCZDBAECAFAGMA + ( cid cdm wss cres wceq cvv ssv dmi sseqtrri ssdmres mpbi ) ABCZDBAECAFAGMA HIJABKL $. $( TODO : replace uses of ~ resid with set.mm/dfrel3 and delete ~ resid $) @@ -43596,7 +43596,7 @@ is most interesting when the natural number is a successor (as seen in $( A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.) $) cnvimass $p |- ( `' A " B ) C_ dom A $= - ( ccnv cima crn cdm imassrn dfdm4 sseqtr4i ) ACZBDJEAFJBGAHI $. + ( ccnv cima crn cdm imassrn dfdm4 sseqtrri ) ACZBDJEAFJBGAHI $. $( The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) $) @@ -44317,7 +44317,7 @@ is most interesting when the natural number is a successor (as seen in 16-Jan-2006.) $) ssrnres $p |- ( B C_ ran ( C |` A ) <-> ran ( C i^i ( A X. B ) ) = B ) $= ( vy vx cxp cin crn wceq wss rnss ax-mp cvv cv wex wa elrn2 bitr2i 3bitri - wcel cres inss2 rnxpss sstri eqss mpbiran ssv xpss12 mp2an sslin sseqtr4i + wcel cres inss2 rnxpss sstri eqss mpbiran ssv xpss12 mp2an sslin sseqtrri ssid df-res sstr mpan2 cop ssel syl6ib ancrd opelxp anbi2i opelres anbi1i vex elin anass exbii 19.41v syl6ibr ssrdv impbii ) CABFZGZHZBIZBVNJZBCAUA ZHZJZVOVNBJVPVNVLHZBVMVLJVNVTJCVLUBVMVLKLABUCUDVNBUEUFVPVSVPVNVRJZVSVMVQJ @@ -44416,7 +44416,7 @@ is most interesting when the natural number is a successor (as seen in $( The double converse of a class strips out all elements that are not ordered pairs. (Contributed by NM, 8-Dec-2003.) $) cnvcnv $p |- `' `' A = ( A i^i ( _V X. _V ) ) $= - ( ccnv cvv cxp cin wceq wrel relcnv df-rel mpbi relxp dfrel2 sseqtr4i cnvin + ( ccnv cvv cxp cin wceq wrel relcnv df-rel mpbi relxp dfrel2 sseqtrri cnvin wss dfss cnveqi inss2 mpbir eqtr3i 3eqtr2i ) ABZBZUCCCDZBZBZEZUBUEEZBZAUDEZ UCUFOUCUGFUCUDUFUCGUCUDOUBHUCIJUDGUFUDFCCKUDLJMUCUFPJUBUENUJBZBZUIUJUKUHAUD NQUJGZULUJFUMUJUDOAUDRUJISUJLJTUA $. @@ -45013,7 +45013,7 @@ is most interesting when the natural number is a successor (as seen in $( The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) $) cnvssrndm $p |- `' A C_ ( ran A X. dom A ) $= - ( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtr4i + ( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtrri wss ) ABZOCZODZEZADZACZEOFORNAGOHISPTQAJAKLM $. $( Composition as a subset of the cross product of factors. (Contributed by @@ -56207,7 +56207,7 @@ associative structure (such as a group). (Contributed by NM, dmmpossx $p |- dom F C_ U_ x e. A ( { x } X. B ) $= ( vu vt vv cv csn csb cxp ciun cfv nfcv nfcsb1v csbeq1a wceq c1st cbvmpox cdm c2nd cmpo cmpt nfcsb sylan9eqr cop vex op1std csbeq1d op2ndd csbeq2dv - eqtrd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtr4i ) FUCHCHK + eqtrd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtrri ) FUCHCHK ZLZAVDDMZNZOZACAKZLZDNZOIVHAIKZUAPZBVLUDPZEMZMZFABCDEUEHJCVFAVDBJKZEMZMZU EFIVHVPUFABHJCDEVFVSHDQAVDDRZHEQJEQAVDVRRBAVDVRBVDQBVQERUGAVDDSZBKVQTVIVD TZEVRVSBVQESAVDVRSUHUBGHJICVFVPVSVLVDVQUITZVPAVDVOMVSWCAVMVDVOVDVQVLHUJZJ @@ -58512,7 +58512,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and tfrcllemssrecs $p |- ( ph -> U. A C_ recs ( G ) ) $= ( cuni cv wf cfv cres wceq wa con0 wrex cab wral crecs word wss wi ssrexv ordsson 3syl ss2abdv eqsstrid unissd wfn ffn anim1i reximi ss2abi df-recs - unissi sseqtr4i syl6ss ) ADKBLZEFLZMZCLZVBNVBVDOGNPCVAUAZQZBRSZFTZKZGUBZA + unissi sseqtrri syl6ss ) ADKBLZEFLZMZCLZVBNVBVDOGNPCVAUAZQZBRSZFTZKZGUBZA DVHADVFBHSZFTVHIAVKVGFAHUCHRUDVKVGUEJHUGVFBHRUFUHUIUJUKVIVBVAULZVEQZBRSZF TZKVJVHVOVGVNFVFVMBRVCVLVEVAEVBUMUNUOUPURBCFGUQUSUT $. $} @@ -59913,7 +59913,7 @@ defined for all sets (being defined for all ordinals might be enough if sucinc.1 $e |- F = ( z e. _V |-> suc z ) $. $( Successor is increasing. (Contributed by Jim Kingdon, 25-Jun-2019.) $) sucinc $p |- A. x x C_ ( F ` x ) $= - ( cv cfv wss csuc sssucid cvv wcel wceq sucex suceq fvmptg mp2an sseqtr4i + ( cv cfv wss csuc sssucid cvv wcel wceq sucex suceq fvmptg mp2an sseqtrri vex ax-gen ) AEZTCFZGATTHZUATITJKUBJKUAUBLARZTUCMBTBEZHUBJJCUDTNDOPQS $. $d A z $. $d B z $. @@ -62589,7 +62589,7 @@ the first case of his notation (simple exponentiation) and subscript it after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) $) mapex $p |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V ) $= - ( wcel wa cv cab cxp cpw wss cvv fssxp ss2abi df-pw sseqtr4i xpexg pwexg + ( wcel wa cv cab cxp cpw wss cvv fssxp ss2abi df-pw sseqtrri xpexg pwexg wf syl ssexg sylancr ) ACFBDFGZABEHZTZEIZABJZKZLUIMFZUGMFUGUEUHLZEIUIUFUK EABUENOEUHPQUDUHMFUJABCDRUHMSUAUGUIMUBUC $. $} @@ -65771,7 +65771,7 @@ the first case of his notation (simple exponentiation) and subscript it exmidpw $p |- ( EXMID <-> ~P 1o ~~ 2o ) $= ( vx wem c1o cpw c2o cen wbr cvv wcel wceq c0 csn df1o2 p0ex eqeltri cpr cv pwex wss wi wo wal exmid01 biimpi 19.21bi pweqi eleq2i velpw bitri vex elpr - 3imtr4g ssrdv pwpw0ss sseqtr4i a1i eqssd df2o2 syl6eqr eqeng mpsyl 0nep0 wa + 3imtr4g ssrdv pwpw0ss sseqtrri a1i eqssd df2o2 syl6eqr eqeng mpsyl 0nep0 wa wne 0ex prss mpbir en2eqpr 3expb mpan2 eleq2d 3bitr3g biimpd alrimiv sylibr mpi impbii ) BCDZEFGZVRHIBVREJVSCCKLZHMNORBVRKVTPZEBVRWABAVRWABAQZVTSZWBKJW BVTJUAZWBVRIZWBWAIZBWCWDTZABWGAUBZAUCZUDUEWEWBVTDZIWCVRWJWBCVTMUFZUGAVTUHUI @@ -68052,7 +68052,7 @@ readily usable (e.g., by ~ djudom and ~ djufun ) while the simpler casefun $p |- ( ph -> Fun case ( F , G ) ) $= ( cinl ccnv ccom cinr wfun cdm cin c0 wceq cvv csn cxp wf1 wss crn cun wf cdjucase wf1o djulf1o f1of1 ax-mp df-f1 simprbi funco syl2anc c1o djurf1o - mp1i dmcoss df-rn sseqtr4i ss2in mp2an rnresv eqcomi ineq12i djuinr eqtri + mp1i dmcoss df-rn sseqtrri ss2in mp2an rnresv eqcomi ineq12i djuinr eqtri cres a1i sseqtrid ss0 syl funun syl21anc df-case funeqi sylibr ) ABFGZHZC IGZHZUAZJZBCUCZJAVPJZVRJZVPKZVRKZLZMNZVTABJVOJZWBDOMPOQZFRZWHAOWIFUDWJUEO WIFUFUGWJOWIFUBWHOWIFUHUIUNBVOUJUKACJVQJZWCEOULPOQZIRZWKAOWLIUDWMUMOWLIUF @@ -68425,7 +68425,7 @@ property of disjoint unions (see ~ updjud in the case of functions). djufun $p |- ( ph -> Fun ( F |_|d G ) ) $= ( cdm cres ccnv ccom wfun cin c0 wceq wf1 wf df-f1 simprbi mp1i funco wss cinl cinr cun cdjud cdju inlresf1 syl2anc inrresf1 crn dmcoss df-rn ss2in - sseqtr4i mp2an djuinr a1i sseqtrid ss0 syl syl21anc df-djud funeqi sylibr + sseqtrri mp2an djuinr a1i sseqtrid ss0 syl syl21anc df-djud funeqi sylibr funun ) ABUABFZGZHZIZCUBCFZGZHZIZUCZJZBCUDZJAVHJZVLJZVHFZVLFZKZLMZVNABJVG JZVPDVEVEVIUEZVFNZWBAVEVIUFWDVEWCVFOWBVEWCVFPQRBVGSUGACJVKJZVQEVIWCVJNZWE AVEVIUHWFVIWCVJOWEVIWCVJPQRCVKSUGAVTLTWAAVFUIZVJUIZKZVTLVRWGTVSWHTVTWITVR @@ -81542,7 +81542,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) $) ressxr $p |- RR C_ RR* $= - ( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtr4i ) AABCDZEFAJGHI $. + ( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtrri ) AABCDZEFAJGHI $. $( The Cartesian product of standard reals are a subset of the Cartesian product of extended reals (common case). (Contributed by David A. @@ -81640,7 +81640,7 @@ product of extended reals (common case). (Contributed by David A. ltrelxr $p |- < C_ ( RR* X. RR* ) $= ( vx vy cv cr wcel copab cmnf csn cun cxp cxr wa eqsstri sstri wss ressxr cpnf unssi xpss12 mp2an clt cltrr wbr w3a df-ltxr df-3an opabbii opabssxp - rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtr4i snsspr1 ) UAACZDEZBCZDEZUPURU + rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtrri snsspr1 ) UAACZDEZBCZDEZUPURU BUCZUDZABFZDGHZIZQHZJZVCDJZIZIKKJZABUEVBVHVIVBDDJZVIVBUQUSLUTLZABFVJVAVKA BUQUSUTUFUGUTABDDUHMUINVFVGVIVDKOVEKOVFVIODVCKPVCQGUJZKQGUKVLDVLIKVLDULUM UNZNZRVEVLKQGUOVMNVDKVEKSTVCKODKOVGVIOVNPVCKDKSTRRM $. @@ -90840,7 +90840,7 @@ Nonnegative integers (as a subset of complex numbers) $( Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) $) nnssnn0 $p |- NN C_ NN0 $= - ( cn cc0 csn cun cn0 ssun1 df-n0 sseqtr4i ) AABCZDEAIFGH $. + ( cn cc0 csn cun cn0 ssun1 df-n0 sseqtrri ) AABCZDEAIFGH $. $( Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) $) @@ -91021,7 +91021,7 @@ Nonnegative integers (as a subset of complex numbers) (Contributed by Mario Carneiro, 17-Jul-2014.) $) un0addcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T ) $= ( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1 - cun sseqtr4i sseldi expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom + cun sseqtrri sseldi expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addid1d simpr jaod oveq2d syl5bi impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZIV OCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVSAV @@ -91034,7 +91034,7 @@ Nonnegative integers (as a subset of complex numbers) $( If ` S ` is closed under multiplication, then so is ` S u. { 0 } ` . (Contributed by Mario Carneiro, 17-Jul-2014.) $) un0mulcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T ) $= - ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtr4i cc sselda csn wi expr + ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtrri cc sselda csn wi expr cun ssun1 sseldi mul02d wss ssun2 c0ex mpbir syl6eqel elsni oveq1d eleq1d snss syl5ibrcom impancom jaodan sylan2b 0cnd snssd eqsstrid mul01d oveq2d unssd jaod syl5bi impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOUDZ @@ -91277,7 +91277,7 @@ finite and infinite sets (and therefore if the set size is a nonnegative $( The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) $) nn0ssxnn0 $p |- NN0 C_ NN0* $= - ( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtr4i ) AABCZDEAIFGH $. + ( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtrri ) AABCZDEAIFGH $. $( A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) $) @@ -98690,7 +98690,7 @@ Infinity and the extended real number system (cont.) $( Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.) $) fzssnn $p |- ( M e. NN -> ( M ... N ) C_ NN ) $= - ( cn wcel cfz co c1 wss cuz cfv fzss1 nnuz eleq2s fzssuz sseqtr4i syl6ss ) + ( cn wcel cfz co c1 wss cuz cfv fzss1 nnuz eleq2s fzssuz sseqtrri syl6ss ) ACDABEFZGBEFZCQRHAGIJZCAGBKLMRSCGBNLOP $. $( Join a successor to the end of a finite set of sequential integers. @@ -99796,7 +99796,7 @@ Finite intervals of nonnegative integers (or "finite sets of sequential 13-May-2018.) $) fzossnn0 $p |- ( M e. NN0 -> ( M ..^ N ) C_ NN0 ) $= ( cn0 wcel cfzo co cc0 cfz fzossfz wss cuz fzss1 nn0uz eleq2s sstrid fzssuz - cfv sseqtr4i syl6ss ) ACDZABEFZGBHFZCTUAABHFZUBABIUCUBJAGKQZCAGBLMNOUBUDCGB + cfv sseqtrri syl6ss ) ACDZABEFZGBHFZCTUAABHFZUBABIUCUBJAGKQZCAGBLMNOUBUDCGB PMRS $. $( One direction of splitting a half-open integer range in half. @@ -117572,7 +117572,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) isumclim3 $p |- ( ph -> F ~~> sum_ k e. Z A ) $= ( vm vx cli cfv csu wcel cc wceq cdm wbr climdm sylib cv cmpt caddc eqidd cseq wa fmpttd ffvelrnda isum wral ralrimiva sumfct syl cio cvv seqex a1i - cfz simpl cres wss cuz fzssuz sseqtr4i resmpt ax-mp fveq1i fvres syl5reqr + cfz simpl cres wss cuz fzssuz sseqtrri resmpt ax-mp fveq1i fvres syl5reqr sumeq2i ssralv mpsyl syl5eq simpr syl6eleq eleq2i biimpri syl2an fsum3ser co 3eqtr2rd climeq iotabidv df-fv 3eqtr4g 3eqtr3d breqtrrd ) AEEOPZGBDQZO AEOUAZREWLOUBJEUCUDAGMUEZDGBUFZPZMQZUGWPFUIZOPZWMWLAWQMWPFGHIAWOGRZUJWQUH @@ -127597,7 +127597,7 @@ According to Wikipedia ("Least common multiple", 27-Aug-2020, 1z syl3an1 syl3an2 ancoms 3adant3 anim12d pm4.38 df-ne nesym ioran bitr4i anbi12i syl6bb syl6 syl2an syld eluzelz caddc zltp1le mpan breq1i syl6bbr imp df-2 zltlem1 anbi12d peano2zm elfz mp3an2 bitr4d bitr3d anasss expcom - pm5.32d fzssuz wss 2eluzge1 uzss ax-mp sstri nnuz sseqtr4i sseli pm4.71ri + pm5.32d fzssuz wss 2eluzge1 uzss ax-mp sstri nnuz sseqtrri sseli pm4.71ri 2z notbid pm5.74da bi2.04 con2b 3bitr3g ralbidv2 pm5.32i bitri ) BUACBDEU BZCZAUCZBUDFZYOGUEZYOBUEZUFZHZAIUGZJYNYPKZADBGUHLZUNLZUGZJABUIYNUUAUUEYNY TUUBAIUUDYNYPYOICZYSHZHYPYOUUDCZKZHUUFYTHUUHUUBHYNYPUUGUUIYNYPJZUUGUUFYSK @@ -129200,7 +129200,7 @@ reduced fraction representation (no common factors, denominator eqtrd syl3anc wf1o crab ssrab2 eqsstri mpbird syl2an cmpt eqtr3d eqeltrrd eqtri sylib cfn 0z fzofig sylancr eqeltrid wdc gcdcld nn0zd zdceq ssfirab 1zzd dfphi2 fveq2i syl6eqr simprbi dvdsmul2 opelxpi eqeltrd wfun cdm crth - dvdstr wfn f1ofn fnfun fndm sseqtrrid funimass4 ccnv mp2an sseqtr4i sseli + dvdstr wfn f1ofn fnfun fndm sseqtrrid funimass4 ccnv mp2an sseqtrri sseli xpss12 f1ocnvfv2 wf f1ocnv f1of ffvelrn opelxp rpmul funfvima2 imp syldan cbvmptv ex ssrdv eqssd wf1 f1of1 a1i zmulcld f1imaeng eqbrtrrd 1z sylancl xpfi hashen hashxp rabeqi oveq12d 3eqtr4d ) AKUCUDZFUCUDZJUCUDZUEUFZHIUEU @@ -133157,7 +133157,7 @@ a topological space (with the topology extractor function coming out the Kingdon, 4-Mar-2023.) $) tgvalex $p |- ( B e. V -> ( topGen ` B ) e. _V ) $= ( vy wcel ctg cfv cpw cin cuni wss cab cvv tgval inss1 unissi sstr ss2abi - cv mpan2 df-pw sseqtr4i uniexg pwexd ssexg sylancr eqeltrd ) ABDZAEFCRZAU + cv mpan2 df-pw sseqtrri uniexg pwexd ssexg sylancr eqeltrd ) ABDZAEFCRZAU HGZHZIZJZCKZLCABMUGUMAIZGZJUOLDUMLDUMUHUNJZCKUOULUPCULUKUNJUPUJAAUINOUHUK UNPSQCUNTUAUGUNLABUBUCUMUOLUDUEUF $. @@ -135380,7 +135380,7 @@ converges to zero (in the standard topology on the reals) with this jcad adantl toptopon resttopon 3adant2 adantr simpr cnf2 syl3anc jca ccnv wi ex cin vex inex1 wrex simpl1 toponmax syl simpl3 elrest syl2anc imaeq2 ssexd eleq1d ralxfr2d wfun simplrr ffun inpreima 3syl cnvimass cnvimarndm - cv cdm sseqtr4i simpll2 imass2 sstrid df-ss eqtrd ralbidva fssd biantrurd + cv cdm sseqtrri simpll2 imass2 sstrid df-ss eqtrd ralbidva fssd biantrurd simprr 3bitrrd bitrd simprl iscn 3bitr4d pm5.21ndd ) DEHIZJZBUAZAKZAEKZUB ZCUCJZCUDZABUEZLZBCDUFMJZBCDAUGMZUFMJZXNXSXOXQXSXOVHXNBCDUHNXNXSBXPUIZXLL XQXNXSYBXLXSYBVHXNXSXPDUDZBBCDXPYCXPUJZYCUJUKULNXJXLXMUMUNXPABUOUPUQXNYAX @@ -139839,7 +139839,7 @@ S C_ ( P ( ball ` D ) T ) ) $= ( vz vw vt cioo wcel cv cin wral wa cxr clt wceq eleq1d mp2an vv cxp cima vu cvv ctb iooex imaex cpr csup cinf sseli anim12i iooinsup syl2an rgen2a co preq12 prcom syl6eq supeq1d rspc2gv mpi ancoms infeq1d cop cfv opelxpi - df-ov wfun cdm wss wi cr cpw wf ioof ffun ax-mp xpss12 sseqtr4i funfvima2 + df-ov wfun cdm wss wi cr cpw wf ioof ffun ax-mp xpss12 sseqtrri funfvima2 fdmi syl eqeltrid an4s eqeltrd ralrimivva wfn wb ffn ineq1 ralbidv ralima fveq2 syl6eqr ineq1d ineq2 ineq2d ralxp bitri syl6bb mpbir fiinbas ) JCCU BZUCZUEKALZBLZMZXFKZBXFNZAXFNZXFUFKJXEUGUHXLGLZHLZJUQZUALZUDLZJUQZMZXFKZU @@ -140171,7 +140171,7 @@ S C_ ( P ( ball ` D ) T ) ) $= cima ctg crn cvv wceq iooex imaex imassrn wf wfn wral cpw ioof ffn simpll co cr w3a elioo1 biimpa simp1d simp2d qbtwnxr syl3anc simplr simp3d df-ov reeanv cop opelxpi 3ad2ant2 wfun cdm ffun qssre ressxr sstri xpss12 mp2an - wi sseqtr4i funfvima2 syl eqeltrid 3ad2ant1 simp3lr simp3rl simp2l sseldi + wi sseqtrri funfvima2 syl eqeltrid 3ad2ant1 simp3lr simp3rl simp2l sseldi fdmi wb simp2r syl2anc mpbir3and cle simp3ll xrltled iooss1 simp3rr sstrd iooss2 eleq2 sseq1 anbi12d rspcev 3exp rexlimdvv syl5bir mp2and ralrimiva syl12anc ctb qtopbas eltg2b sylibr rgen2a ffnov mpbir2an frn mp3an eqtr2i diff --git a/mmset.raw.html b/mmset.raw.html index a9656da40d..207b8ec05d 100644 --- a/mmset.raw.html +++ b/mmset.raw.html @@ -459,9 +459,11 @@

Inspired by Whitehead and Russell's monumental Principia Mathematica, -the Metamath Proof Explorer has over 23,000 completely worked out proofs, -starting from the very foundation that mathematics is built on and eventually -arriving at familiar mathematical facts and beyond. +the Metamath Proof Explorer has over 26,000 completely worked out proofs in its +main sections (and over 41,000 counting "mathboxes", which are annexes where +contributors can develop additional topics), starting from the very foundation +that mathematics is built on and eventually arriving at familiar mathematical +facts and beyond. Each proof is pieced together with razor-sharp precision using a simple @@ -1139,9 +1141,11 @@ calculus, and set theory. Each axiom is a string of mathematical symbols of two kinds: constants, also called connectives, which we show in black; and variables, which we show in color. The constants -that occur in the axioms are ` ( ` , ` ) ` , ` -> ` , ` -. ` , ` = ` , ` e. ` , -and ` A. ` (left parenthesis, right parenthesis, implies, not, equals, is a -member of, for all). +that occur in the axioms of predicate calculus are ` ( ` , ` ) ` , ` -> ` , +` -. ` , ` = ` , ` e. ` , and ` A. ` (left parenthesis, right parenthesis, +implies, not, equals, is a member of, for all). The axioms of set theory may +also use some defined constants (like ` /\ ` and ` E. ` ) when this results in +a more readable assertion.

@@ -6073,7 +6077,7 @@   -This page was last updated on 4-Aug-2021. +This page was last updated on 14-Jan-2024.
Your comments are welcome: Norman Megill F/ y ps ) $. + cbv2w.4 $e |- ( ph -> F/ x ch ) $. + cbv2w.5 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $. + $( Version of ~ cbv2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbv2w $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= + ( wal weq wb wi biimp syl6 cbv1v equcomi biimpr syl56 impbid ) ABDKCEKABC + DEFGHIADELZBCMZBCNJBCOPQACBEDGFIHEDLUBAUCCBNEDRJBCSTQUA $. + $} + ${ $d x y $. cbv3hv.nf1 $e |- ( ph -> A. y ph ) $. @@ -20024,6 +20040,36 @@ Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, ACJBDJSUARTCDADEKBCFKCDLABGMNOACPBDPQ $. $} + ${ + $d x y z w $. + cbval2v.1 $e |- F/ z ph $. + cbval2v.2 $e |- F/ w ph $. + cbval2v.3 $e |- F/ x ps $. + cbval2v.4 $e |- F/ y ps $. + cbval2v.5 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. + $( Version of ~ cbval2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) (Proof shortened + by Gino Giotto, 10-Jan-2024.) $) + cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $= + ( wal nfal weq nfv wnf a1i wb ex cbv2w cbvalv1 ) ADLBFLCEAEDGMBCFIMCENZAB + DFUBDOUBFOAFPUBHQBDPUBJQUBDFNABRKSTUA $. + + $( Obsolete version of ~ cbval2v as of 14-Jan-2024. (Contributed by BJ, + 16-Jan-2019.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + cbval2vOLD $p |- ( A. x A. y ph <-> A. z A. w ps ) $= + ( wal nfal weq wi nfv nfim wb cbvalv1 19.21v pm5.74d 3bitr3i pm5.74ri + expcom ) ADLZBFLZCEAEDGMBCFIMCENZUEUFUGAOZDLUGBOZFLUGUEOUGUFOUHUIDFUGAFUG + FPHQUGBDUGDPJQDFNZUGABUGUJABRKUDUASUGADTUGBFTUBUCS $. + + $( Version of ~ cbvex2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) $) + cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $= + ( wn wal wex nfn weq wa notbid cbval2v 2exnaln notbii 3bitr4i ) ALZDMCMZL + BLZFMEMZLADNCNBFNENUDUFUCUECDEFAEGOAFHOBCIOBDJOCEPDFPQABKRSUAACDTBEFTUB + $. + $} + ${ $d x z $. $d y z $. dvelimhw.1 $e |- ( ph -> A. x ph ) $. @@ -20253,6 +20299,30 @@ Corresponds to the dual of Axiom (B) of modal logic. (Contributed by NM, ( weq wal wi ax-12 sps pm2.27 al2imi syld ) CBDZCEABEZLAFZCEZACELMOFCACBGHL NACLAIJK $. + ${ + $d x y $. + dral1v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $. + $( Version of ~ dral1 with a disjoint variable condition, which does not + require ~ ax-13 . Remark: the corresponding versions for ~ dral2 and + ~ drex2 are instances of ~ albidv and ~ exbidv respectively. + (Contributed by BJ, 17-Jun-2019.) $) + dral1v $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $= + ( weq wal nfa1 albid axc11v axc11r impbid bitrd ) CDFZCGZACGBCGZBDGZOABCN + CHEIOPQBCDJBDCKLM $. + + $( Version of ~ drex1 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by BJ, 17-Jun-2019.) $) + drex1v $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $= + ( weq wal wn wex notbid dral1v df-ex 3bitr4g ) CDFCGZAHZCGZHBHZDGZHACIBDI + NPROQCDNABEJKJACLBDLM $. + + $( Version of ~ drnf1 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by BJ, 17-Jun-2019.) $) + drnf1v $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $= + ( weq wal wi wnf dral1v imbi12d nf5 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDGACI + BDIPRCDNABOQEABCDEJKJACLBDLM $. + $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= @@ -20796,17 +20866,17 @@ theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the ${ $d z w ph $. $d x y ps $. $d x w $. $d z y $. - cbval2v.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. + cbval2vv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. $( Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.) Remove dependency on ~ ax-10 . (Revised by Wolf Lammen, 18-Jul-2021.) $) - cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $= + cbval2vv $p |- ( A. x A. y ph <-> A. z A. w ps ) $= ( wal weq cbvaldva cbvalv ) ADHBFHCECEIABDFGJK $. $( Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) Remove dependency on ~ ax-10 . (Revised by Wolf Lammen, 18-Jul-2021.) $) - cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $= + cbvex2vv $p |- ( E. x E. y ph <-> E. z E. w ps ) $= ( wex weq cbvexdva cbvexv ) ADHBFHCECEIABDFGJK $. $} @@ -20823,8 +20893,8 @@ theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the $( Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.) $) cbvex4v $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $= - ( wex weq wa 2exbidv cbvex2v 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZINH - NUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. + ( wex weq wa 2exbidv cbvex2vv 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN + HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. $} $( Lemma used in proofs of implicit substitution properties. The converse @@ -23383,15 +23453,6 @@ of the unique existential quantifier (note that ` y ` and ` z ` need not ( wn weu wo biorf eubidv biimpa ) ADZBCEABFZCEJBKCABGHI $. $} - ${ - $d x ph $. - $( Obsolete version of ~ euorv as of 14-Jan-2023. (Contributed by NM, - 23-Mar-1995.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - euorvOLD $p |- ( ( -. ph /\ E! x ps ) -> E! x ( ph \/ ps ) ) $= - ( nfv euor ) ABCACDE $. - $} - $( Introduce or eliminate a disjunct in a unique existential quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.) $) @@ -23510,15 +23571,6 @@ of the unique existential quantifier (note that ` y ` and ` z ` need not wa ) ABPZCDZABCDZPRASRQCEAQCFQACABGHIASRABQCABJKZLMASRTNO $. $} - ${ - $d x ph $. - $( Obsolete version of ~ euanv as of 14-Jan-2023. (Contributed by NM, - 23-Mar-1995.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - euanvOLD $p |- ( E! x ( ph /\ ps ) <-> ( ph /\ E! x ps ) ) $= - ( nfv euan ) ABCACDE $. - $} - ${ $d x y $. $d y ph $. $d y ps $. $( "At most one" picks a variable value, eliminating an existential @@ -24996,9 +25048,9 @@ yield an eliminable and weakly (that is, object-level) conservative $} $( Every setvar is a member of ` { x | T. } ` , which is therefore "a" - universal class. Once class extensionality is available, we can say "the" - universal class (see ~ df-v ). This is ~ sbtru expressed using class - abstractions. (Contributed by BJ, 2-Sep-2023.) $) + universal class. Once class extensionality ~ dfcleq is available, we can + say "the" universal class (see ~ df-v ). This is ~ sbtru expressed using + class abstractions. (Contributed by BJ, 2-Sep-2023.) $) vextru $p |- y e. { x | T. } $= ( wtru tru vexw ) CABDE $. @@ -30356,16 +30408,6 @@ Such interpretation is rarely needed (see also ~ df-ral ). (Contributed DEKCDEKAQRDAPBCFLMBDENCDENO $. $} - ${ - $d x ph $. - reubidvaOLD.1 $e |- ( ( ph /\ x e. A ) -> ( ps <-> ch ) ) $. - $( Obsolete version of ~ reubidva as of 14-Jan-2023. (Contributed by NM, - 13-Nov-2004.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - reubidvaOLD $p |- ( ph -> ( E! x e. A ps <-> E! x e. A ch ) ) $= - ( nfv reubida ) ABCDEADGFH $. - $} - ${ $d x ph $. reubidv.1 $e |- ( ph -> ( ps <-> ch ) ) $. @@ -36542,11 +36584,11 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $} ${ - sseqtr4.1 $e |- A C_ B $. - sseqtr4.2 $e |- C = B $. + sseqtrri.1 $e |- A C_ B $. + sseqtrri.2 $e |- C = B $. $( Substitution of equality into a subclass relationship. (Contributed by NM, 4-Apr-1995.) $) - sseqtr4i $p |- A C_ C $= + sseqtrri $p |- A C_ C $= ( eqcomi sseqtri ) ABCDCBEFG $. $} @@ -36742,7 +36784,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $( Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.) $) eqimss2i $p |- B C_ A $= - ( ssid sseqtr4i ) BBABDCE $. + ( ssid sseqtrri ) BBABDCE $. $} $( Two classes are different if they don't include the same class. @@ -38250,7 +38292,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $( The symmetric difference contains one of the differences. (Proposed by BJ, 18-Aug-2022.) (Contributed by AV, 19-Aug-2022.) $) difsssymdif $p |- ( A \ B ) C_ ( A /_\ B ) $= - ( cdif cun csymdif ssun1 df-symdif sseqtr4i ) ABCZIBACZDABEIJFABGH $. + ( cdif cun csymdif ssun1 df-symdif sseqtrri ) ABCZIBACZDABEIJFABGH $. ${ difsymssdifssd.1 $e |- ( ph -> ( A /_\ B ) C_ C ) $. @@ -42691,29 +42733,29 @@ will do (e.g., ` O = (/) ` and ` T = { (/) } ` , see ~ 0nep0 ). $( A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 27-Aug-2004.) $) snsspr1 $p |- { A } C_ { A , B } $= - ( csn cun cpr ssun1 df-pr sseqtr4i ) ACZIBCZDABEIJFABGH $. + ( csn cun cpr ssun1 df-pr sseqtrri ) ACZIBCZDABEIJFABGH $. $( A singleton is a subset of an unordered pair containing its member. (Contributed by NM, 2-May-2009.) $) snsspr2 $p |- { B } C_ { A , B } $= - ( csn cun cpr ssun2 df-pr sseqtr4i ) BCZACZIDABEIJFABGH $. + ( csn cun cpr ssun2 df-pr sseqtrri ) BCZACZIDABEIJFABGH $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp1 $p |- { A } C_ { A , B , C } $= - ( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtr4i ) ADZABEZCDZFZABCGMNPA + ( csn cpr cun ctp snsspr1 ssun1 sstri df-tp sseqtrri ) ADZABEZCDZFZABCGMNPA BHNOIJABCKL $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp2 $p |- { B } C_ { A , B , C } $= - ( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtr4i ) BDZABEZCDZFZABCGMNPA + ( csn cpr cun ctp snsspr2 ssun1 sstri df-tp sseqtrri ) BDZABEZCDZFZABCGMNPA BHNOIJABCKL $. $( A singleton is a subset of an unordered triple containing its member. (Contributed by NM, 9-Oct-2013.) $) snsstp3 $p |- { C } C_ { A , B , C } $= - ( csn cpr cun ctp ssun2 df-tp sseqtr4i ) CDZABEZKFABCGKLHABCIJ $. + ( csn cpr cun ctp ssun2 df-tp sseqtrri ) CDZABEZKFABCGKLHABCIJ $. $( A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 22-Mar-2006.) (Proof shortened by @@ -44112,7 +44154,7 @@ either the empty set or a singleton ( ~ uniintsn ). (Contributed by NM, 30-Oct-2010.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) $) unissint $p |- ( U. A C_ |^| A <-> ( A = (/) \/ U. A = |^| A ) ) $= ( cuni cint wss c0 wo wn wa simpl wne df-ne intssuni sylbir adantl eqssd ex - wceq orrd cvv ssv int0 sseqtr4i inteq sseqtrrid eqimss jaoi impbii ) ABZACZ + wceq orrd cvv ssv int0 sseqtrri inteq sseqtrrid eqimss jaoi impbii ) ABZACZ DZAEQZUHUIQZFUJUKULUJUKGZULUJUMHUHUIUJUMIUMUIUHDZUJUMAEJUNAEKALMNOPRUKUJULU KECZUHUIUHSUOUHTUAUBAEUCUDUHUIUEUFUG $. @@ -44797,7 +44839,7 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and 10-Dec-2004.) $) iunxdif2 $p |- ( A. x e. A E. y e. ( A \ B ) C C_ D -> U_ y e. ( A \ B ) D = U_ x e. A C ) $= - ( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtr4i + ( wss cdif wrex wral ciun wceq iunss2 difss iunss1 ax-mp cbviunv sseqtrri wa jctil eqss sylibr ) EFHBCDIZJACKZBUDFLZACELZHZUGUFHZTUFUGMUEUIUHABCUDE FNUFBCFLZUGUDCHUFUJHCDOBUDCFPQABCEFGRSUAUFUGUBUC $. $} @@ -45266,7 +45308,7 @@ same distinct variable group (meaning ` A ` cannot depend on ` x ` ) and the double power class of that class. (Contributed by BJ, 29-Apr-2021.) $) pwpwssunieq $p |- { x | U. x = A } C_ ~P ~P A $= - ( cv cuni wceq cab wss cpw eqimss ss2abi pwpwab sseqtr4i ) ACDZBEZAFMBGZA + ( cv cuni wceq cab wss cpw eqimss ss2abi pwpwab sseqtrri ) ACDZBEZAFMBGZA FBHHNOAMBIJABKL $. $} @@ -47538,7 +47580,7 @@ holding in an empty domain (see Axiom A5 and Rule R2 of [LeBlanc] abstraction. (Contributed by NM, 3-Jul-2005.) $) intabs $p |- |^| { x | ( x C_ A /\ ph ) } = |^| { x | ph } $= ( cv wss wa cab cint cvv wcel wceq sseq1 anbi12d intmin3 intnex ssv sseq2 - wn mpbiri sylbi pm2.61i cbvabv inteqi sseqtr4i simpr ss2abi intss ax-mp + wn mpbiri sylbi pm2.61i cbvabv inteqi sseqtrri simpr ss2abi intss ax-mp eqssi ) DJZFKZALZDMZNZADMZNZUTBEMZNZVBVDOPZUTVDKZURVDFKZCLDVDOUPVDQUQVGAC UPVDFRHSITVEUDVDOQZVFVCUAVHVFUTOKUTUBVDOUTUCUEUFUGVAVCABDEGUHUIUJUSVAKVBU TKURADUQAUKULUSVAUMUNUO $. @@ -47820,11 +47862,28 @@ This theorem is proved directly from set theory axioms (no set theory ${ $d x y $. - $( Demonstration of a theorem (scheme) that requires (meta)variables ` x ` - and ` y ` to be distinct, but no others. It bundles the theorem schemes - ` E. x ( x = y -> x e. x ) ` and ` E. x ( x = y -> y e. x ) ` . Compare - ~ dvdemo2 . ("Bundles" is a term introduced by Raph Levien.) - (Contributed by NM, 1-Dec-2006.) $) + $( Demonstration of a theorem that requires the setvar variables ` x ` and + ` y ` to be disjoint (but without any other disjointness conditions, and + in particular, none on ` z ` ). + + That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with + ` x , y , z ` disjoint), often called its "principal instance", and the + two "degenerate instances" ( ` |- E. x ( x = y -> x e. x ) ` with + ` x , y ` disjoint) and ( ` |- E. x ( x = y -> y e. x ) ` with ` x , y ` + disjoint). + + Compare with ~ dvdemo2 , which has the same principal instance and one + common degenerate instance but crucially differs in the other degenerate + instance. + + See ~ https://us.metamath.org/mpeuni/mmset.html#distinct for details on + the "disjoint variable" mechanism. (The verb "bundle" to express this + phenomenon was introduced by Raph Levien.) + + Note that ~ dvdemo1 is partially bundled, in that the pairs of setvar + variables ` x , z ` and ` y , z ` need not be disjoint, and in spite of + that, its proof does not require ~ ax-11 nor ~ ax-13 . (Contributed by + NM, 1-Dec-2006.) (Revised by BJ, 13-Jan-2024.) $) dvdemo1 $p |- E. x ( x = y -> z e. x ) $= ( weq wn wel wi wex wal dtru exnal mpbir pm2.21 eximii ) ABDZEZOCAFZGAPAH OAIEABJOAKLOQMN $. @@ -47832,10 +47891,28 @@ This theorem is proved directly from set theory axioms (no set theory ${ $d x z $. - $( Demonstration of a theorem (scheme) that requires (meta)variables ` x ` - and ` z ` to be distinct, but no others. It bundles the theorem schemes - ` E. x ( x = x -> z e. x ) ` and ` E. x ( x = y -> y e. x ) ` . Compare - ~ dvdemo1 . (Contributed by NM, 1-Dec-2006.) $) + $( Demonstration of a theorem that requires the setvar variables ` x ` and + ` z ` to be disjoint (but without any other disjointness conditions, and + in particular, none on ` y ` ). + + That theorem bundles the theorems ( ` |- E. x ( x = y -> z e. x ) ` with + ` x , y , z ` disjoint), often called its "principal instance", and the + two "degenerate instances" ( ` |- E. x ( x = x -> z e. x ) ` with + ` x , z ` disjoint) and ( ` |- E. x ( x = z -> z e. x ) ` with ` x , z ` + disjoint). + + Compare with ~ dvdemo1 , which has the same principal instance and one + common degenerate instance but crucially differs in the other degenerate + instance. + + See ~ https://us.metamath.org/mpeuni/mmset.html#distinct for details on + the "disjoint variable" mechanism. + + Note that ~ dvdemo2 is partially bundled, in that the pairs of setvar + variables ` x , y ` and ` y , z ` need not be disjoint, and in spite of + that, its proof does not require any of the auxiliary axioms ~ ax-10 , + ~ ax-11 , ~ ax-12 , ~ ax-13 . (Contributed by NM, 1-Dec-2006.) + (Revised by BJ, 13-Jan-2024.) $) dvdemo2 $p |- E. x ( x = y -> z e. x ) $= ( wel weq wi el ax-1 eximii ) CADZABEZJFACAGJKHI $. $} @@ -51103,7 +51180,7 @@ Contrast with domain (defined in ~ df-dm ). For alternate definitions, $( Intersection with a Cartesian product is a subclass of restriction. (Contributed by Peter Mazsa, 19-Jul-2019.) $) inxpssres $p |- ( R i^i ( A X. B ) ) C_ ( R |` A ) $= - ( cxp cin cvv cres wss ssid ssv xpss12 mp2an sslin ax-mp df-res sseqtr4i + ( cxp cin cvv cres wss ssid ssv xpss12 mp2an sslin ax-mp df-res sseqtrri ) CABDZEZCAFDZEZCAGQSHZRTHAAHBFHUAAIBJAABFKLQSCMNCAOP $. $( A Cartesian product is a relation. Theorem 3.13(i) of [Monk1] p. 37. @@ -51707,7 +51784,7 @@ Contrast with domain (defined in ~ df-dm ). For alternate definitions, (Contributed by NM, 16-Jul-1995.) $) opabssxp $p |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ph ) } C_ ( A X. B ) $= - ( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtr4i ) BFDGCFEGHZAHZBCIOB + ( cv wcel wa copab cxp simpl ssopab2i df-xp sseqtrri ) BFDGCFEGHZAHZBCIOB CIDEJPOBCOAKLBCDEMN $. $} @@ -54194,7 +54271,7 @@ the restriction (of the relation) to the singleton containing this $( The domain of a restricted identity function. (Contributed by NM, 27-Aug-2004.) $) dmresi $p |- dom ( _I |` A ) = A $= - ( cid cdm wss cres wceq cvv ssv dmi sseqtr4i ssdmres mpbi ) ABCZDBAECAFAGMA + ( cid cdm wss cres wceq cvv ssv dmi sseqtrri ssdmres mpbi ) ABCZDBAECAFAGMA HIJABKL $. ${ @@ -54399,7 +54476,7 @@ the restriction (of the relation) to the singleton containing this $( A preimage under any class is included in the domain of the class. (Contributed by FL, 29-Jan-2007.) $) cnvimass $p |- ( `' A " B ) C_ dom A $= - ( ccnv cima crn cdm imassrn dfdm4 sseqtr4i ) ACZBDJEAFJBGAHI $. + ( ccnv cima crn cdm imassrn dfdm4 sseqtrri ) ACZBDJEAFJBGAHI $. $( The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) $) @@ -55317,7 +55394,7 @@ the restriction (of the relation) to the singleton containing this sofld $p |- ( ( R Or A /\ R C_ ( A X. A ) /\ R =/= (/) ) -> A = ( dom R u. ran R ) ) $= ( vx vy wor cxp wss c0 cdm crn wa wrel ad2antlr wcel ssun1 syl6sseq unssd - cun cv ex wne w3a wn wceq relxp relss mpi cop wbr df-br csn cdif sseqtr4i + cun cv ex wne w3a wn wceq relxp relss mpi cop wbr df-br csn cdif sseqtrri undif1 simpll dmxpid releldm sylancom sseldd sossfld syl2anc sseldi snssd dmss sstrid syl5bir con3dimp pm2.21d relssdv ss0 syl necon1ad 3impia rnss rnxpid 3ad2ant2 eqssd ) ABEZBAAFZGZBHUAZUBABIZBJZRZVRVTWAAWDGZVRVTKZWEBHW @@ -55368,7 +55445,7 @@ the restriction (of the relation) to the singleton containing this ordered pairs. (Contributed by NM, 8-Dec-2003.) (Proof shortened by BJ, 26-Nov-2021.) $) cnvcnv $p |- `' `' A = ( A i^i ( _V X. _V ) ) $= - ( ccnv cvv cxp cin cnvin cnveqi wceq wrel relcnv df-rel mpbi relxp sseqtr4i + ( ccnv cvv cxp cin cnvin cnveqi wceq wrel relcnv df-rel mpbi relxp sseqtrri wss dfrel2 dfss 3eqtr4ri relinxp eqtri ) ABZBZACCDZEZBZBZUDUAUCBZEZBUBUGBZE ZUFUBUAUGFUEUHAUCFGUBUIOUBUJHUBUCUIUBIUBUCOUAJUBKLUCIUIUCHCCMUCPLNUBUIQLRUD IUFUDHCCASUDPLT $. @@ -55958,7 +56035,7 @@ singleton of the first member (with no sethood assumptions on ` B ` ). $( The converse is a subset of the cartesian product of range and domain. (Contributed by Mario Carneiro, 2-Jan-2017.) $) cnvssrndm $p |- `' A C_ ( ran A X. dom A ) $= - ( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtr4i + ( ccnv cdm crn cxp wrel relcnv relssdmrn ax-mp df-rn dfdm4 xpeq12i sseqtrri wss ) ABZOCZODZEZADZACZEOFORNAGOHISPTQAJAKLM $. $( Composition as a subset of the Cartesian product of factors. (Contributed @@ -57257,7 +57334,7 @@ with a prime (apostrophe-like) symbol, such as Definition 6 of [Suppes] [TakeutiZaring] p. 41 (generalized to arbitrary classes). (Contributed by NM, 31-May-1994.) $) sssucid $p |- A C_ suc A $= - ( csn cun csuc ssun1 df-suc sseqtr4i ) AAABZCADAHEAFG $. + ( csn cun csuc ssun1 df-suc sseqtrri ) AAABZCADAHEAFG $. $( Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, @@ -64049,8 +64126,8 @@ from the cartesian product of two singletons onto a singleton (case where fnprb.b $e |- B e. _V $. $( A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) - (Proof shortened by Scott Fenton, 12-Oct-2017.) Revised to eliminate - unnecessary antecedent ` A =/= B ` . (Revised by NM, 29-Dec-2018.) $) + (Proof shortened by Scott Fenton, 12-Oct-2017.) Eliminate unnecessary + antecedent ` A =/= B ` . (Revised by NM, 29-Dec-2018.) $) fnprb $p |- ( F Fn { A , B } <-> F = { <. A , ( F ` A ) >. , <. B , ( F ` B ) >. } ) $= ( vx cpr wfn cfv cop wceq wb csn dfsn2 fveq2 wa cdm fvex adantl wcel wral @@ -64391,7 +64468,7 @@ from the cartesian product of two singletons onto a singleton (case where 9-Nov-1995.) $) fvclss $p |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } ) $= ( cv cfv wceq wex cab crn wcel c0 csn wo cun wn wne eqcom tz6.12i syl6ibr - wbr syl5bi eximdv elrn com12 necon1bd velsn orrd ss2abi df-un sseqtr4i + wbr syl5bi eximdv elrn com12 necon1bd velsn orrd ss2abi df-un sseqtrri vex ) BDZADZCEZFZAGZBHULCIZJZULKLZJZMZBHUQUSNUPVABUPURUTUPUROULKFUTUPURUL KULKPZUPURVBUPUMULCTZAGURVBUOVCAUOUNULFVBVCULUNQUMULCRUAUBAULCBUKUCSUDUEB KUFSUGUHBUQUSUIUJ $. @@ -67502,9 +67579,9 @@ result of an operator (deduction version). (Contributed by Paul $d x y z w ps $. eloprabga.1 $e |- ( ( x = A /\ y = B /\ z = C ) -> ( ph <-> ps ) ) $. $( The law of concretion for operation class abstraction. Compare - ~ elopab . (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct - variable restrictions were removed by David Abernethy, 19-Jun-2012.) - (Revised by Mario Carneiro, 19-Dec-2013.) $) + ~ elopab . (Contributed by NM, 14-Sep-1999.) Remove unnecessary + distinct variable conditions. (Revised by David Abernethy, + 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) $) eloprabga $p |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( <. <. A , B >. , C >. e. { <. <. x , y >. , z >. | ph } <-> ps ) ) $= ( vw wcel cvv cop cv wceq wa wex coprab wb elex w3a wi simpr eqeq1d eqcom @@ -67934,8 +68011,8 @@ result of an operator (deduction version). (Contributed by Paul ovig.3 $e |- F = { <. <. x , y >. , z >. | ( ( x e. R /\ y e. S ) /\ ph ) } $. $( The value of an operation class abstraction (weak version). - (Unnecessary distinct variable restrictions were removed by David - Abernethy, 19-Jun-2012.) (Contributed by NM, 14-Sep-1999.) (Revised by + (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable + conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) $) ovig $p |- ( ( A e. R /\ B e. S /\ C e. D ) -> ( ps -> ( A F B ) = C ) ) $= @@ -71268,9 +71345,9 @@ numbers have the property (conclusion). Exercise 25 of [Enderton] The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. The basis of this version is an - arbitrary ordinal ` suc B ` instead of zero. (Unnecessary distinct - variable restrictions were removed by David Abernethy, 19-Jun-2012.) - (Contributed by NM, 5-Jan-2005.) $) + arbitrary ordinal ` suc B ` instead of zero. (Contributed by NM, + 5-Jan-2005.) Remove unnecessary distinct variable conditions. (Revised + by David Abernethy, 19-Jun-2012.) $) tfindsg2 $p |- ( ( A e. On /\ B e. A ) -> ta ) $= ( con0 wcel wa wi csuc wss onelon sucelon sylib eloni ordsucss syl sylbir word imp cv wb ordelsuc sylan2 ancoms ex adantr sylbird sylan2br wlim cvv @@ -73365,8 +73442,8 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) dfoprab4f.y $e |- F/ y ph $. dfoprab4f.1 $e |- ( w = <. x , y >. -> ( ph <-> ps ) ) $. $( Operation class abstraction expressed without existential quantifiers. - (Unnecessary distinct variable restrictions were removed by David - Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by + (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable + conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 31-Aug-2015.) $) dfoprab4f $p |- { <. w , z >. | ( w e. ( A X. B ) /\ ph ) } = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } $= @@ -73525,7 +73602,7 @@ Power Set ( ~ ax-pow ). (Contributed by Mario Carneiro, 20-May-2013.) dmmpossx $p |- dom F C_ U_ x e. A ( { x } X. B ) $= ( vu vt vv cv csn csb cxp ciun cfv nfcv nfcsb1v csbeq1a wceq c1st cbvmpox cdm c2nd cmpo cmpt nfcsb sylan9eqr cop vex op1std csbeq1d op2ndd csbeq2dv - eqtrd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtr4i ) FUCHCHK + eqtrd mpomptx 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtrri ) FUCHCHK ZLZAVDDMZNZOZACAKZLZDNZOIVHAIKZUAPZBVLUDPZEMZMZFABCDEUEHJCVFAVDBJKZEMZMZU EFIVHVPUFABHJCDEVFVSHDQAVDDRZHEQJEQAVDVRRBAVDVRBVDQBVQERUGAVDDSZBKVQTVIVD TZEVRVSBVQESAVDVRSUHUBGHJICVFVPVSVLVDVQUITZVPAVDVOMVSWCAVMVDVOVDVQVLHUJZJ @@ -75104,7 +75181,7 @@ any sets (which usually are functions) and any element (even not 4-Jun-2019.) $) suppun $p |- ( ph -> ( F supp Z ) C_ ( ( F u. G ) supp Z ) ) $= ( cvv wcel wa csupp co cun wss wi ccnv csn cdif cima wceq suppimacnv a1i - ssun1 cnvun imaeq1i imaundir eqtri sseqtr4i adantr adantlr sylan2 syl2anc + ssun1 cnvun imaeq1i imaundir eqtri sseqtrri adantr adantlr sylan2 syl2anc unexg simplr 3sstr4d ex wn c0 supp0prc 0ss syl6eqss a1d pm2.61i ) BGHZEGH ZIZABEJKZBCLZEJKZMZNVEAVIVEAIZBOZGEPQZRZVGOZVLRZVFVHVMVOMVJVMVMCOZVLRZLZV OVMVQUBVOVKVPLZVLRVRVNVSVLBCUCUDVKVPVLUEUFUGUAVEVFVMSABGGETUHVJVGGHZVDVHV @@ -76520,47 +76597,6 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and NWOUIWPWQWRWSWTXAUJXNYFXBJCXQEXTXCXDXEXFXEXGXHXIWG $. $} - ${ - $d A a b c f g h x y $. $d B a $. $d F a b c f g h x y $. - $d R a b c f g h x y $. - wfrlem4OLD.1 $e |- R We A $. - wfrlem4OLD.2 $e |- B = { f | E. x ( f Fn x /\ ( x C_ A /\ - A. y e. x Pred ( R , A , y ) C_ x ) /\ - A. y e. x ( f ` y ) = ( F ` ( f |` Pred ( R , A , y ) ) ) ) } $. - $( Obsolete version of ~ wfrlem4 as of 18-Jul-2022. (Contributed by Scott - Fenton, 21-Apr-2011.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - wfrlem4OLD $p |- ( ( g e. B /\ h e. B ) -> - ( ( g |` ( dom g i^i dom h ) ) Fn ( dom g i^i dom h ) /\ - A. a e. ( dom g i^i dom h ) - ( ( g |` ( dom g i^i dom h ) ) ` a ) = - ( F ` ( ( g |` ( dom g i^i dom h ) ) |` - Pred ( R , ( dom g i^i dom h ) , a ) ) ) ) ) $= - ( vb vc wcel wa cfv wceq wral wss cv cdm cin cres wfn cpred wfrlem2 funfn - wfun sylib fnresin1 syl adantr inss1 sseli w3a wex wi wfrlem1 abeq2i fndm - raleqdv biimpar rsp 3adant2 exlimiv sylbi imp adantlr fvres adantl resres - syl5 predss sseqin2 mpbi 3an6 2exbii eeanv bitri ssinss1 ad2antrr syl5com - nfra1 nfan inss2 anim12d biimpi syl6com ralrimi ad2ant2l jca wb ineqan12d - ssin sseq1 sseq2 raleqbi1dv anbi12d imbi2d mpbiri 3adant3 exlimivv sylbir - syl2anb simpr preddowncl sylc syl5eq reseq2d fveq2d 3eqtr4d ralrimiva ) G - UAZDOZHUAZDOZPZXNXNUBZXPUBZUCZUDZYAUEZJUAZYBQZYBYAEYDUFZUDZIQZRZJYASXOYCX - QXOXNXSUEZYCXOXNUIYJABCDEFGILUGXNUHUJXSXTXNUKULUMXRYIJYAXRYDYAOZPZYDXNQZX - NCEYDUFZUDZIQZYEYHXOYKYMYPRZXQXOYKYQYKYDXSOZXOYQYAXSYDXSXTUNUOXOXNMUAZUEZ - YSCTZYNYSTZJYSSZPZYQJYSSZUPZMUQZYRYQURZUUGGDABMJCDEFGILUSUTZUUFUUHMYTUUEU - UHUUDYTUUEPYQJXSSZUUHYTUUJUUEYTYQJXSYSYSXNVAZVBVCYQJXSVDULVEVFVGVMVHVIYKY - EYMRXRYDYAXNVJVKYLYGYOIYLYGXNYAYFUCZUDYOXNYAYFVLYLUULYNXNYLUULYFYNYFYATUU - LYFRYAEYDVNYFYAVOVPYLYACTZYNYATZJYASZPZYKYFYNRXRUUPYKXOUUGXPNUAZUEZUUQCTZ - YNUUQTZJUUQSZPZYDXPQXPYNUDIQRJUUQSZUPZNUQZUUPXQUUIUVEHDABNJCDEFHILUSUTUUG - UVEPZYTUURPZUUDUVBPZUUEUVCPZUPZNUQMUQZUUPUVKUUFUVDPZNUQMUQUVFUVJUVLMNYTUU - RUUDUVBUUEUVCVQVRUUFUVDMNVSVTUVJUUPMNUVGUVHUUPUVIUVGUVHUUPUVGUVHUUPURZUVH - YSUUQUCZCTZYNUVNTZJUVNSZPZURZUVHUVOUVQUUAUVOUUCUVBYSUUQCWAWBUUCUVAUVQUUAU - USUUCUVAPZUVPJUVNUUCUVAJUUBJYSWDUUTJUUQWDWEYDUVNOZUVTUUBUUTPZUVPUWAUUCUUB - UVAUUTUWAYDYSOUUCUUBUVNYSYDYSUUQUNUOUUBJYSVDWCUWAYDUUQOUVAUUTUVNUUQYDYSUU - QWFUOUUTJUUQVDWCWGUWBUVPYNYSUUQWOWHWIWJWKWLUVGYAUVNRZUVMUVSWMYTUURXSYSXTU - UQUUKUUQXPVAWNUWCUUPUVRUVHUWCUUMUVOUUOUVQYAUVNCWPUUNUVPJYAUVNYAUVNYNWQWRW - SWTULXAVHXBXCXDXEUMXRYKXFJCYAEYDXGXHXIXJXIXKXLXMWL $. - $} - ${ $d A a f g h x y $. $d B a $. $d F a f g h x y $. $d R a f g h x y $. $d g h u v x $. @@ -76741,7 +76777,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( y e. ( dom F u. { z } ) -> ( C ` y ) = ( G ` ( C |` Pred ( R , A , y ) ) ) ) ) $= ( cv wcel csn cun cfv cres wceq ax-mp c0 cdm wfn cpred wi wfrlem13 weq wo - cdif elun velsn orbi2i bitri wa wfrlem12 wfun wb fnfun wss ssun1 sseqtr4i + cdif elun velsn orbi2i bitri wa wfrlem12 wfun wb fnfun wss ssun1 sseqtrri cop funssfv wfrdmcl fun2ssres syl3an3 fveq2d eqeq12d mp3an2 sylan syl5ibr w3a ex pm2.43d vsnid elun2 reseq1i resundir wfr wn wwe predfrirr ressnop0 wefr mp2b uneq2i un0 eqtri 3eqtri fveq2i opex elsn mpbir eleqtrri fnopfvb @@ -77613,7 +77649,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( B e. suc dom recs ( F ) -> ( C ` B ) = ( F ` ( C |` B ) ) ) ) $= ( wcel wceq con0 cfv cres wi wa syl wss cop csn cdm csuc wo elsuci wfun crecs wfn tfrlem10 fnfun cun ssun1 tfrlem9 funssfv 3expa adantrl onelss - sseqtr4i imp fun2ssres fveq2d sylan2 eqeq12d syl5ibr mpanl2 sylan exp32 + sseqtrri imp fun2ssres fveq2d sylan2 eqeq12d syl5ibr mpanl2 sylan exp32 pm2.43i pm2.43d opex snid opeq1 adantl eqimss mp3an2 syl2an reseq2 wrel resdm ax-mp syl6eq eqtrd opeq2d sneqd eleqtrid elun2 syl6eleqr wb simpr tfrlem6 sucidg adantr eqeltrd fnopfvb syl2an2r mpbird ex jaod syl5 ) DG @@ -77854,9 +77890,9 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and tz7.44.4 $e |- F Fn X $. tz7.44.5 $e |- Ord X $. $( The value of ` F ` at a successor ordinal. Part 2 of Theorem 7.44 of - [TakeutiZaring] p. 49. (Unnecessary distinct variable restrictions were - removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, - 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) $) + [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) Remove + unnecessary distinct variable conditions. (Revised by David Abernethy, + 19-Jun-2012.) (Revised by Mario Carneiro, 14-Nov-2014.) $) tz7.44-2 $p |- ( suc B e. X -> ( F ` suc B ) = ( H ` ( F ` B ) ) ) $= ( wcel cfv c0 wceq cuni fveq2d cvv csuc cres cdm wlim crn cv fveq2 reseq2 cif eqeq12d vtoclga eleq1d noel dmeq dm0 syl6eq con0 word wss ordsson wtr @@ -79314,7 +79350,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and sseq1d mpbiri oneli wn crab inteqi eleq2i onnminsb syl5bi wb oacl sylancr ontri1 con2bid sylibrd word onordi ordsucss 3syl adantl eqsstrd rexlimiva mpcom a1d ciun oalim iunss onelssi syl mprgbir syl6eqss 3jaoi rspcev nfcv - nfrab1 nfint nfov nfss onminsb oveq2i sseqtr4i eqss sylanblrc eqtr3 oacan + nfrab1 nfint nfov nfss onminsb oveq2i sseqtrri eqss sylanblrc eqtr3 oacan eqeq1d mp3an1 syl5ib rgen2a reu4 ) CDJZCAKZLMZDNZAOUAZYCCBKZLMZDNZPZYAYEN ZUBZBOUCAOUCYCAOUDXTEUFZOQZCYKLMZDNZYDEOJERUEYLDYFJZBOEHUGDEDEQDOQZDCDLMZ JZGYPCOQZYRGFDCUHUIZYOYRBDOEYEDNYFYQDYEDCLSUJZHURUKULEUMUIZXTYMDJZDYMJYNY @@ -82804,7 +82840,7 @@ the first case of his notation (simple exponentiation) and subscript it after Definition 10.24 of [Kunen] p. 31. (Contributed by Raph Levien, 4-Dec-2003.) $) mapex $p |- ( ( A e. C /\ B e. D ) -> { f | f : A --> B } e. _V ) $= - ( wcel wa cv cab cxp cpw wss cvv fssxp ss2abi df-pw sseqtr4i xpexg pwexd + ( wcel wa cv cab cxp cpw wss cvv fssxp ss2abi df-pw sseqtrri xpexg pwexd wf ssexg sylancr ) ACFBDFGZABEHZTZEIZABJZKZLUHMFUFMFUFUDUGLZEIUHUEUIEABUD NOEUGPQUCUGMABCDRSUFUHMUAUB $. $} @@ -93912,7 +93948,7 @@ a Cantor normal form (and injectivity, together with coherence trcl $p |- ( A C_ C /\ Tr C /\ A. x ( ( A C_ x /\ Tr x ) -> C C_ x ) ) $= ( vv vu wss cv com cfv wrex c0 wcel wceq fveq2 wtr wa wal ciun peano1 cvv wi cuni cun cmpt crdg cres fveq1i fr0g ax-mp eqtr2i eqimssi sseq2d rspcev - mp2an ssiun sseqtr4i dftr2 eliun anbi2i r19.42v bitr4i elunii ssun2 uniex + mp2an ssiun sseqtrri dftr2 eliun anbi2i r19.42v bitr4i elunii ssun2 uniex csuc fvex unex unieq uneq12d frsucmpt2 mpan2 sseqtrrid sseld syl5 reximia weq id sylbi peano2 eleq2d syl rexlimiv cbvrexv sylibr ax-gen mpgbir treq ex mpbir wral sseq1d eqtri sseq1i biimpri adantr uniss df-tr sstr2 syl5bi @@ -96125,7 +96161,7 @@ of wff (meta)variables, see ~ htalem . (Contributed by NM, djuunxp $p |- ( ( A |_| B ) u. ( B |_| A ) ) = ( { (/) , 1o } X. ( A u. B ) ) $= ( vy vz cun c0 c1o cv wcel wa wo elun anim1i ancoms opelxp sylibr orcd ex - cxp olcd vx cdju cpr djuss uncom xpeq2i sseqtr4i unssi cop wceq wex elxpi + cxp olcd vx cdju cpr djuss uncom xpeq2i sseqtrri unssi cop wceq wex elxpi csn vex elpr wi velsn biimpri jaoi com12 imp syl2anb df-dju bitri orbi12i eleq2i adantl wb eleq1 adantr mpbird exlimivv syl ssriv eqssi ) ABUBZBAUB ZEZFGUCZABEZSZVPVQWAABUDVQVSBAEZSWABAUDVTWBVSABUEUFUGUHUAWAVRUAHZWAIWCCHZ @@ -98994,7 +99030,7 @@ is defined as the Axiom of Choice (first form) of [Enderton] p. 49. The A. x E. f A. z e. x ( z =/= (/) -> ( f ` z ) e. z ) ) $= ( vy vw vv vu cv wss cdm wfn wa wex wal cfv wcel wel vex wceq cab wac wne c0 wral df-ac copab cuni cxp vuniex xpex simpl elunii ancoms jca ssopab2i - wi df-xp sseqtr4i ssexi sseq2 dmeq fneq2d anbi12d exbidv spcv csn cima wb + wi df-xp sseqtrri ssexi sseq2 dmeq fneq2d anbi12d exbidv spcv csn cima wb fndm eleq2 dmopab eleq2i weq elequ1 elab 19.42v n0 anbi2i bitr4i syl6rbbr 3bitrri syl adantl wfun fnfun funfvima3 sylan2 sylbid imp abbi2dv wbr cvv ibar imasng elv anbi2d eqid abbii eqtri syl6reqr eleq2d ad2antrl ralrimiv @@ -101170,7 +101206,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality ( vx vy vz vw vv con0 wcel cfv cv ccrd wceq wss wrex wral wa wex c1o wn c0 csuc ccf cab sucelon cfval sylbi csn cardsn eqcomd snidg elsuci onelss cint wo wi eqimss a1i jaod syl5 sseq2 rspcev syl6an ralrimiv ssun2 df-suc - cun sseqtr4i jctil fveq2 eqeq2d sseq1 rexeq ralbidv anbi12d spcev syl2anc + cun sseqtrri jctil fveq2 eqeq2d sseq1 rexeq ralbidv anbi12d spcev syl2anc snex 1oex eqeq1 anbi1d exbidv elab sylibr el1o eqcom cdm cvv onssnum mpan wb vex cardnueq0 syl syl5bb biimpa rex0 wne nsuceq0 r19.2z mtbiri intnand nrex mto imnan mpbi w3a suceloni onss sylan2 ancoms adantrr 3adant2 simp2 @@ -102146,7 +102182,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality isfin4p1 $p |- ( A e. Fin4 <-> A ~< ( A |_| 1o ) ) $= ( cfin4 wcel c1o cdju csdm wbr cdom cen wn con0 csn cxp mp2an ax-mp syl cvv c0 brrelex1i sylancr 1on djudoml mpan2 wpss cop cun 1oex snid 0lt1o opelxpi - elun2 df-dju eleqtrri wne 1n0 wceq opelxp1 elsni necon3ai wa ssun1 sseqtr4i + elun2 df-dju eleqtrri wne 1n0 wceq opelxp1 elsni necon3ai wa ssun1 sseqtrri wss wi ssnelpss relen xpsnen2g entr mpancom fin4i fin4en1 mtod con2i brsdom 0ex sylanbrc com sdomnen infdju1 ensymd nsyl relsdom isfin4-2 mpbird impbii wb ) ABCZAADEZFGZWGAWHHGZAWHIGZJWIWGDKCWJUAADBKUBUCWKWGWKWGWHBCZWKRLZAMZWHU @@ -102181,7 +102217,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality A. w e. { c e. ~P A | ( A \ c ) e. B } -. ph -> E. z e. B A. v e. B -. ps ) ) $= ( wss wn cv cdif wcel wceq cpw crab wral wa wi difeq2 eleq1d elrab simp2r - wrex w3a notbid simpl3 difss cvv wb ssun1 undif1 sseqtr4i simpl2r simpl2l + wrex w3a notbid simpl3 difss cvv wb ssun1 undif1 sseqtrri simpl2r simpl2l unexg syl2anc ssexg sylancr elpw2g mpbiri simpl1 simpr sseldd dfss4 sylib cun elpwid eqeltrd elrabd rspcdva simplrl ssel2 3adantl3 mpbird ralrimiva syl adantlr ralbidv rspcev 3exp syl5bi rexlimdv ) JIUAZOZAPZGIKQZRZJSZKWJ @@ -104338,7 +104374,7 @@ enough that it can be proven using DC (see ~ axcc ). It is, however, $( Lemma for ~ domtriom . (Contributed by Mario Carneiro, 9-Feb-2013.) $) domtriomlem $p |- ( -. A e. Fin -> _om ~<_ A ) $= ( vc wcel cv com wral wex wbr wi wa cen vm vj cfn wn cfv cdom wfn wne wss - c0 cpw cab cvv pwex simpl ss2abi df-pw sseqtr4i ssexi eqeltri enref axcc3 + c0 cpw cab cvv pwex simpl ss2abi df-pw sseqtrri ssexi eqeltri enref axcc3 omex nfv nfra1 nfan ccrd nnfi pwfi sylib ficardom isinf wceq breq2 anbi2d exbidv rspcv syl5 3syl finnum cardid2 entr expcom 4syl anim2d eximdv syld cdm neeq1i abn0 bitri syl6ibr com12 adantr rsp adantl mpdd ralrimi 3expib @@ -104560,10 +104596,9 @@ that every (nonempty) pruned tree has a branch. This axiom is redundant ( s ` (/) ) = C /\ A. k e. n ( s ` suc k ) e. ( F ` ( s ` k ) ) ) } $. $( The class ` S ` of finite approximations to the DC sequence is a set. (We derive here the stronger statement that ` S ` is a subset of a - specific set, namely ` ~P ( _om X. A ) ` .) (Unnecessary distinct - variable restrictions were removed by David Abernethy, 18-Mar-2014.) - (Contributed by Mario Carneiro, 27-Jan-2013.) (Revised by Mario - Carneiro, 18-Mar-2014.) $) + specific set, namely ` ~P ( _om X. A ) ` .) (Contributed by Mario + Carneiro, 27-Jan-2013.) Remove unnecessary distinct variable + conditions. (Revised by David Abernethy, 18-Mar-2014.) $) axdc3lem $p |- S e. _V $= ( com cxp cpw dcomex xpex pwex cv csuc cfv wcel wss wf wceq wral w3a wrex c0 cab wa fssxp peano2 cvv con0 omelon2 ax-mp onelssi xpss1 3syl sylan9ss @@ -105849,7 +105884,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario 15-May-2015.) $) ttukeylem1 $p |- ( ph -> ( C e. A <-> ( ~P C i^i Fin ) C_ A ) ) $= ( cvv wcel cpw cfn cin wss cdom cun ccrd ssexg wb wi elex a1i wa wbr cuni - cdif ssun1 undif1 sseqtr4i cfv wfo fvex wf1o f1ofo syl fornex mpsyl unexg + cdif ssun1 undif1 sseqtrri cfv wfo fvex wf1o f1ofo syl fornex mpsyl unexg id syl2anc sylancr uniexb sylibr syl2anr infpwfidom reldom brrelex1i 3syl ex cv wal wceq eleq1 pweq ineq1d sseq1d bibi12d spcgv syl5com pm5.21ndd ) AEJKZECKZELZMNZCOZWCWBUAAECUBUCAWFWBAWFUDWEJKZEWEPUEWBWFWFCJKZWGAWFUTACUF @@ -105981,7 +106016,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario ttukeylem7 $p |- ( ph -> E. x e. A ( B C_ x /\ A. y e. A -. x C. y ) ) $= ( cfv wcel wss wn wa c0 con0 wceq cuni cdif ccrd wpss wral wrex csuc fvex va cv sucid ttukeylem6 mpan2 ttukeylem4 w3a cardon 0ss 3pm3.2i ttukeylem5 - 0elon eqsstrrd wi simprr cun ssun1 undif1 sseqtr4i ccnv simpl wf1o f1ocnv + 0elon eqsstrrd wi simprr cun ssun1 undif1 sseqtrri ccnv simpl wf1o f1ocnv wf f1of 3syl adantr eldifi ad2antll simprll elunii eldifn eldifd ffvelrnd syl2anc onelon sylancr suceloni syl a1i word onordi ordsucss syl13anc csn mpsyl ssun2 eloni ordunisuc fveq2d f1ocnvfv2 eqtr2d velsn ordelss eqsstrd @@ -106269,7 +106304,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario c0 syl6rbb df-br a1i anbi12d rexbidva rexbidv rexcom zfpair2 eqeq1 anbi1d 2rexbidv elab bitr4i adantr breq1 breq2 ceqsrex2v rmobidva ralbiia ancoms bitrd eqcom ancom 3bitr4g bicomi syl5bb csn snex simpl ss2abi df-sn ssexi - sseqtr4i ab2rexex2 eleq2 rmobidv ralbidv spcev exlimiv copab preq1 eleq1d + sseqtrri ab2rexex2 eleq2 rmobidv ralbidv spcev exlimiv copab preq1 eleq1d syl2anbr preq2 eqid brab rmobii ralbii rexbii df-opab vuniex prid1 elunii cvv cuni mpan adantl prid2 eqeltrri abexex eqeltri breq impbii bitri ) CD UALAMZBMZUFMZLZBCUBZADNZYFYEYGLZBDOZACNZPZUFUCZYEYFQZEMZRZBCUBZADNZYFYEQZ @@ -106308,7 +106343,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario elab incom eqtri disjne mp3an1 vex opthpr syl breq12 biimprd impd adantrd syl6bi ex rexlimdvv syl5bi moimdv ralimia ancoms eqcom 3bitr4g bicomi a1i ancom anbi12d rexbidva rexbidv syl5bb breq2 breq1 ceqsrex2v bitrd ralbiia - adantr biimpri csn snex simpl ss2abi df-sn sseqtr4i ssexi ab2rexex2 eleq2 + adantr biimpri csn snex simpl ss2abi df-sn sseqtrri ssexi ab2rexex2 eleq2 mobidv ralbidv spcev syl2an exlimiv copab preq1 eleq1d preq2 mobii ralbii eqid brab rexbii cvv df-opab cuni vuniex prid1 elunii mpan prid2 eqeltrri adantl abexex eqeltri breq syl2anbr impbii bitri ) CDUALAMZBMZUFMZLZBUBZA @@ -107842,7 +107877,7 @@ it contains AC as a simple corollary (letting ` m ( i ) = (/) ` , this $( A finite set is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.) $) fingch $p |- Fin C_ GCH $= - ( vx vy cfn cv csdm wbr cpw wa wn wal cab cun cgch ssun1 df-gch sseqtr4i + ( vx vy cfn cv csdm wbr cpw wa wn wal cab cun cgch ssun1 df-gch sseqtrri ) CCADZBDZEFRQGEFHIBJAKZLMCSNABOP $. $( The only GCH-sets which have other sets between it and its power set are @@ -108296,7 +108331,7 @@ it contains AC as a simple corollary (letting ` m ( i ) = (/) ` , this un0 crn uneq12d cpw wfun fpwwe2lem11 ffun funfvbrb 3jca fpwwe2lem5 syldan snssd syl6ss wal cdif ssdif0 simpllr ovex breq2 rexsn simplrr neneqd sssn wf ord raleqdv breq1 rexeqbidv syl5bir mp1i wefr simplrl ssundif syl22anc - difexg fri eldifn pm2.21d syl5 con3d ralimdv jctird undif1 sseqtr4i ralun + difexg fri eldifn pm2.21d syl5 con3d ralimdv jctird undif1 sseqtrri ralun jaod ssralv mpsyl eldifi expimpd reximdv2 pm2.61dne alrimiv df-fr anbi12i jctild weso solin sylan 3orim123d ancomd 3mix3 3mix1 syl2an 3mix2d ccased eqtr3 ralrimivv dfwe2 fpwwe2cbv fpwwe2lem3 fvex xpexg sylancl unexg dmexd @@ -108680,7 +108715,7 @@ set maps to an element of the set (so that it cannot be extended without pssssd f1fveq syl12anc ex necon3ad npss sylib wral pm3.2i elinel1 ffvelrn wex syl2an fpwwe mpbiri simpld fpwwelem simprld weeq1 spcev ween eqeltrrd domtri2 infdju1 syl6bir ensym syl6 mt3d 2onn nnsdom isfinite csuc sssucid - djufi sseqtr4i xpss2 unss2 mp1i ssun2 sucid eleqtrri opelxpi mp2an sselii + djufi sseqtrri xpss2 unss2 mp1i ssun2 sucid eleqtrri opelxpi mp2an sselii snid wo 1n0 neii opelxp1 elsni word 1onn ordirr mp2b opelxp2 pm3.2ni elun nnord mtbir ssnelpss mp2ani psseq12i php3 canthp1lem1 sdomdomtr pm2.65i ) ADQUAZDUBZUCUDZAVUBVUAAVUBUERVUBVUAFUFZVUBVUAUCUDZADUEAQDUGUDZDUERZKQDUGU @@ -109942,7 +109977,7 @@ prove that every set is contained in a weak universe in ZF (see c1o sstri wceq simprl fvex uniex unex prex mptex rnex iunex unieq uneq12d weq pweq preq12d preq2 cbvmptv preq1 mpteq2dv syl5eq rneqd cbviunv mpteq1 id uneq2d iuneq12d frsucmpt2 sylancl sseqtr4d fvssunirn rexlimdvaa syl5bi - sseqtr4i ralrimiv dftr3 sylibr con0 1on unexg mpan2 fveq1i fr0g syl6eqssr + sseqtrri ralrimiv dftr3 sylibr con0 1on unexg mpan2 fveq1i fr0g syl6eqssr syl unssbd 1n0 ssn0 sseqtrrid sstrd unssad vpwex vuniex prss simprd fveq2 ssiun2s sseq2d vtoclga findsg sseldd wi imbi12d eqeltri simpld word ordom simplrl ordunel mp3an2i ssidd suceq fveq2d sseq12d ad2antrr sstr2 syl5com @@ -110020,7 +110055,7 @@ prove that every set is contained in a weak universe in ZF (see Mario Carneiro, 2-Jan-2017.) $) wunex3 $p |- ( A e. V -> ( U e. WUni /\ A C_ U ) ) $= ( wcel wss cwun crnk cfv cr1 r1rankid com coa co con0 rankon omelon mp2an - oacl wlim c0 peano1 wb oaord1 mpbi r1ord2 sseqtr4i syl6ss wa limom pm3.2i + oacl wlim c0 peano1 wb oaord1 mpbi r1ord2 sseqtrri syl6ss wa limom pm3.2i mp2 oalimcl r1limwun eqeltri jctil ) ACEZABFBGEUQAAHIZJIZBACKUSURLMNZJIZB UTOEZURUTEZUSVAFUROEZLOEZVBAPZQURLSRZUALEZVCUBVDVEVHVCUCVFQURLUDRUEURUTUF ULDUGUHBVAGDVBUTTZVAGEVGVDVELTZUIVIVFVEVJQUJUKURLOUMRUTOUNRUOUP $. @@ -116468,7 +116503,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a $( The standard reals are a subset of the extended reals. (Contributed by NM, 14-Oct-2005.) $) ressxr $p |- RR C_ RR* $= - ( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtr4i ) AABCDZEFAJGHI $. + ( cr cpnf cmnf cpr cun cxr ssun1 df-xr sseqtrri ) AABCDZEFAJGHI $. $( The Cartesian product of standard reals are a subset of the Cartesian product of extended reals. (Contributed by David A. Wheeler, @@ -116570,7 +116605,7 @@ this axiom (with the defined operation in place of ` x. ` ) follows as a ltrelxr $p |- < C_ ( RR* X. RR* ) $= ( vx vy cv cr wcel copab cmnf csn cun cxp cxr wa eqsstri sstri wss ressxr cpnf unssi xpss12 mp2an clt cltrr wbr w3a df-ltxr df-3an opabbii opabssxp - rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtr4i snsspr1 ) UAACZDEZBCZDEZUPURU + rexpssxrxp cpr snsspr2 ssun2 df-xr sseqtrri snsspr1 ) UAACZDEZBCZDEZUPURU BUCZUDZABFZDGHZIZQHZJZVCDJZIZIKKJZABUEVBVHVIVBDDJZVIVBUQUSLUTLZABFVJVAVKA BUQUSUTUFUGUTABDDUHMUINVFVGVIVDKOVEKOVFVIODVCKPVCQGUJZKQGUKVLDVLIKVLDULUM UNZNZRVEVLKQGUOVMNVDKVEKSTVCKODKOVGVIOVNPVCKDKSTRRM $. @@ -118192,14 +118227,6 @@ it could represent the (meaningless) operation of ( cc wcel cmin co caddc wceq subcl w3a eqid subadd mpbii mpd3an3 ancoms ) B CDZACDZABAEFZGFBHZPQRCDZSBAIPQTJRRHSRKBARLMNO $. - $( Obsolete version of ~ pncan3 as of 8-Jan-2023. Subtraction and addition - of equals. (Contributed by NM, 14-Mar-2005.) (New usage is discouraged.) - (Proof modification is discouraged.) $) - pncan3OLD $p |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B - A ) ) = B ) $= - ( cc wcel wa cmin co wceq caddc eqid simpr simpl subcl ancoms syl3anc mpbii - wb subadd ) ACDZBCDZEZBAFGZUBHZAUBIGBHZUBJUATSUBCDZUCUDQSTKSTLTSUEBAMNBAUBR - OP $. - $( Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) $) npcan $p |- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A ) $= @@ -126412,7 +126439,7 @@ Nonnegative integers (as a subset of complex numbers) $( Positive naturals are a subset of nonnegative integers. (Contributed by Raph Levien, 10-Dec-2002.) $) nnssnn0 $p |- NN C_ NN0 $= - ( cn cc0 csn cun cn0 ssun1 df-n0 sseqtr4i ) AABCZDEAIFGH $. + ( cn cc0 csn cun cn0 ssun1 df-n0 sseqtrri ) AABCZDEAIFGH $. $( Nonnegative integers are a subset of the reals. (Contributed by Raph Levien, 10-Dec-2002.) $) @@ -126595,7 +126622,7 @@ Nonnegative integers (as a subset of complex numbers) (Contributed by Mario Carneiro, 17-Jul-2014.) $) un0addcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M + N ) e. T ) $= ( wcel caddc co cc0 wo wa eleq2i elun bitri cc sselda eqeltrd csn ssun1 - cun sseqtr4i sseldi expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom + cun sseqtrri sseldi expr addid2d wss a1i elsni oveq1d eleq1d syl5ibrcom wi impancom jaodan sylan2b 0cnd snssd unssd eqsstrid addid1d simpr jaod oveq2d syl5bi impr ) ADCIZECIZDEJKZCIZVIEBIZELUAZIZMZAVHNZVKVIEBVMUCZIV OCVQEGOEBVMPQVPVLVKVNVHADBIZDVMIZMZVLVKUNZVHDVQIVTCVQDGODBVMPQAVRWAVSAV @@ -126608,7 +126635,7 @@ Nonnegative integers (as a subset of complex numbers) $( If ` S ` is closed under multiplication, then so is ` S u. { 0 } ` . (Contributed by Mario Carneiro, 17-Jul-2014.) $) un0mulcl $p |- ( ( ph /\ ( M e. T /\ N e. T ) ) -> ( M x. N ) e. T ) $= - ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtr4i cc sselda csn wi expr + ( wcel cmul co cc0 wo wa eleq2i elun bitri sseqtrri cc sselda csn wi expr cun ssun1 sseldi mul02d wss ssun2 c0ex mpbir syl6eqel elsni oveq1d eleq1d snss syl5ibrcom impancom jaodan sylan2b 0cnd snssd eqsstrid mul01d oveq2d unssd jaod syl5bi impr ) ADCIZECIZDEJKZCIZVKEBIZELUAZIZMZAVJNZVMVKEBVOUDZ @@ -126924,7 +126951,7 @@ distinguish between finite and infinite sets (and therefore if the set size $( The standard nonnegative integers are a subset of the extended nonnegative integers. (Contributed by AV, 10-Dec-2020.) $) nn0ssxnn0 $p |- NN0 C_ NN0* $= - ( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtr4i ) AABCZDEAIFGH $. + ( cn0 cpnf csn cun cxnn0 ssun1 df-xnn0 sseqtrri ) AABCZDEAIFGH $. $( A standard nonnegative integer is an extended nonnegative integer. (Contributed by AV, 10-Dec-2020.) $) @@ -142467,51 +142494,51 @@ seq M ( .+ , F ) ) $= wa wi wcel c1 caddc oveq2 f1oeq23 syl2anc anbi12d eqeq12d imbi12d 2albidv wb feq2d fveq2 imbi2d weq cz f1of adantr elfz3 fvco3 syl2anr ffvelrn fzsn csn eleq2d elsni syl6bi syldan adantrr fveq2d eqtrd seq1 3eqtr4d alrimivv - imp a1d f1oeq1 feq1 bi2anan9r coeq1 coeq2 sylan9eq seqeq3d fveq1d cbval2v - ex simpl ccnv clt wbr cif simplll sylan w3a simpllr wss syl simprl simprr - eqid simplr sylib seqf1olem2 exp31 syl5bir syl5bi expcom a2d uzind4 mpcom - alrimdv cfn cvv wfn fvex fnmpti fzfi fvmpt seqfveq fnfi mp2an fex2 spc2gv - ovexd syl3anc sylancr mpd mp2and ffvelrnda adantl 3eqtr3d ) AMFBLMUGUHZBU - IZJUJZUKZIULZLUMZUJZMFUUPLUMZUJZMFKLUMUJMFJLUMUJAUUMUUMIUNZUUMEUUPUOZUUSU - VAUPZSABUUMUUOETUQAUUMUUMUBUIZUNZUUMEUCUIZUOZUTZMFUVGUVEULZLUMZUJZMFUVGLU - MZUJZUPZVAZUBURUCURZUVBUVCUTZUVDVAZMLUSUJZVBAUVQQALUUNUGUHZUWAUVEUNZUWAEU - VGUOZUTZUUNUVKUJZUUNUVMUJZUPZVAZUBURUCURZVAALLUGUHZUWJUVEUNZUWJEUVGUOZUTZ - LUVKUJZLUVMUJZUPZVAZUBURUCURZVAALHUIZUGUHZUWTUVEUNZUWTEUVGUOZUTZUWSUVKUJZ - 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Fin ) -> ph ) $= ( vx cfn wcel wbr chash cfv cn0 wi hashcl cv wceq wa wex dfclel wal caddc cc0 c1 co eqeq2 anbi2d imbi1d 2albidv gen2 breq12 wb fveq2 eqeq1d anbi12d - adantr imbi12d cbval2v clt nn0re cr 1re a1i 0lt1 addgegt0d simpr breqtrrd - nn0ge0 adantrl cvv vex hashgt0elex csn cdif simpl hashdifsnp1 syl3anc imp - w3a peano2nn0 ad2antrr ad2antlr simplrr simprlr difexg ax-mp spc2gv mp2an - jca expdimp syl6an exp41 com15 com23 mpcom ex mpd com4l exlimiv syl com25 - 3jca impcom impancom alrimivv syl5bi nn0ind brrelex1i brrelex2i expd syl5 - expcom eqcoms sylbi ) NUDUEZNKMUFZAYKNUGUHZUIUEZYLAUJZNUKYNJULZYMUMZYPUIU - EZUNZJUOYOJYMUIUPYSYOJYQYRYOYRYOUJYMYPYMYPUMZYLYRAYLYTYRAUJYRGULZHULZMUFZ - UUAUGUHZYPUMZUNZBUJZHUQGUQZYLYTUNZAUUCUUDUCULZUMZUNZBUJZHUQGUQUUCUUDUSUMZ - UNZBUJZHUQGUQUUCUUDEULZUMZUNZBUJZHUQGUQZUUCUUDUUQUTURVAZUMZUNZBUJZHUQGUQZ - UUHUCEYPUUJUSUMZUUMUUPGHUVGUULUUOBUVGUUKUUNUUCUUJUSUUDVBVCVDVEUUJUUQUMZUU - MUUTGHUVHUULUUSBUVHUUKUURUUCUUJUUQUUDVBVCVDVEUUJUVBUMZUUMUVEGHUVIUULUVDBU - 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(Contributed by Stefan O'Rear, - 15-Aug-2015.) $) - ccatlen $p |- ( ( S e. Word B /\ T e. Word B ) -> + 15-Aug-2015.) (Revised by JJ, 1-Jan-2024.) $) + ccatlen $p |- ( ( S e. Word A /\ T e. Word B ) -> ( # ` ( S ++ T ) ) = ( ( # ` S ) + ( # ` T ) ) ) $= - ( vx cword wcel wa cconcat co chash cfv cc0 caddc cfzo cmin wceq fvex cn0 - cv lencl cif cmpt ccatfval fveq2d wfn ifex eqid fnmpti hashfn mp1i syl2an - nn0addcl hashfzo0 syl 3eqtrd ) BAEZFZCUPFZGZBCHIZJKDLBJKZCJKZMIZNIZDSZLVA - NIFZVEBKZVEVAOIZCKZUAZUBZJKZVDJKZVCUSUTVKJDBCUPUPUCUDVKVDUEVLVMPUSDVDVJVK - VFVGVIVEBQVHCQUFVKUGUHVDVKUIUJUSVCRFZVMVCPUQVARFVBRFVNURABTACTVAVBULUKVCU - MUNUO $. + ( vx cword wcel wa cconcat co chash cfv cc0 caddc cfzo cv wceq fvex lencl + cn0 cmin cif cmpt ccatfval fveq2d wfn ifex eqid fnmpti hashfn mp1i syl2an + nn0addcl hashfzo0 syl 3eqtrd ) CAFZGZDBFZGZHZCDIJZKLEMCKLZDKLZNJZOJZEPZMV + COJGZVGCLZVGVCUAJZDLZUBZUCZKLZVFKLZVEVAVBVMKECDUQUSUDUEVMVFUFVNVOQVAEVFVL + VMVHVIVKVGCRVJDRUGVMUHUIVFVMUJUKVAVETGZVOVEQURVCTGVDTGVPUTACSBDSVCVDUMULV + EUNUOUP $. $( The concatenation of two words is empty iff the two words are empty. (Contributed by AV, 4-Mar-2022.) $) @@ -149526,9 +149553,9 @@ computer programs (as last() or lastChar()), the terminology used for ( wcel wa co chash cfv cc0 wceq c0 wb hasheq0 syl cle wbr cn0 lencl nn0re cr cword cconcat ccatcl caddc ccatlen eqeq1d nn0ge0 add20 syl2an bi2anan9 jca 3bitrd bitr3d ) BAUAZDZCUNDZEZBCUBFZGHZIJZURKJZBKJZCKJZEZUQURUNDUTVAL - ABCUCURUNMNUQUTBGHZCGHZUDFZIJZVEIJZVFIJZEZVDUQUSVGIABCUEUFUOVETDZIVEOPZEZ - VFTDZIVFOPZEZVHVKLUPUOVEQDZVNABRVRVLVMVESVEUGUKNUPVFQDZVQACRVSVOVPVFSVFUG - UKNVEVFUHUIUOVIVBUPVJVCBUNMCUNMUJULUM $. + ABCUCURUNMNUQUTBGHZCGHZUDFZIJZVEIJZVFIJZEZVDUQUSVGIAABCUEUFUOVETDZIVEOPZE + ZVFTDZIVFOPZEZVHVKLUPUOVEQDZVNABRVRVLVMVESVEUGUKNUPVFQDZVQACRVSVOVPVFSVFU + GUKNVEVFUHUIUOVIVBUPVJVCBUNMCUNMUJULUM $. $d x I $. $( Value of a symbol in the left half of a concatenated word. (Contributed @@ -149582,7 +149609,7 @@ computer programs (as last() or lastChar()), the terminology used for -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) ) $= ( cword wcel wa cc0 chash cfv cfz co caddc cconcat wi lencl elfz0add syl2an cn0 ccatlen oveq2d eleq2d sylibrd ) ADEZFZBUDFZGZCHAIJZKLFZCHUHBIJZMLZKLZFZ - CHABNLIJZKLZFUEUHSFUJSFUIUMOUFDAPDBPUHUJCQRUGUOULCUGUNUKHKDABTUAUBUC $. + CHABNLIJZKLZFUEUHSFUJSFUIUMOUFDAPDBPUHUJCQRUGUOULCUGUNUKHKDDABTUAUBUC $. ${ $d A x $. $d B x $. $d V x $. @@ -149602,29 +149629,28 @@ computer programs (as last() or lastChar()), the terminology used for ccatsymb $p |- ( ( A e. Word V /\ B e. Word V /\ I e. ZZ ) -> ( ( A ++ B ) ` I ) = if ( I < ( # ` A ) , ( A ` I ) , ( B ` ( I - ( # ` A ) ) ) ) ) $= - ( wcel cz w3a cfv wbr co cc0 cle wa wceq 3adant3 wb syl cr c0 adantr clt wi - cword chash cmin cif cconcat id ad2antrl simpr anim2i simp3 0zd lencl nn0zd - cfzo 3ad2ant1 3jca elfzo mpbird df-3an sylanbrc ccatval1 eqcomd ex zre 0red - wn jca 3ad2ant3 ltnle wo 3adant2 animorrl wrdsymb0 sylc ccatcl eqtr4d com12 - sylbird pm2.61i nn0red syl2an biimpar ancomd zaddcl ccatval2 readdcl lenltd - caddc lenlt ccatlen olcd simp2 zsubcl sylan2 ancoms leaddsub2 syl3an biimpa - eqbrtrd ifeqda ) ADUCZEZBXCEZCFEZGZCAUDHZUAIZCAHZCXHUEJZBHZUFCABUGJZHZXGXIX - JXLXNKCLIZXGXIMZXJXNNZUBXOXPXQXOXPMZXDXECKXHUPJEZGZXQXRXDXEMZXSXTXGYAXOXIXD - XEYAXFYAUHOZUIXRXSXOXIMZXPXIXOXGXIUJUKXRXFKFEZXHFEZGZXSYCPXGYFXOXIXGXFYDYEX - DXEXFULZXGUMXDXEYEXFXDXHDAUNZUOZUQZURUICKXHUSQUTXDXEXSVAVBXTXNXJDABCVCVDQVE - XPXOVHZXQXGYKXQUBXIXGYKCKUAIZXQXGCREZKREZMZYLYKPXFXDYOXEXFYMYNCVFZXFVGVIVJC - KVKQXGYLXQXGYLMZXJSXNYQXDXFMZYLXHCLIZVLXJSNXGYRYLXDXFYRXEYRUHVMTXGYLYSVNCDA - VOVPYQXMXCEZXFMZYLXMUDHZCLIZVLZXNSNZXGUUAYLXGYTXFXDXEYTXFDABVQOYGVIZTXGYLUU - CVNCDXMVOZVPVRVEVTTVSWAXGXIVHZMZXNXLCXHBUDHZWJJZUAIZUUIXNXLNZUBUULUUIUUMUUL - UUIMZXDXECXHUUKUPJEZGZUUMUUNYAUUOUUPXGYAUULUUHYBUIUUNUUOYSUULMZUUNUULYSUUIY - SUULXGYSUUHXDXFYSUUHPZXEXDXHREZYMUURXFXDXHYHWBZYPXHCWKWCVMWDUKWEUUNXFYEUUKF - EZGZUUOUUQPXGUVBUULUUHXGXFYEUVAYGYJXDXEUVAXFXDYEUUJFEUVAXEYIXEUUJDBUNZUOXHU - UJWFWCOURUICXHUUKUSQUTXDXEUUOVAVBDABCWGQVEUUIUULVHZUUMXGUVDUUMUBUUHXGUVDUUK - CLIZUUMXGUUKCXDXEUUKREZXFXDUUSUUJREZUVFXEUUTXEUUJUVCWBZXHUUJWHWCOXFXDYMXEYP - VJWIXGUVEUUMXGUVEMZXNSXLUVIUUAUUDUUEXGUUAUVEUUFTUVIUUCYLUVIUUBUUKCLXGUUBUUK - NZUVEXDXEUVJXFDABWLOTXGUVEUJXAWMUUGVPUVIXEXKFEZMZXKKUAIZUUJXKLIZVLXLSNXGUVL - UVEXGXEUVKXDXEXFWNXDXFUVKXEXFXDUVKXDXFYEUVKYICXHWOWPWQVMVITUVIUVNUVMXGUVEUV - NXDUUSXEUVGXFYMUVEUVNPUUTUVHYPXHUUJCWRWSWTWMXKDBVOVPVRVEVTTVSWAVDXBVD $. + ( wcel cz co cfv clt wbr wceq wa cc0 cle simpr wb ex c0 adantr sylc cconcat + cword chash cmin cif cfzo w3a simprll anim2i 0zd lencl nn0zd ad2antrr elfzo + syl3anc ad2antrl mpbird df-3an sylanbrc ccatval1 eqcomd syl zre 0red ltnled + wi wn adantl wo simpl anim1i animorrl wrdsymb0 ccatcl sylbird com12 adantrd + eqtr4d pm2.61i caddc id nn0red lenlt syl2an adantlr biimpar anim12ci zaddcl + ccatval2 readdcl simplr zsubcld jca ad2antlr leaddsub2d biimpa olcd ccatlen + cr eqbrtrd ifeqda 3impa ) ADUBZEZBXCEZCFEZCABUAGZHZCAUCHZIJZCAHZCXIUDGZBHZU + EZKXDXELZXFLZXNXHXPXJXKXMXHMCNJZXPXJLZXKXHKZVFXQXRXSXQXRLZXDXECMXIUFGEZUGZX + SXTXOYAYBXQXOXFXJUHXTYAXQXJLZXRXJXQXPXJOUIXPYAYCPZXQXJXPXFMFEXIFEZYDXOXFOZX + PUJXDYEXEXFXDXIDAUKZULZUMZCMXIUNUOUPUQXDXEYAURUSYBXHXKDABCUTVAVBQXQVGZXPXSX + JXPYJXSXPYJCMIJZXSXFYKYJPXOXFCMCVCZXFVDVEVHXPYKXSXPYKLZXKRXHYMXDXFLZYKXICNJ + ZVIXKRKXPYNYKXOXDXFXDXEVJVKSXPYKYOVLCDAVMTYMXGXCEZXFLZYKXGUCHZCNJZVIZXHRKZX + PYQYKXOYPXFDABVNVKZSXPYKYSVLCDXGVMZTVRQVOVPVQVSCXIBUCHZVTGZIJZXPXJVGZLZXMXH + KZVFUUFUUHUUIUUFUUHLZXDXECXIUUEUFGEZUGZUUIUUJXOUUKUULUUFXOXFUUGUHUUJUUKYOUU + FLZUUFUUFUUHYOUUFWAXPYOUUGXDXFYOUUGPZXEXDXIWSEZCWSEZUUNXFXDXIYGWBZYLXICWCWD + WEWFWGXPUUKUUMPZUUFUUGXPXFYEUUEFEZUURYFYIXOUUSXFXDYEUUDFEUUSXEYHXEUUDDBUKZU + LXIUUDWHWDSCXIUUEUNUOUPUQXDXEUUKURUSUULXHXMDABCWIVAVBQUUFVGZXPUUIUUGXPUVAUU + IXPUVAUUECNJZUUIXOUUEWSEZUUPUVBUVAPXFXDUUOUUDWSEZUVCXEUUQXEUUDUUTWBZXIUUDWJ + WDYLUUECWCWDXPUVBUUIXPUVBLZXMRXHUVFXEXLFEZLZXLMIJZUUDXLNJZVIXMRKXPUVHUVBXPX + EUVGXDXEXFWKXDXFUVGXEYNCXIXDXFOXDYEXFYHSWLWEWMSUVFUVJUVIXPUVBUVJXPXIUUDCXDU + UOXEXFUUQUMXEUVDXDXFUVEWNXFUUPXOYLVHWOWPWQXLDBVMTUVFYQYTUUAXPYQUVBUUBSUVFYS + YKUVFYRUUECNXOYRUUEKXFUVBDDABWRUMXPUVBOWTWQUUCTVRQVOVPVQVSXAVAXB $. $( The first symbol of a concatenation of two words is the first symbol of the first word if the first word is not empty. (Contributed by Alexander @@ -149688,42 +149714,41 @@ computer programs (as last() or lastChar()), the terminology used for ccatass $p |- ( ( S e. Word B /\ T e. Word B /\ U e. Word B ) -> ( ( S ++ T ) ++ U ) = ( S ++ ( T ++ U ) ) ) $= - ( wcel cc0 chash cfv caddc co cfzo cconcat wfn ccatcl wceq oveq2d syl3anc - adantr syl cmin vx cword w3a wf stoic3 wrdf ffn 3syl ccatlen oveq1d eqtrd - 3adant3 fneq2d mpbid simp1 3adant1 syl2anc lencl 3ad2ant1 nn0cnd 3ad2ant2 - cn0 3ad2ant3 addassd 3eqtr4d cv wo cz nn0zd fzospliti ancoms sylan simpl1 - simpl2 simpr ccatval1 simpl3 cuz wss uzid uzaddcl fzoss2 sseqtr4d zaddcld - sselda ccatval2 fzosubel3 eqtr4d fzoss1 nn0uz eleq2s elfzoelz zcnd adantl - wa subsub4d fveq2d eleq2d biimpa fzosubel2 syl13anc oveq12d jaodan syldan - cc biimpar eqfnfvd ) BAUBZEZCXHEZDXHEZUCZUAFBGHZCGHZIJZDGHZIJZKJZBCLJZDLJ - ZBCDLJZLJZXLXTFXTGHZKJZMZXTXRMXLXTXHEZYDAXTUDYEXIXJXSXHEZXKYFABCNZAXSDNUE - AXTUFYDAXTUGUHXLYDXRXTXLYCXQFKXLYCXSGHZXPIJZXQXIXJYGXKYCYJOYHAXSDUIUEXLYI - XOXPIXIXJYIXOOXKABCUIULZUJZUKPUMUNXLYBFYBGHZKJZMZYBXRMXLYBXHEZYNAYBUDYOXL - XIYAXHEZYPXIXJXKUOZXJXKYQXIACDNUPZABYANUQAYBUFYNAYBUGUHXLYNXRYBXLYMXQFKXL - XMYAGHZIJZXMXNXPIJZIJZYMXQXLYTUUBXMIXJXKYTUUBOXIACDUIUPPZXLXIYQYMUUAOYRYS - ABYAUIUQXLXMXNXPXLXMXIXJXMVBEZXKABURUSZUTZXLXNXJXIXNVBEZXKACURVAZUTZXLXPX - KXIXPVBEZXJADURVCZUTVDZVEPUMUNXLUAVFZXREZUUNFXMKJZEZUUNXMXQKJZEZVGZUUNXTH - ZUUNYBHZOZXLXMVHEZUUOUUTXLXMUUFVIZUUOUVDUUTUUNFXQXMVJVKVLXLUUQUVCUUSXLUUQ - WOZUUNXSHZUUNBHZUVAUVBUVFXIXJUUQUVGUVHOXIXJXKUUQVMZXIXJXKUUQVNXLUUQVOZABC - UUNVPQUVFYGXKUUNFYIKJZEZUVAUVGOZXLYGUUQXIXJYGXKYHULZRXIXJXKUUQVQXLUUPUVKU - UNXLUUPFXOKJZUVKXLXOXMVRHZEZUUPUVOVSXLXMUVPEZUUHUVQXLUVDUVRUVEXMVTSUUIXNX - MXMWAUQZXMFXOWBSXLYIXOFKYKPZWCWEAXSDUUNVPZQUVFXIYQUUQUVBUVHOUVIXLYQUUQYSR - UVJABYAUUNVPQVEXLUUSUUNXMXOKJZEZUUNXOXQKJZEZVGZUVCXLXOVHEZUUSUWFXLXMXNUVE - XLXNUUIVIZWDZUUSUWGUWFUUNXMXQXOVJVKVLXLUWCUVCUWEXLUWCWOZUVGUUNXMTJZYAHZUV - AUVBUWJUVGUWKCHZUWLUWJXIXJUWCUVGUWMOXIXJXKUWCVMZXIXJXKUWCVNZXLUWCVOABCUUN - WFQUWJXJXKUWKFXNKJEZUWLUWMOUWOXIXJXKUWCVQZXLXNVHEZUWCUWPUWHUWCUWRUWPUUNXM - XNWGVKVLACDUWKVPQWHUWJYGXKUVLUVMXLYGUWCUVNRUWQXLUWBUVKUUNXLUWBUVOUVKXLUUE - UWBUVOVSZUUFUWSXMFVRHVBXMFXOWIWJWKSUVTWCWEUWAQUWJXIYQUUNXMUUAKJZEZUVBUWLO - ZUWNXLYQUWCYSRXLUWBUWTUUNXLUWBUURUWTXLXQXOVRHZEZUWBUURVSXLXOUXCEZUUKUXDXL - UWGUXEUWIXOVTSUULXPXOXOWAUQXOXMXQWBSXLUUAXQXMKXLUUAUUCXQUUDUUMWHPZWCWEABY - AUUNWFZQVEXLUWEWOZUUNYITJZDHZUWLUVAUVBUXHUXJUWKXNTJZDHZUWLUXHUXIUXKDUXHUX - IUUNXOTJZUXKXLUXIUXMOUWEXLYIXOUUNTYKPRUXHUUNXMXNUWEUUNXEEXLUWEUUNUUNXOXQW - LWMWNXLXMXEEUWEUUGRXLXNXEEUWEUUJRWPWHWQUXHXJXKUWKXNUUBKJEZUWLUXLOXIXJXKUW - EVNXIXJXKUWEVQZUXHUUNXOUUCKJZEZUVDUWRUUBVHEZUXNXLUWEUXQXLUWDUXPUUNXLXQUUC - XOKUUMPWRWSXLUVDUWEUVERXLUWRUWEUWHRXLUXRUWEXLXNXPUWHXLXPUULVIWDRUUNXMXNUU - BWTXAACDUWKWFQWHUXHYGXKUUNYIYJKJZEZUVAUXJOXLYGUWEUVNRUXOXLUXTUWEXLUXSUWDU - UNXLYIXOYJXQKYKYLXBWRXFAXSDUUNWFQUXHXIYQUXAUXBXIXJXKUWEVMXLYQUWEYSRXLUWDU - WTUUNXLUWDUURUWTXLUVQUWDUURVSUVSXOXMXQWISUXFWCWEUXGQVEXCXDXCXDXG $. + ( wcel cc0 chash cfv caddc co cfzo cconcat wfn wceq oveq2d syl2anc adantr + syl syl3anc cmin vx cword w3a ccatcl stoic3 ccatlen 3adant3 oveq1d fneq2d + wrdfn eqtrd mpbid simp1 3adant1 3ad2ant1 nn0cnd 3ad2ant2 3ad2ant3 addassd + cn0 lencl 3eqtr4d cv wo cz nn0zd fzospliti ex mpan9 wa simp2 id syl2an3an + ccatval1 simpl3 cuz uzidd uzaddcl fzoss2 sseqtr4d sselda zaddcld ccatval2 + wss simpl2 fzosubel3 eqtr4d fzoss1 nn0uz eleq2s simpl1 cc elfzoelz adantl + zcnd subsub4d fveq2d eleq2d biimpa 3jca fzosubel2 oveq12d biimpar eqfnfvd + jaodan syldan ) BAUBZEZCXGEZDXGEZUCZUAFBGHZCGHZIJZDGHZIJZKJZBCLJZDLJZBCDL + JZLJZXKXSFXSGHZKJZMZXSXQMXKXSXGEZYDXHXIXRXGEZXJYEABCUDZAXRDUDUEAXSUJRXKYC + XQXSXKYBXPFKXKYBXRGHZXOIJZXPXHXIYFXJYBYINYGAAXRDUFUEXKYHXNXOIXHXIYHXNNXJA + ABCUFUGZUHZUKOUIULXKYAFYAGHZKJZMZYAXQMXKYAXGEZYNXKXHXTXGEZYOXHXIXJUMZXIXJ + YPXHACDUDUNZABXTUDPAYAUJRXKYMXQYAXKYLXPFKXKXLXTGHZIJZXLXMXOIJZIJZYLXPXKYS + UUAXLIXIXJYSUUANXHAACDUFUNOZXKXHYPYLYTNYQYRAABXTUFPXKXLXMXOXKXLXHXIXLUTEZ + XJABVAUOZUPZXKXMXIXHXMUTEZXJACVAUQZUPZXKXOXJXHXOUTEZXIADVAURZUPUSZVBOUIUL + XKUAVCZXQEZUUMFXLKJZEZUUMXLXPKJZEZVDZUUMXSHZUUMYAHZNZXKXLVEEZUUNUUSXKXLUU + EVFZUUNUVCUUSUUMFXPXLVGVHVIXKUUPUVBUURXKUUPVJZUUMXRHZUUMBHZUUTUVAXKXHXIUU + PUUPUVFUVGNYQXHXIXJVKZUUPVLZABCUUMVNVMUVEYFXJUUMFYHKJZEZUUTUVFNZXKYFUUPXH + XIYFXJYGUGZQXHXIXJUUPVOXKUUOUVJUUMXKUUOFXNKJZUVJXKXNXLVPHZEZUUOUVNWDXKXLU + VOEUUGUVPXKXLUVDVQUUHXMXLXLVRPZXLFXNVSRXKYHXNFKYJOZVTWAAXRDUUMVNZSXKXHYPU + UPUUPUVAUVGNYQYRUVIABXTUUMVNVMVBXKUURUUMXLXNKJZEZUUMXNXPKJZEZVDZUVBXKXNVE + EZUURUWDXKXLXMUVDXKXMUUHVFZWBZUURUWEUWDUUMXLXPXNVGVHVIXKUWAUVBUWCXKUWAVJZ + UVFUUMXLTJZXTHZUUTUVAUWHUVFUWICHZUWJXKXHXIUWAUWAUVFUWKNYQUVHUWAVLABCUUMWC + VMUWHXIXJUWIFXMKJEZUWJUWKNXHXIXJUWAWEXHXIXJUWAVOZXKXMVEEZUWAUWLUWFUWAUWNU + WLUUMXLXMWFVHVIACDUWIVNSWGUWHYFXJUVKUVLXKYFUWAUVMQUWMXKUVTUVJUUMXKUVTUVNU + VJXKUUDUVTUVNWDZUUEUWOXLFVPHUTXLFXNWHWIWJRUVRVTWAUVSSUWHXHYPUUMXLYTKJZEZU + VAUWJNZXHXIXJUWAWKXKYPUWAYRQXKUVTUWPUUMXKUVTUUQUWPXKXPXNVPHZEZUVTUUQWDXKX + NUWSEUUJUWTXKXNUWGVQUUKXOXNXNVRPXNXLXPVSRXKYTXPXLKXKYTUUBXPUUCUULWGOZVTWA + ABXTUUMWCZSVBXKUWCVJZUUMYHTJZDHZUWJUUTUVAUXCUXEUWIXMTJZDHZUWJUXCUXDUXFDUX + CUXDUUMXNTJZUXFXKUXDUXHNUWCXKYHXNUUMTYJOQUXCUUMXLXMUWCUUMWLEXKUWCUUMUUMXN + XPWMWOWNXKXLWLEUWCUUFQXKXMWLEUWCUUIQWPWGWQUXCXIXJUWIXMUUAKJEZUWJUXGNXHXIX + JUWCWEXHXIXJUWCVOZUXCUUMXNUUBKJZEZUVCUWNUUAVEEZUCZUXIXKUWCUXLXKUWBUXKUUMX + KXPUUBXNKUULOWRWSXKUXNUWCXKUVCUWNUXMUVDUWFXKXMXOUWFXKXOUUKVFWBWTQUUMXLXMU + UAXAPACDUWIWCSWGUXCYFXJUUMYHYIKJZEZUUTUXENXKYFUWCUVMQUXJXKUXPUWCXKUXOUWBU + UMXKYHXNYIXPKYJYKXBWRXCAXRDUUMWCSUXCXHYPUWQUWRXHXIXJUWCWKXKYPUWCYRQXKUWBU + WPUUMXKUWBUUQUWPXKUVPUWBUUQWDUVQXNXLXPWHRUXAVTWAUXBSVBXEXFXEXFXD $. $( The range of a concatenated word. (Contributed by Stefan O'Rear, 15-Aug-2015.) $) @@ -149768,18 +149793,18 @@ computer programs (as last() or lastChar()), the terminology used for shortened by AV, 1-May-2020.) $) lswccatn0lsw $p |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) ) -> ( lastS ` ( A ++ B ) ) = ( lastS ` B ) ) $= - ( wcel w3a cconcat co clsw cfv chash c1 cmin cvv wceq lsw wa oveq1d 3adant3 - cz eqtrd cword c0 wne ovex mp1i caddc cfzo ccatlen clt lencl nn0zd 3ad2ant1 - wbr cn lennncl 3adant1 simpl nnz adantl zaddcld crp zre nnrp ltaddrp syl2an - cr 3jca syl2anc fzolb sylibr fzoend syl eqeltrd ccatval2 syld3an3 cc nn0cnd - addcl 1cnd sub32d pncan2 fveq2d eqcomd 3ad2ant2 ) ACUAZDZBWEDZBUBUCZEZABFGZ - HIZBJIZKLGZBIZBHIZWIWKWJJIZKLGZWJIZWNWJMDWKWRNWIABFUDWJMOUEWIWRWQAJIZLGZBIZ - WNWFWGWHWQWSWSWLUFGZUGGZDWRXANWIWQXBKLGZXCWFWGWQXDNWHWFWGPZWPXBKLCABUHQZRWI - WSXCDZXDXCDWIWSSDZXBSDZWSXBUIUMZEZXGWIXHWLUNDZXKWFWGXHWHWFWSCAUJZUKULWGWHXL - WFCBUOUPXHXLPZXHXIXJXHXLUQZXNWSWLXOXLWLSDXHWLURUSUTXHWSVFDWLVADXJXLWSVBWLVC - WSWLVDVEVGVHWSXBVIVJWSXBVKVLVMCABWQVNVOWIWTWMBWFWGWTWMNWHXEWTXDWSLGZWMXEWQX - DWSLXFQWFWSVPDZWLVPDZXPWMNWGWFWSXMVQWGWLCBUJVQXQXRPZXPXBWSLGZKLGWMXSXBKWSWS - WLVRXSVSXQXRUQVTXSXTWLKLWSWLWAQTVETRWBTTWGWFWNWONWHWGWOWNBWEOWCWDT $. + ( wcel w3a cconcat co chash cfv c1 cmin clsw wa oveq1d 3adant3 lencl syl2an + wceq cz eqtrd cword c0 wne caddc ccatlen clt wbr cn nn0zd lennncl simpl nnz + cfzo zaddcl sylan2 cr crp zre nnrp ltaddrp 3jca 3impb sylibr fzoend eqeltrd + fzolb syl ccatval2 syld3an3 nn0cnd addcl 1cnd sub32d pncan2 fveq2d cvv ovex + cc lsw mp1i 3ad2ant2 3eqtr4d ) ACUAZDZBWCDZBUBUCZEZABFGZHIZJKGZWHIZBHIZJKGZ + BIZWHLIZBLIZWGWKWJAHIZKGZBIZWNWDWEWFWJWQWQWLUDGZUMGZDWKWSRWGWJWTJKGZXAWDWEW + JXBRWFWDWEMZWIWTJKCCABUENZOWGWQXADZXBXADWGWQSDZWTSDZWQWTUFUGZEZXEWDWEWFXIWD + XFWLUHDZXIWEWFMWDWQCAPZUICBUJXFXJMXFXGXHXFXJUKXJXFWLSDXGWLULWQWLUNUOXFWQUPD + WLUQDXHXJWQURWLUSWQWLUTQVAQVBWQWTVFVCWQWTVDVGVECABWJVHVIWGWRWMBWDWEWRWMRWFX + CWRXBWQKGZWMXCWJXBWQKXDNWDWQVRDZWLVRDZXLWMRWEWDWQXKVJWEWLCBPVJXMXNMZXLWTWQK + GZJKGWMXOWTJWQWQWLVKXOVLXMXNUKVMXOXPWLJKWQWLVNNTQTOVOTWHVPDWOWKRWGABFVQWHVP + VSVTWEWDWPWNRWFBWCVSWAWB $. $( The last symbol of a word concatenated with the empty word is the last symbol of the word. (Contributed by AV, 22-Oct-2018.) (Proof shortened @@ -150053,9 +150078,9 @@ computer programs (as last() or lastChar()), the terminology used for 4-Mar-2022.) $) ccatws1len $p |- ( W e. Word V -> ( # ` ( W ++ <" X "> ) ) = ( ( # ` W ) + 1 ) ) $= - ( cword wcel cs1 cconcat co chash cfv caddc wceq wrdv s1cli ccatlen sylancl - c1 cvv s1len a1i oveq2d eqtrd ) BADEZBCFZGHIJZBIJZUDIJZKHZUFQKHUCBRDZEUDUIE - UEUHLABMCNRBUDOPUCUGQUFKUGQLUCCSTUAUB $. + ( cword wcel cs1 cconcat co chash cfv caddc c1 cvv wceq s1cli ccatlen mpan2 + s1len oveq2i syl6eq ) BADEZBCFZGHIJZBIJZUBIJZKHZUDLKHUAUBMDEUCUFNCOAMBUBPQU + ELUDKCRST $. $( The length of a word is ` N ` iff the length of the concatenation of the word with a singleton word is ` N + 1 ` . (Contributed by AV, @@ -150077,12 +150102,11 @@ computer programs (as last() or lastChar()), the terminology used for NOUCUEQEUGUERUCUEBCPSUETUAUB $. $( The length of the concatenation of two singleton words. (Contributed by - Alexander van der Vekens, 22-Sep-2018.) $) - ccat2s1len $p |- ( ( X e. V /\ Y e. V ) - -> ( # ` ( <" X "> ++ <" Y "> ) ) = 2 ) $= - ( wcel cs1 cword cconcat co chash cfv c2 wceq s1cl wa caddc ccatlen oveq12i - c1 s1len 1p1e2 eqtri syl6eq syl2an ) BADBEZAFZDZCEZUEDZUDUGGHIJZKLCADBAMCAM - UFUHNUIUDIJZUGIJZOHZKAUDUGPULRROHKUJRUKROBSCSQTUAUBUC $. + Alexander van der Vekens, 22-Sep-2018.) (Revised by JJ, 14-Jan-2024.) $) + ccat2s1len $p |- ( # ` ( <" X "> ++ <" Y "> ) ) = 2 $= + ( cs1 cvv cword wcel cconcat co chash cfv c2 wceq s1cli wa caddc ccatlen c1 + s1len oveq12i 1p1e2 eqtri syl6eq mp2an ) ACZDEZFZBCZUEFZUDUGGHIJZKLAMBMUFUH + NUIUDIJZUGIJZOHZKDDUDUGPULQQOHKUJQUKQOARBRSTUAUBUC $. $( The concatenation of a word with two singleton words is a word. (Contributed by Alexander van der Vekens, 22-Sep-2018.) $) @@ -150699,34 +150723,34 @@ computer programs (as last() or lastChar()), the terminology used for ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) = ( S substr <. X , Z >. ) ) $= ( wcel cc0 cfz co chash cfv wa cmin cfzo adantr syl2anc caddc wceq oveq2d - syl3anc vx cword w3a cop csubstr cconcat wfn wf swrdcl ccatcl ffn ccatlen - wrdf simpl simpr1 simpr2 simpr3 fzass4 biimpri swrdlen oveq12d cz elfzelz - 3syl simpld syl npncan3d eqtrd fneq2d mpbid cv wo simpr zsubcld fzospliti - eleq2d biimpar ccatval1 simpll simplr1 swrdfv syl31anc ccatval2 fzosubel3 - simplr2 simplr3 eqeltrd oveq1d cc elfzoelz adantl subcld subadd23d nncand - zcnd 3eqtrd fveq2d jaodan syldan eqtr4d eqfnfvd ) BAUBZFZCGDHIFZDGEHIZFZE - GBJKZHIZFZUCZLZUAGECMIZNIZBCDUDUEIZBDEUDUEIZUFIZBCEUDUEIZXKXPGXPJKZNIZUGZ - XPXMUGXKXPXBFZXSAXPUHXTXKXNXBFZXOXBFZYAXCYBXJABCDUIOZXCYCXJABDEUIOZAXNXOU - JPAXPUMXSAXPUKVDXKXSXMXPXKXRXLGNXKXRXNJKZXOJKZQIZXLXKYBYCXRYHRYDYEAXNXOUL - PXKYHDCMIZEDMIZQIZXLXKYFYIYGYJQXKXCXDDXHFZYFYIRXCXJUNZXCXDXFXIUOZXKXFXIYL - XCXDXFXIUPZXCXDXFXIUQZXFXILZYLEDXGHIFZYLYRLYQGDEXGURUSVEPZABCDUTTZXKXCXFX - IYGYJRYMYOYPABDEUTTVAXKDCEXKDXKXFDVBFYODGEVCVFZWOZXKCXKXDCVBFYNCGDVCVFZWO - ZXKEXKXIEVBFYPEGXGVCVFZWOVGZVHZVHSVIVJXKXQGXQJKZNIZUGZXQXMUGXKXQXBFZUUIAX - QUHUUJXCUUKXJABCEUIOAXQUMUUIAXQUKVDXKUUIXMXQXKUUHXLGNXKXCCXEFZXIUUHXLRYMX - KXDXFUULYNYOXDXFLZUULDCEHIFZUULUUNLUUMGCDEURUSVEPZYPABCEUTTSVIVJXKUAVKZXM - FZLZUUPXPKZUUPCQIZBKZUUPXQKZXKUUQUUPGYINIZFZUUPYIXLNIZFZVLZUUSUVARZUURUUQ - YIVBFZUVGXKUUQVMZXKUVIUUQXKDCUUAUUCVNOUUPGXLYIVOPXKUVDUVHUVFXKUVDLZUUSUUP - XNKZUVAUVKYBYCUUPGYFNIZFZUUSUVLRXKYBUVDYDOXKYCUVDYEOXKUVNUVDXKUVMUVCUUPXK - YFYIGNYTSVPVQAXNXOUUPVRTUVKXCXDYLUVDUVLUVARXCXJUVDVSXDXFXIXCUVDVTXKYLUVDY - SOXKUVDVMABCDUUPWAWBVHXKUVFLZUUSUUPYFMIZXOKZUVPDQIZBKZUVAUVOYBYCUUPYFYHNI - ZFZUUSUVQRXKYBUVFYDOXKYCUVFYEOXKUWAUVFXKUVTUVEUUPXKYFYIYHXLNYTUUGVAVPVQAX - NXOUUPWCTUVOXCXFXIUVPGYJNIZFUVQUVSRXCXJUVFVSXDXFXIXCUVFWEXDXFXIXCUVFWFUVO - UVPUUPYIMIZUWBXKUVPUWCRUVFXKYFYIUUPMYTSZOUVOUUPYIYKNIZFZYJVBFZUWCUWBFXKUW - FUVFXKUWEUVEUUPXKYKXLYINUUFSVPVQXKUWGUVFXKEDUUEUUAVNOUUPYIYJWDPWGABDEUVPW - AWBUVOUVRUUTBUVOUVRUWCDQIZUUPDYIMIZQIZUUTXKUVRUWHRUVFXKUVPUWCDQUWDWHOUVOU - UPYIDUVFUUPWIFXKUVFUUPUUPYIXLWJWOWKXKYIWIFUVFXKDCUUBUUDWLOXKDWIFUVFUUBOWM - XKUWJUUTRUVFXKUWICUUPQXKDCUUBUUDWNSOWPWQWPWRWSUURXCUULXIUUQUVBUVARXCXJUUQ - VSXKUULUUQUUOOXDXFXIXCUUQWFUVJABCEUUPWAWBWTXA $. + syl vx cword w3a cop csubstr cconcat wfn swrdcl ccatcl wrdfn simpl simpr1 + ccatlen simpr2 simpr3 fzass4 biimpri swrdlen syl3anc 3adant3r1 oveq12d cz + simpld elfzelz zcnd npncan3d 3eqtrd mpbid cv wo zsubcld anim1ci fzospliti + fneq2d ad2antrr eleq2d biimpar ccatval1 simpll simplr1 simpr swrdfv eqtrd + syl31anc ccatval2 simplr2 simplr3 fzosubel3 oveq1d elfzoelz adantl subcld + eqeltrd cc subadd23d nncand fveq2d jaodan syldan eqtr4d eqfnfvd ) BAUBZFZ + CGDHIFZDGEHIZFZEGBJKZHIZFZUCZLZUAGECMIZNIZBCDUDUEIZBDEUDUEIZUFIZBCEUDUEIZ + XKXPGXPJKZNIZUGZXPXMUGXKXPXBFZXTXKXNXBFZXOXBFZYAXCYBXJABCDUHZOZXCYCXJABDE + UHZOZAXNXOUIPAXPUJTXKXSXMXPXKXRXLGNXKXRXNJKZXOJKZQIZDCMIZEDMIZQIZXLXKYBYC + XRYJRYEYGAAXNXOUMPXKYHYKYIYLQXKXCXDDXHFZYHYKRXCXJUKZXCXDXFXIULZXKXFXIYNXC + XDXFXIUNZXCXDXFXIUOZXFXILZYNEDXGHIFZYNYTLYSGDEXGUPUQVCPZABCDURUSZXCXFXIYI + YLRXDABDEURUTVAZXKDCEXKDXKXFDVBFYQDGEVDTZVEZXKCXKXDCVBFYPCGDVDTZVEZXKEXKX + IEVBFYREGXGVDTZVEVFZVGSVNVHXKXQGXQJKZNIZUGZXQXMUGXKXQXBFZUULXCUUMXJABCEUH + OAXQUJTXKUUKXMXQXKUUJXLGNXKXCCXEFZXIUUJXLRYOXKXDXFUUNYPYQXDXFLZUUNDCEHIFZ + UUNUUPLUUOGCDEUPUQVCPZYRABCEURUSSVNVHXKUAVIZXMFZLZUURXPKZUURCQIZBKZUURXQK + ZXKUUSUURGYKNIZFZUURYKXLNIZFZVJZUVAUVCRZUUTUUSYKVBFZLUVIXKUVKUUSXKDCUUDUU + FVKVLUURGXLYKVMTXKUVFUVJUVHXKUVFLZUVAUURXNKZUVCUVLYBYCUURGYHNIZFZUVAUVMRX + CYBXJUVFYDVOXCYCXJUVFYFVOXKUVOUVFXKUVNUVEUURXKYHYKGNUUBSVPVQAXNXOUURVRUSU + VLXCXDYNUVFUVMUVCRXCXJUVFVSXDXFXIXCUVFVTXKYNUVFUUAOXKUVFWAABCDUURWBWDWCXK + UVHLZUVAUURYHMIZXOKZUVQDQIZBKZUVCUVPYBYCUURYHYJNIZFZUVAUVRRXCYBXJUVHYDVOX + CYCXJUVHYFVOXKUWBUVHXKUWAUVGUURXKYHYKYJXLNUUBXKYJYMXLUUCUUIWCVAVPVQAXNXOU + URWEUSUVPXCXFXIUVQGYLNIZFUVRUVTRXCXJUVHVSXDXFXIXCUVHWFXDXFXIXCUVHWGUVPUVQ + UURYKMIZUWCXKUVQUWDRUVHXKYHYKUURMUUBSZOUVPUURYKYMNIZFZYLVBFZUWDUWCFXKUWGU + VHXKUWFUVGUURXKYMXLYKNUUISVPVQXKUWHUVHXKEDUUHUUDVKOUURYKYLWHPWMABDEUVQWBW + DUVPUVSUVBBUVPUVSUWDDQIZUURDYKMIZQIZUVBXKUVSUWIRUVHXKUVQUWDDQUWEWIOUVPUUR + YKDUVHUURWNFXKUVHUURUURYKXLWJVEWKXKYKWNFUVHXKDCUUEUUGWLOXKDWNFUVHUUEOWOXK + UWKUVBRUVHXKUWJCUURQXKDCUUEUUGWPSOVGWQVGWRWSUUTXCUUNXIUUSUVDUVCRXCXJUUSVS + XKUUNUUSUUQOXDXFXIXCUUSWGXKUUSWAABCEUURWBWDWTXA $. $} ${ @@ -150735,20 +150759,19 @@ computer programs (as last() or lastChar()), the terminology used for Carneiro, 27-Sep-2015.) $) swrdccat2 $p |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) = T ) $= - ( vk wcel wa cc0 chash cfv cfzo co caddc wfn wf cfz cuz cn0 adantr oveq2d - wceq cword cconcat cop csubstr ccatcl swrdcl wrdf ffn 4syl lencl syl6eleq - cmin nn0uz cz nn0zd syl adantl uzaddcl syl2anc elfzuzb sylanbrc nn0addcld - uzid ccatlen eleqtrrd swrdlen syl3anc nn0cnd pncan2d eqtrd fneq2d ffnd cv - mpbid eleq2d biimpar swrdfv syl31anc ccatval3 3expa eqfnfvd ) BAUAZEZCWBE - ZFZDGCHIZJKZBCUBKZBHIZWIWFLKZUCUDKZCWEWKGWKHIZJKZMZWKWGMWEWHWBEZWKWBEWMAW - KNWNABCUEZAWHWIWJUFAWKUGWMAWKUHUIWEWMWGWKWEWLWFGJWEWLWJWIULKZWFWEWOWIGWJO - KZEZWJGWHHIZOKZEZWLWQTWPWEWIGPIZEWJWIPIZEZWSWEWIQXCWCWIQEWDABUJRZUMUKWEWI - XDEZWFQEZXEWEWIUNEXGWEWIXFUOWIVCUPWDXHWCACUJUQZWFWIWIURUSWIGWJUTVAZWEWJWR - XAWEWJXCEWJWJPIEZWJWREWEWJQXCWEWIWFXFXIVBZUMUKWEWJUNEXKWEWJXLUOWJVCUPWJGW - JUTVAWEWTWJGOABCVDSVEZAWHWIWJVFVGWEWIWFWEWIXFVHWEWFXIVHVIZVJSVKVNWEWGACWD - WGACNWCACUGUQVLWEDVMZWGEZFZXOWKIZXOWILKWHIZXOCIZXQWOWSXBXOGWQJKZEZXRXSTWE - WOXPWPRWEWSXPXJRWEXBXPXMRWEYBXPWEYAWGXOWEWQWFGJXNSVOVPAWHWIWJXOVQVRWCWDXP - XSXTTABCXOVSVTVJWA $. + ( vk wcel wa cc0 chash cfv cfzo co caddc wfn wrdfn cfz wceq syl2an oveq2d + cuz cn0 cword cconcat cop csubstr ccatcl swrdcl 3syl lencl nn0uz syl6eleq + cmin adantr nn0zd uzidd uzaddcl elfzuzb sylanbrc nn0addcl ccatlen swrdlen + eleqtrrd syl3anc cc nn0cnd pncan2 eqtrd fneq2d mpbid adantl cv w3a eleq2d + 3jca biimpar swrdfv syl2an2r ccatval3 3expa eqfnfvd ) BAUAZEZCVTEZFZDGCHI + ZJKZBCUBKZBHIZWGWDLKZUCUDKZCWCWIGWIHIZJKZMZWIWEMWCWFVTEZWIVTEWLABCUEZAWFW + GWHUFAWINUGWCWKWEWIWCWJWDGJWCWJWHWGUKKZWDWCWMWGGWHOKZEZWHGWFHIZOKZEZWJWOP + WNWCWGGSIZEZWHWGSIZEZWQWAXBWBWAWGTXAABUHZUIUJULWAWGXCEWDTEZXDWBWAWGWAWGXE + UMUNACUHZWDWGWGUOQWGGWHUPUQZWCWHWPWSWCWHXAEWHWHSIEWHWPEWCWHTXAWAWGTEXFWHT + EWBXEXGWGWDURQZUIUJWCWHWCWHXIUMUNWHGWHUPUQWCWRWHGOAABCUSRVAZAWFWGWHUTVBWA + WGVCEWDVCEWOWDPWBWAWGXEVDWBWDXGVDWGWDVEQZVFRVGVHWBCWEMWAACNVIWCDVJZWEEZFX + LWIIZXLWGLKWFIZXLCIZWCWMWQWTVKXMXLGWOJKZEZXNXOPWCWMWQWTWNXHXJVMWCXRXMWCXQ + WEXLWCWOWDGJXKRVLVNAWFWGWHXLVOVPWAWBXMXOXPPABCXLVQVRVFVS $. $} @@ -151120,31 +151143,30 @@ computer programs (as last() or lastChar()), the terminology used for ccatpfx $p |- ( ( S e. Word A /\ Y e. ( 0 ... Z ) /\ Z e. ( 0 ... ( # ` S ) ) ) -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) ) $= - ( wcel cc0 cfz co chash cfzo syl2anc caddc wceq cmin wa zcnd eqtrd adantr - cfv syl3anc vx cword w3a cop csubstr cconcat wfn wf pfxcl 3ad2ant1 swrdcl - cpfx ccatcl wrdf 3syl ccatlen simp1 fzass4 biimpri simpld 3adant1 swrdlen - pfxlen oveq12d cc cz elfzelz ad2antrl 3impb ad2antll pncan3d oveq2d mpbid - ffn fneq2d pfxfn 3adant2 cv wo 3ad2ant2 fzospliti eleq2d biimpar ccatval1 - simpr pfxfv ccatval2 anim1ci fzosubel syl wb subidd eqcomd oveq1d eqeltrd - mpbird swrdfv syldan elfzoelz adantl npcand fveq2d 3eqtrd jaodan eqfnfvd - simpl3 eqtr4d ) BAUBZEZCFDGHEZDFBISZGHZEZUCZUAFDJHZBCULHZBCDUDUEHZUFHZBDU - LHZXNXRFXRISZJHZUGZXRXOUGXNXRXHEZYAAXRUHYBXNXPXHEZXQXHEZYCXIXJYDXMABCUIUJ - ZXIXJYEXMABCDUKUJZAXPXQUMKAXRUNYAAXRVNUOXNYAXOXRXNXTDFJXNXTXPISZXQISZLHZD - XNYDYEXTYJMYFYGAXPXQUPKXNYJCDCNHZLHDXNYHCYIYKLXNXICXLEZYHCMXIXJXMUQZXJXMY - LXIXJXMOZYLDCXKGHEZYLYOOYNFCDXKURUSUTVAZABCVCKZABCDVBVDXNCDXIXJXMCVEEZXIY - NOZCXJCVFEZXIXMCFDVGZVHPVIZXIXJXMDVEEYSDXMDVFEXIXJDFXKVGVJPVIVKQZQVLVOVMX - IXMXSXOUGXJBDAVPVQXNUAVRZXOEZOZUUDXRSZUUDBSZUUDXSSZXNUUEUUDFCJHZEZUUDCDJH - ZEZVSZUUGUUHMZUUFUUEYTUUNXNUUEWEZXNYTUUEXJXIYTXMUUAVTZRUUDFDCWAKXNUUKUUOU - UMXNUUKOZUUGUUDXPSZUUHUURYDYEUUDFYHJHZEZUUGUUSMXNYDUUKYFRXNYEUUKYGRXNUVAU - UKXNUUTUUJUUDXNYHCFJYQVLWBWCAXPXQUUDWDTUURXIYLUUKUUSUUHMXNXIUUKYMRXNYLUUK - YPRXNUUKWEUUDCABWFTQXNUUMOZUUGUUDYHNHZXQSZUVCCLHZBSZUUHUVBYDYEUUDYHYJJHZE - ZUUGUVDMXNYDUUMYFRXNYEUUMYGRXNUVHUUMXNUVGUULUUDXNYHCYJDJYQUUCVDWBWCAXPXQU - UDWGTXNUUMUVCFYKJHZEUVDUVFMUVBUVCUUDCNHZUVIXNUVCUVJMUUMXNYHCUUDNYQVLRZUVB - UVJUVIEZUVJCCNHZYKJHZEZUVBUUMYTOUVOXNYTUUMUUQWHUUDCDCWIWJXNUVLUVOWKUUMXNU - VIUVNUVJXNFUVMYKJXJXIFUVMMXMXJUVMFXJCXJCUUAPWLWMVTWNWBRWPWOABCDUVCWQWRUVB - UVEUUDBUVBUVEUVJCLHUUDUVBUVCUVJCLUVKWNUVBUUDCUUMUUDVEEXNUUMUUDUUDCDWSPWTX - NYRUUMUUBRXAQXBXCXDWRUUFXIXMUUEUUIUUHMXNXIUUEYMRXIXJXMUUEXFUUPUUDDABWFTXG - XE $. + ( wcel cc0 cfz co chash cfv wceq wa cfzo adantr caddc cmin cc id ad2antrr + wfn vx cword cpfx cop csubstr cconcat pfxcl swrdcl ccatcl syl2anc ccatlen + wrdfn syl fzass4 biimpri simpld pfxlen swrdlen 3expb oveq12d elfzelz zcnd + sylan2 pncan3 syl2an adantl 3eqtrd oveq2d mpbid pfxfn adantrl cv ad2antrl + fneq2d wo fzospliti syl2anr eleq2d biimpar ccatval1 simpl pfxfv syl2an3an + syl3anc eqtrd ccatval2 w3a fzosubel subidd oveq1d eqeltrd swrdfv syl2an2r + cz wb elfzoelz npcan fveq2d jaodan syldan 3expa adantlrl eqtr4d eqfnfvd + 3impb ) BAUBZEZCFDGHEZDFBIJZGHZEZBCUCHZBCDUDUEHZUFHZBDUCHZKXGXHXKLZLZUAFD + MHZXNXOXQXNFXNIJZMHZTZXNXRTXGYAXPXGXNXFEZYAXGXLXFEZXMXFEZYBABCUGZABCDUHZA + XLXMUIUJAXNULUMNXQXTXRXNXQXSDFMXQXSXLIJZXMIJZOHZCDCPHZOHZDXGXSYIKZXPXGYCY + DYLYEYFAAXLXMUKUJNXQYGCYHYJOXPXGCXJEZYGCKXPYMDCXIGHEZYMYNLXPFCDXIUNUOUPZA + BCUQVCZXGXHXKYHYJKABCDURUSUTZXPYKDKZXGXHCQEZDQEYRXKXHCCFDVAZVBZXKDDFXIVAV + BCDVDVEVFZVGVHVNVIXGXKXOXRTXHBDAVJVKXQUAVLZXREZLUUCXNJZUUCBJZUUCXOJZXQUUD + UUCFCMHZEZUUCCDMHZEZVOZUUEUUFKZUUDUUDCWNEZUULXQUUDRXHUUNXGXKYTVMZUUCFDCVP + VQXQUUIUUMUUKXQUUILZUUEUUCXLJZUUFUUPYCYDUUCFYGMHZEZUUEUUQKXGYCXPUUIYESXGY + DXPUUIYFSXQUUSUUIXQUURUUHUUCXQYGCFMYPVHVRVSAXLXMUUCVTWDXQXGYMUUIUUIUUQUUF + KXGXPWAXPYMXGYOVFUUIRUUCCABWBWCWEXQUUKLZUUEUUCYGPHZXMJZUVACOHZBJZUUFUUTYC + YDUUCYGYIMHZEZUUEUVBKXGYCXPUUKYESXGYDXPUUKYFSXQUVFUUKXQUVEUUJUUCXQYGCYIDM + YPXQYIYKDYQUUBWEUTVRVSAXLXMUUCWFWDXQXGXHXKWGZUUKUVAFYJMHZEUVBUVDKXGXHXKUV + GUVGRUSUUTUVAUUCCPHZUVHXQUVAUVIKUUKXQYGCUUCPYPVHZNUUTUVICCPHZYJMHZEZUVIUV + HEZUUKUUKUUNUVMXQUUKRUUOUUCCDCWHVQXQUVMUVNWOZUUKXHUVOXGXKXHUVLUVHUVIXHUVK + FYJMXHCUUAWIWJVRVMNVIWKABCDUVAWLWMUUTUVCUUCBUUTUVCUVICOHZUUCXQUVCUVPKUUKX + QUVAUVICOUVJWJNUUKUUCQEYSUVPUUCKXQUUKUUCUUCCDWPVBXHYSXGXKUUAVMUUCCWQVQWEW + RVGWSWTXGXKUUDUUGUUFKZXHXGXKUUDUVQUUCDABWBXAXBXCXDXE $. $} ${ @@ -151153,16 +151175,16 @@ computer programs (as last() or lastChar()), the terminology used for Carneiro, 27-Sep-2015.) (Revised by AV, 6-May-2020.) $) pfxccat1 $p |- ( ( S e. Word B /\ T e. Word B ) -> ( ( S ++ T ) prefix ( # ` S ) ) = S ) $= - ( vk wcel wa chash cfv cc0 cfzo cfz wceq cn0 lencl adantr syl2anc wfn syl - co eqtrd cword cconcat cpfx cres ccatcl caddc anim12i sylib elfz0add sylc - nn0fz0 ccatlen oveq2d eleqtrrd pfxres wss ccatvalfn cz nn0zd uzid uzaddcl - cuz syl2an fzoss2 fnssres wrdfn cv fvres adantl ccatval1 3expa eqfnfvd ) - BAUAZEZCVMEZFZBCUBSZBGHZUCSZVQIVRJSZUDZBVPVQVMEVRIVQGHZKSZEVSWALABCUEVPVR - IVRCGHZUFSZKSZWCVPVRMEZWDMEZFVRIVRKSEZVRWFEVNWGVOWHABNZACNZUGVNWIVOVNWGWI - WJVRUKUHOVRWDVRUIUJVPWBWEIKABCULUMUNAVQVRUOPVPDVTWABVPVQIWEJSZQVTWLUPZWAV - TQBCAUQVPWEVRVBHZEZWMVNVRWNEZWHWOVOVNVRUREWPVNVRWJUSVRUTRWKWDVRVRVAVCVRIW - EVDRWLVTVQVEPVNBVTQVOABVFOVPDVGZVTEZFWQWAHZWQVQHZWQBHZWRWSWTLVPWQVTVQVHVI - VNVOWRWTXALABCWQVJVKTVLT $. + ( vk cword wcel wa cconcat co chash cfv cpfx cc0 cfzo cfz wceq cn0 adantr + lencl eqtrd cres ccatcl caddc anim12i nn0fz0 elfz0add sylc ccatlen oveq2d + sylib eleqtrrd pfxres syl2anc ccatvalfn cuz wss nn0zd uzidd syl2an fzoss2 + uzaddcl syl fnssresd wfn wrdfn cv fvres adantl ccatval1 3expa eqfnfvd ) B + AEZFZCVLFZGZBCHIZBJKZLIZVPMVQNIZUAZBVOVPVLFVQMVPJKZOIZFVRVTPABCUBVOVQMVQC + JKZUCIZOIZWBVOVQQFZWCQFZGVQMVQOIFZVQWEFVMWFVNWGABSZACSZUDVMWHVNVMWFWHWIVQ + UEUJRVQWCVQUFUGVOWAWDMOAABCUHUIUKAVPVQULUMVODVSVTBVOMWDNIZVSVPBCAUNVOWDVQ + UOKZFZVSWKUPVMVQWLFWGWMVNVMVQVMVQWIUQURWJWCVQVQVAUSVQMWDUTVBVCVMBVSVDVNAB + VERVODVFZVSFZGWNVTKZWNVPKZWNBKZWOWPWQPVOWNVSVPVGVHVMVNWOWQWRPABCWNVIVJTVK + T $. $} $( The prefix of length one of a nonempty word expressed as a singleton word. @@ -151369,9 +151391,9 @@ computer programs (as last() or lastChar()), the terminology used for -> ( # ` ( ( W substr <. M , ( # ` W ) >. ) ++ ( W prefix M ) ) ) = ( # ` W ) ) $= ( cword wcel cc0 chash cfv cfz co wa csubstr cpfx cconcat caddc wceq swrdcl - cop pfxcl jca adantr ccatlen syl addlenrevpfx eqtrd ) CBDZEZAFCGHZIJEZKZCAU - HRLJZCAMJZNJGHZUKGHULGHOJZUHUJUKUFEZULUFEZKZUMUNPUGUQUIUGUOUPBCAUHQBCASTUAB - UKULUBUCABCUDUE $. + cop pfxcl ccatlen syl2anc adantr addlenrevpfx eqtrd ) CBDZEZAFCGHZIJEZKCAUG + RLJZCAMJZNJGHZUIGHUJGHOJZUGUFUKULPZUHUFUIUEEUJUEEUMBCAUGQBCASBBUIUJTUAUBABC + UCUD $. $( Reconstruct a nonempty word from its prefix and last symbol. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 9-May-2020.) $) @@ -151426,14 +151448,14 @@ computer programs (as last() or lastChar()), the terminology used for ( C e. Word X /\ D e. Word X ) /\ ( # ` A ) = ( # ` C ) ) -> ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) ) $= ( wcel wa chash cfv wceq cconcat co cpfx pfxccat1 caddc cop csubstr ccatlen - syl 3eqtr3d cword oveq1 3ad2ant1 simp3 oveq2d 3ad2ant2 eqtrd eqeq12d syl5ib - w3a simpr simpl3 fveq2d simpl1 simpl2 opeq12d oveq12d swrdccat2 jcad oveq12 - ex impbid1 ) AEUAZFBVCFGZCVCFDVCFGZAHIZCHIZJZUJZABKLZCDKLZJZACJZBDJZGVIVLVM - VNVLVJVFMLZVKVFMLZJVIVMVJVKVFMUBVIVOAVPCVDVEVOAJVHEABNUCVIVPVKVGMLZCVIVFVGV - KMVDVEVHUDUEVEVDVQCJVHECDNUFUGUHUIVIVLVNVIVLGZVJVFVFBHIOLZPZQLZVKVGVGDHIOLZ - PZQLZBDVRVJVKVTWCQVIVLUKZVRVFVGVSWBVDVEVHVLULVRVJHIZVKHIZVSWBVRVJVKHWEUMVRV - DWFVSJVDVEVHVLUNZEABRSVRVEWGWBJVDVEVHVLUOZECDRSTUPUQVRVDWABJWHEABURSVRVEWDD - JWIECDURSTVAUSACBDKUTVB $. + syl 3eqtr3d cword w3a wi oveq1 oveq2 sylan9eqr eqeqan12d syl5ib 3impb simpr + simpl3 fveq2d simpl1 simpl2 opeq12d oveq12d swrdccat2 jcad oveq12 impbid1 + ex ) AEUAZFBVBFGZCVBFDVBFGZAHIZCHIZJZUBZABKLZCDKLZJZACJZBDJZGVHVKVLVMVCVDVG + VKVLUCVKVIVEMLZVJVEMLZJVCVDVGGZGVLVIVJVEMUDVCVPVNAVOCEABNVGVDVOVJVFMLCVEVFV + JMUEECDNUFUGUHUIVHVKVMVHVKGZVIVEVEBHIOLZPZQLZVJVFVFDHIOLZPZQLZBDVQVIVJVSWBQ + VHVKUJZVQVEVFVRWAVCVDVGVKUKVQVIHIZVJHIZVRWAVQVIVJHWDULVQVCWEVRJVCVDVGVKUMZE + EABRSVQVDWFWAJVCVDVGVKUNZEECDRSTUOUPVQVCVTBJWGEABUQSVQVDWCDJWHECDUQSTVAURAC + BDKUSUT $. $( An ~ opth -like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) $) @@ -151445,10 +151467,10 @@ computer programs (as last() or lastChar()), the terminology used for simp2l simp2r bitrd syl5ib ccatopth biimpd 3expia com23 3adant3 mpdd oveq12 impbid1 ) AEUAZFZBVCFZGZCVCFZDVCFZGZBHIZDHIZJZKZABLMZCDLMZJZACJBDJGZVMVPAHI ZCHIZJZVQVPVNHIZVOHIZJZVMVTVNVOHUBVMWCVRVKNMZVSVKNMZJVTVMWAWDWBWEVMWAVRVJNM - ZWDVFVIWAWFJVLEABOUCVMVJVKVRNVFVIVLUDUEUFVIVFWBWEJVLECDOUGUHVMVRVSVKVMVRVMV - DVRPFVDVEVIVLUIEAQRSVMVSVMVGVSPFVFVGVHVLUKECQRSVMVKVMVHVKPFVFVGVHVLULEDQRSU - JUMUNVFVIVPVTVQTTVLVFVIGVTVPVQVFVIVTVPVQTVFVIVTKVPVQABCDEUOUPUQURUSUTACBDLV - AVB $. + ZWDVFVIWAWFJVLEEABOUCVMVJVKVRNVFVIVLUDUEUFVIVFWBWEJVLEECDOUGUHVMVRVSVKVMVRV + MVDVRPFVDVEVIVLUIEAQRSVMVSVMVGVSPFVFVGVHVLUKECQRSVMVKVMVHVKPFVFVGVHVLULEDQR + SUJUMUNVFVIVPVTVQTTVLVFVIGVTVPVQVFVIVTVPVQTVFVIVTKVPVQABCDEUOUPUQURUSUTACBD + LVAVB $. $( Concatenation of words is left-cancellative. (Contributed by Mario Carneiro, 2-Oct-2015.) $) @@ -151651,10 +151673,10 @@ computer programs (as last() or lastChar()), the terminology used for /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( ( # ` A ) + ( # ` B ) ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0 ..^ ( N - M ) ) ) $= ( cword wcel wa cc0 cfz chash cfv caddc cconcat cop csubstr cmin cfzo wfn - co ccatcl adantr simprl ccatlen eqcomd oveq2d eleq2d biimpcd adantl syl3anc - wi impcom swrdvalfn ) AEFZGBUNGHZCIDJTGZDIAKLBKLMTZJTZGZHZHABNTZUNGZUPDIVAK - LZJTZGZVACDOPTIDCQTRTSUOVBUTEABUAUBUOUPUSUCUTUOVEUSUOVEUKUPUOUSVEUOURVDDUOU - QVCIJUOVCUQEABUDUEUFUGUHUIULVACDEUMUJ $. + co ccatcl adantr simprl ccatlen oveq2d biimpar adantrl swrdvalfn syl3anc + eleq2d ) AEFZGBUKGHZCIDJTGZDIAKLBKLMTZJTZGZHZHABNTZUKGZUMDIURKLZJTZGZURCDOP + TIDCQTRTSULUSUQEABUAUBULUMUPUCULUPVBUMULVBUPULVAUODULUTUNIJEEABUDUEUJUFUGUR + CDEUHUI $. ${ $d A k $. $d B k $. $d M k $. $d N k $. $d V k $. @@ -151753,52 +151775,48 @@ computer programs (as last() or lastChar()), the terminology used for -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( B substr <. ( M - L ) , ( N - L ) >. ) ) ) $= - ( wcel wa cfz co cfv cmin wceq cc0 wi syl adantr wbr adantl cword cconcat - vk chash caddc cop csubstr cfzo wfn wb oveq1 eleq2d oveq12d anbi12d ax-mp - id cn0 lencl cuz wss elnn0uz biimpi fzss1 sseld anim12d syl5bi swrdccatfn - imp syldan cc w3a cz cle elfz2 3anim123i 3comr sylbi oveq2d fneq2d mpbird - zcn nnncan2 simpr elfzmlbm ad2antrl elfzmlbp nn0zd syl11 impcom swrdvalfn - ex syl3anc cv clt cif simpl elfzelz zaddcl expcom elfzoelz impel sylanbrc - df-3an ccatsymb wn cr zre anim12i elnn0z 0red anim1i ancoms anim2i le2add - syl2an2 recn addid2d breq1d sylibd readdcl lenltd expd com12 mpan9 eqcomi - breq2i notbii syl6ibr com23 3adant2 elfzonn0 iffalsed ad2antrr addsubassd - oveq2 eqeq1d syl5ib fveq2d 3eqtrd swrdfv ccatcl eqeltrid ccatlen fzmmmeqm - 3syl biimpd syl31anc 3jca sylan 3eqtr4d eqfnfvd ) AFUAZHZBUULHZIZDCEJKZHZ - ECCBUDLZUEKZJKZHZIZABUBKZDEUFUGKZBDCMKZECMKZUFUGKZNUUOUVBIZUCOUVFUVEMKZUH - KZUVDUVGUVHUVDUVJUIUVDOEDMKZUHKZUIZUUOUVBDOEJKZHZEOAUDLZUURUEKZJKZHZIZUVM - UUOUVBUVTUVBDUVPEJKZHZEUVPUVQJKZHZIZUUOUVTCUVPNZUVBUWEUJGUWFUUQUWBUVAUWDU - WFUUPUWADCUVPEJUKULUWFUUTUWCEUWFCUVPUUSUVQJUWFUPCUVPUURUEUKUMULZUNUOUUMUW - EUVTPZUUNUUMUVPUQHZUWHFAURZUWIUWBUVOUWDUVSUWIUWAUVNDUWIUVPOUSLZHZUWAUVNUT - UWIUWLUVPVAVBZUVPOEVCQVDUWIUWCUVREUWIUWLUWCUVRUTZUWMUVPOUVQVCZQVDVEQRVFVH - ABDEFVGVIUVHUVJUVLUVDUVHUVIUVKOUHUVBUVIUVKNZUUOUVBEVJHZDVJHZCVJHZVKZUWPUU - QUWTUVAUUQCVLHZEVLHZDVLHZVKZCDVMSZDEVMSZIZIZUWTDCEVNZUXDUWTUXGUXBUXCUXAUW - TUXBUWQUXCUWRUXAUWSEWADWAZCWAZVOVPRVQREDCWBQTVRVSVTUVHUUNUVEOUVFJKHZUVFOU - URJKHZUVGUVJUIUUOUUNUVBUUMUUNWCRZUUQUXLUUOUVADCEWDWEZUVBUUOUXMUVAUUOUXMPZ - UUQUURVLHZUVAUXMUUOUXQUVAUXMECUURWFWKZUUNUXQUUMUUNUURFBURWGZTWHTWIBUVEUVF - FWJWLUVHUCWMZUVJHZIZUXTDUEKZUVCLZUXTUVEUEKZBLZUXTUVDLZUXTUVGLZUYBUYDUYCUV - PWNSZUYCALZUYCUVPMKZBLZWOZUYLUYFUYBUUMUUNUYCVLHZVKZUYDUYMNUYBUUOUYNUYOUVH - UUOUYAUUOUVBWPRUVHUXTVLHZUYNUYAUUQUYPUYNPZUUOUVAUUQUXCUYQDCEWQUYPUXCUYNUX - TDWRWSQWEUXTOUVIWTZXAUUMUUNUYNXCXBABUYCFXDQUYBUYIUYJUYLUVHUXTUQHZUYIXEZUY - AUUQUYSUYTPZUUOUVAUUQUXHVUAUXIUXGUXDVUAUXEUXDVUAPUXFUXDUXEVUAUXAUXCUXEVUA - PUXBUXAUXCIZUYSUXEUYTVUBUYSUXEUYTPVUBUYSIUXEUYCCWNSZXEZUYTVUBCXFHZDXFHZIZ - UYSUXEVUDPZUXAVUEUXCVUFCXGDXGXHUYSUYPOUXTVMSZIVUGVUHPZUXTXIUYPUXTXFHZVUIV - UJUXTXGVUIVUKVUGVUHVUKVUGIZVUIVUHVULVUIUXEVUDVULVUIUXEIZCUYCVMSZVUDVULVUM - OCUEKZUYCVMSZVUNVUGOXFHZVUEIZVUKVUKVUFIZVUMVUPPVUFVUEVURVUFVUQVUEVUFXJXKX - LVUGVUFVUKVUEVUFWCXMZOCUXTDXNXOVULVUOCUYCVMVUEVUOCNVUKVUFVUECCXPXQWEXRXSV - ULCUYCVUGVUEVUKVUEVUFWPTVULVUSUYCXFHVUTUXTDXTQYAXSYBYCYBYDVQYDUYIVUCUVPCU - YCWNCUVPGYEYFYGYHWKYIYJYCRWIVQWEUXTUVIYKXAYLUYBUYKUYEBUVHUYPUYKUYENZUYAUU - QUYPVVAPZUUOUVAUUQUXHVVBUXIUXDVVBUXGUXAUXCVVBUXBVUBUYPVVAUWFVUBUYPIZVVAPG - VVCUYCCMKZUYENUWFVVAVVCUXTDCUYPUXTVJHVUBUXTWATVUBUWRUYPUXCUWRUXAUXJTRUXAU - WSUXCUYPUXKYMYNUWFVVDUYKUYECUVPUYCMYOYPYQUOWKYJRVQWEUYRXAYRYSUYBUVCUULHZU - VOEOUVCUDLZJKZHZUXTUVLHZUYGUYDNUUOVVEUVBUYAFABUUAYMUVHUVOUYAUVBUUOUVOUUQU - UOUVOPUVAUUOUUQUVOUUMUUQUVOPUUNUUMUUPUVNDUUMUWICUWKHUUPUVNUTUWJUWICUVPUWK - GUWMUUBCOEVCUUEVDRYCRWIRUVHVVHUYAUVBUUOVVHUVAUUOVVHPZUUQUVAUWDVVJUWFUVAUW - DUJGUWGUOUWDUUOVVHUWDUUOIVVHUVSUUOUWDUVSUUOUWCUVREUUOUWLUWNUUMUWLUUNUUMUW - IUWLUWJUWMQRUWOQVDWIUUOVVHUVSUJUWDUUOVVGUVREUUOVVFUVQOJFABUUCVRULTVTWKVQT - WIRUVHUYAVVIUUQUYAVVIPUUOUVAUUQUYAVVIUUQUVJUVLUXTUUQUVIUVKOUHCDEUUDVRULUU - FWEVHFUVCDEUXTYTUUGUVHUUNUXLUXMVKUYAUYHUYFNUVHUUNUXLUXMUXNUXOUVBUUOUXMUVA - UXPUUQUUOUVAUXMUUNUVAUXMPZUUMUUNUXQVVKUXSUXRQTYCTWIUUHFBUVEUVFUXTYTUUIUUJ - UUKWK $. + ( wcel wa cfz co cfv caddc cmin wceq cc0 wi sylbi ad2antrl wbr vk cconcat + cword chash cop csubstr cfzo wfn wb oveq1 eleq2d id oveq12d anbi12d ax-mp + cn0 lencl cuz elnn0uz fzss1 sseld anim12d adantr syl5bi swrdccatfn syldan + wss syl imp fzmmmeqm oveq2d fneq2d mpbird simplr elfzmlbm cz nn0zd adantl + elfzmlbp sylan adantrl swrdvalfn syl3anc cv clt cif simpll elfzelz zaddcl + w3a expcom elfzoelz impel df-3an sylanbrc ccatsymb wn cle elfz2 cr elnn0z + zre anim12i jctl le2add syl2anc recn addid2d breq1d sylibd simprl readdcl + 0re lenltd expd com12 mpan9 breq2i notbii syl6ib ex com23 3adant2 adantrr + elfzonn0 iffalsed cc zcn ad2antlr ad2antrr addsubassd oveq2 eqeq1d syl5ib + fveq2d 3eqtrd 3syl impcom 3jca swrdfv ccatcl biimpi sselda ccatlen biimpa + eqeltrid ad2ant2r syl2an2r ad2ant2l 3eqtr4d eqfnfvd ) AFUCZHZBUULHZIZDCEJ + KZHZECCBUDLZMKZJKZHZIZABUBKZDEUEUFKZBDCNKZECNKZUEUFKZOUUOUVBIZUAPUVFUVENK + ZUGKZUVDUVGUVHUVDUVJUHZUVDPEDNKZUGKZUHZUUOUVBDPEJKZHZEPAUDLZUURMKZJKZHZIZ + UVNUUOUVBUWAUVBDUVQEJKZHZEUVQUVRJKZHZIZUUOUWACUVQOZUVBUWFUIGUWGUUQUWCUVAU + WEUWGUUPUWBDCUVQEJUJUKUWGUUTUWDEUWGCUVQUUSUVRJUWGULCUVQUURMUJUMUKZUNUOUUM + UWFUWAQZUUNUUMUVQUPHZUWIFAUQZUWJUWCUVPUWEUVTUWJUWBUVODUWJUVQPURLZHZUWBUVO + VGUVQUSZUVQPEUTRVAUWJUWDUVSEUWJUWMUWDUVSVGZUWNUVQPUVRUTZRVAVBVHVCVDVIABDE + FVEVFUUQUVKUVNUIUUOUVAUUQUVJUVMUVDUUQUVIUVLPUGCDEVJVKZVLSVMUVHUUNUVEPUVFJ + KHZUVFPUURJKHZUVGUVJUHUUMUUNUVBVNZUUQUWRUUOUVADCEVOSZUUOUVAUWSUUQUUOUURVP + HZUVAUWSUUNUXBUUMUUNUURFBUQVQZVRECUURVSZVTWABUVEUVFFWBWCUVHUAWDZUVJHZIZUX + EDMKZUVCLZUXEUVEMKZBLZUXEUVDLZUXEUVGLZUXGUXIUXHUVQWETZUXHALZUXHUVQNKZBLZW + FZUXQUXKUXGUUMUUNUXHVPHZWJZUXIUXROUXGUUOUXSUXTUUOUVBUXFWGUVHUXEVPHZUXSUXF + UUQUYAUXSQZUUOUVAUUQDVPHZUYBDCEWHUYAUYCUXSUXEDWIWKVHSUXEPUVIWLZWMUUMUUNUX + SWNWOABUXHFWPVHUXGUXNUXOUXQUVHUXEUPHZUXNWQZUXFUUQUYEUYFQZUUOUVAUUQCVPHZEV + PHZUYCWJZCDWRTZDEWRTZIZIZUYGDCEWSZUYJUYKUYGUYLUYJUYKUYGUYHUYCUYKUYGQUYIUY + HUYCIZUYEUYKUYFUYPUYEUYKUYFQUYPUYEIUYKUXHCWETZWQZUYFUYPCWTHZDWTHZIZUYEUYK + UYRQZUYHUYSUYCUYTCXBDXBXCUYEUYAPUXEWRTZIVUAVUBQZUXEXAUYAUXEWTHZVUCVUDUXEX + BVUCVUEVUAVUBVUEVUAIZVUCVUBVUFVUCUYKUYRVUFVUCUYKIZCUXHWRTZUYRVUFVUGPCMKZU + XHWRTZVUHVUFPWTHZUYSIZVUEUYTIZVUGVUJQUYSVULVUEUYTUYSVUKXMXDSVUEUYTVUMUYSV + UMULWAPCUXEDXEXFVUFVUICUXHWRUYSVUICOVUEUYTUYSCCXGXHSXIXJVUFCUXHVUEUYSUYTX + KVUEUYTUXHWTHUYSUXEDXLWAXNXJXOXPXOXQRXQUYQUXNCUVQUXHWEGXRXSXTYAYBYCVIYDRS + UXEUVIYEWMYFUXGUXPUXJBUVHUYAUXPUXJOZUXFUUQUYAVUNQZUUOUVAUUQUYNVUOUYOUYJVU + OUYMUYHUYCVUOUYIUYPUYAVUNUWGUYPUYAIZVUNQGVUPUXHCNKZUXJOUWGVUNVUPUXEDCUYAU + XEYGHUYPUXEYHVRUYCDYGHUYHUYADYHYIUYHCYGHUYCUYACYHYJYKUWGVUQUXPUXJCUVQUXHN + YLYMYNUOYAYCVCRSUYDWMYOYPUVHUVCUULHZUVPEPUVCUDLZJKZHZWJUXFUXEUVMHZUXLUXIO + UVHVURUVPVVAUUOVURUVBFABUUAVCUUMUUQUVPUUNUVAUUMUUPUVODUUMUWJCUWLHUUPUVOVG + UWKUWJCUVQUWLGUWJUWMUWNUUBZUUFCPEUTYQUUCUUGUUOUVAVVAUUQUVAUUOVVAUVAUWEUUO + VVAQUWGUVAUWEUIGUWHUOUWEUUOVVAUWEUUOIVVAUVTUUOUWEUVTUUOUWDUVSEUUMUWOUUNUU + MUWJUWMUWOUWKVVCUWPYQVCVAYRUUOVVAUVTUIUWEUUOVUTUVSEUUOVUSUVRPJFFABUUDVKUK + VRVMYARYRWAYSUVHUXFVVBUUQUXFVVBUIUUOUVAUUQUVJUVMUXEUWQUKSUUEFUVCDEUXEYTUU + HUVHUUNUWRUWSWJUXFUXMUXKOUVHUUNUWRUWSUWTUXAUUNUVAUWSUUMUUQUUNUXBUVAUWSUXC + UXDVTUUIYSFBUVEUVFUXEYTVTUUJUUKYA $. $( Lemma for ~ pfxccatin12lem2 and ~ pfxccatin12lem3 . (Contributed by AV, 30-Mar-2018.) (Revised by AV, 27-May-2018.) $) @@ -151806,12 +151824,12 @@ computer programs (as last() or lastChar()), the terminology used for /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) ) -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N ) /\ N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) ) $= - ( cword wcel wa cc0 cfz co chash cfv caddc cconcat ccatcl adantr adantl - elfz0fzfz0 cuz wss elfzuz2 fzss1 sseld impr wb ccatlen wceq eqcomi oveq1d - syl a1i eqtrd oveq2d eleq2d mpbird 3jca ) AFHZIBUTIJZDKCLMIZECCBNOZPMZLMZ - IZJZJZABQMZUTIZDKELMIZEKVINOZLMZIZVAVJVGFABRSVGVKVACDEVDUATVHVNEKVDLMZIZV - AVBVFVPVAVBJZVEVOEVQCKUBOIZVEVOUCVBVRVADKCUDTCKVDUEUMUFUGVAVNVPUHVGVAVMVO - EVAVLVDKLVAVLANOZVCPMVDFABUIVAVSCVCPVSCUJVACVSGUKUNULUOUPUQSURUS $. + ( cword wcel wa cc0 cfz chash cfv caddc cconcat ccatcl adantr elfz0fzfz0 + co adantl cuz wss elfzuz2 syl sselda ccatlen oveq1i syl6eqr oveq2d eleq2d + fzss1 syl5ibr imp 3jca ) AFHZIBUPIJZDKCLTIZECCBMNZOTZLTZIJZJABPTZUPIZDKEL + TIZEKVCMNZLTZIZUQVDVBFABQRVBVEUQCDEUTSUAUQVBVHVBVHUQEKUTLTZIURVAVIEURCKUB + NIVAVIUCDKCUDCKUTULUEUFUQVGVIEUQVFUTKLUQVFAMNZUSOTUTFFABUGCVJUSOGUHUIUJUK + UMUNUO $. $( Lemma 2 for ~ pfxccatin12 . (Contributed by AV, 30-Mar-2018.) (Revised by AV, 9-May-2020.) $) @@ -151901,42 +151919,34 @@ computer programs (as last() or lastChar()), the terminology used for -> ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) -> ( ( A ++ B ) substr <. M , N >. ) = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) ) $= - ( wcel wa cc0 cfz co cfv caddc wceq cfzo adantl syl w3a ad2antrl vk cword - chash cconcat cop csubstr cmin cpfx wfn ccatcl elfz0fzfz0 cuz wss elfzuz2 - adantr fzss1 ccatlen eqcomi oveq1i syl6eq oveq2d sseqtr4d sseld swrdvalfn - impr syl3anc swrdcl pfxcl anim12i ccatvalfn simpll simprl elnn0uz eluzfz2 - cn0 lencl sylbi eqeltrid ad2antrr swrdlen simplr cz nn0zd elfzmlbp pfxlen - simpr syl2anc oveq12d cc cle wbr wi elfz2nn0 nn0cn zcn 3jca elfzelz syl11 - ex 3adant3 imp npncan3 eqtr2d fneq2d mpbird anim2i ancomd pfxccatin12lem3 - sylc simpl nn0fz0 sylib eleqtrrd df-3an sylanbrc ccatval1 pfxccatin12lem2 - eqtr4d elfzuz eluzelz ancoms syl5com impcom pfxccatin12lem4 jca pm2.61ian - cv wn ccatval2 eqfnfvd ) AFUBZHZBYKHZIZDJCKLHZECCBUCMZNLZKLZHZIZABUDLZDEU - EUFLZADCUEUFLZBECUGLZUHLZUDLZOYNYTIZUAJEDUGLZPLZUUBUUFUUGUUAYKHZDJEKLHZEJ - UUAUCMZKLZHZUUBUUIUIYNUUJYTFABUJUOYTUUKYNCDEYQUKQYNYOYSUUNYNYOIZYRUUMEUUO - YRJYQKLZUUMUUOCJULMZHZYRUUPUMYOUURYNDJCUNQCJYQUPRUUOUULYQJKYNUULYQOYOYNUU - 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F , T , R >. ) ) = ( ( # ` S ) + ( ( # ` R ) - ( T - F ) ) ) ) $= ( co chash cfv caddc cmin wcel cc0 wceq syl syl2anc cotp csplice cpfx cop - cconcat csubstr cword cfz splval syl13anc fveq2d pfxcl ccatcl ccatlen cn0 - swrdcl nn0cnd cz elfzelz zcnd addcld elfzel2 addsub12d cuz elfzuz elfzuz3 - lencl uztrn elfzuzb sylanbrc pfxlen oveq1d addcomd 3eqtrd elfzuz2 eluzfz2 - swrdlen syl3anc oveq12d subsub3d oveq2d 3eqtr4d ) ADFECUAUBKZLMDFUCKZCUEK - ZDEDLMZUDUFKZUEKZLMZWELMZWGLMZNKZWFCLMZEFOKOKZNKZAWCWHLADBUGZPZFQEUHKZPZE - QWFUHKZPZCWPPZWCWHRGHIJCDEFWPWRWTWPUIUJUKAWEWPPZWGWPPZWIWLRAWDWPPZXBXCAWQ - XEGBDFULSZJBWDCUMTAWQXDGBDEWFUPSBWEWGUNTAWMFNKZWFEOKZNKWFXGEOKZNKWLWOAXGW - FEAWMFAWMAXBWMUOPJBCVGSUQZAFAWSFURPHFQEUSSUTZVAAWFAXAWFURPIEQWFVBSUTAEAXA - EURPIEQWFUSSUTZVCAWJXGWKXHNAWJWDLMZWMNKZFWMNKXGAXEXBWJXNRXFJBWDCUNTAXMFWM - NAWQFWTPZXMFRGAFQVDMZPZWFFVDMZPZXOAWSXQHFQEVESAWFEVDMPZEXRPZXSAXAXTIEQWFV - FSAWSYAHFQEVFSEWFFVHTFQWFVIVJBDFVKTVLAFWMXKXJVMVNAWQXAWFWTPZWKXHRGIAWFXPP - ZYBAXAYCIEQWFVOSQWFVPSBDEWFVQVRVSAWNXIWFNAWMEFXJXLXKVTWAWBVN $. + cconcat csubstr cword cfz splval syl13anc fveq2d ccatcl swrdcl ccatlen cc + pfxcl lencl nn0cnd elfzelz zcnd addcld elfzel2 addsub12d cuz elfzuz uztrn + elfzuz3 elfzuzb pfxlen oveq1d addcomd 3eqtrd elfzuz2 eluzfz2 3syl swrdlen + sylanbrc syl3anc oveq12d subsub3d oveq2d 3eqtr4d ) ADFECUAUBKZLMDFUCKZCUE + KZDEDLMZUDUFKZUEKZLMZWELMZWGLMZNKZWFCLMZEFOKOKZNKZAWCWHLADBUGZPZFQEUHKZPZ + EQWFUHKZPZCWPPZWCWHRGHIJCDEFWPWRWTWPUIUJUKAWEWPPZWGWPPZWIWLRAWDWPPZXBXCAW + QXEGBDFUPSZJBWDCULTAWQXDGBDEWFUMSBBWEWGUNTAWMFNKZWFEOKZNKWFXGEOKZNKWLWOAX + GWFEAWMFAXBWMUOPJXBWMBCUQURSZAWSFUOPHWSFFQEUSUTSZVAAXAWFUOPIXAWFEQWFVBUTS + AXAEUOPIXAEEQWFUSUTSZVCAWJXGWKXHNAWJWDLMZWMNKZFWMNKXGAXEXBWJXNRXFJBBWDCUN + TAXMFWMNAWQFWTPZXMFRGAFQVDMZPZWFFVDMZPZXOAWSXQHFQEVESAWFEVDMPZEXRPZXSAXAX + TIEQWFVGSAWSYAHFQEVGSEWFFVFTFQWFVHVQBDFVITVJAFWMXKXJVKVLAWQXAWFWTPZWKXHRG + IAXAWFXPPYBIEQWFVMQWFVNVOBDEWFVPVRVSAWNXIWFNAWMEFXJXLXKVTWAWBVL $. ${ splfv1.x $e |- ( ph -> X e. ( 0 ..^ F ) ) $. @@ -152393,16 +152403,16 @@ Splicing words (substring replacement) ( S ` X ) ) $= ( co cfv chash wcel cc0 wceq cfzo syl2anc cotp csplice cpfx cconcat cop csubstr cword cfz splval syl13anc fveq1d pfxcl syl ccatcl caddc cuz wss - swrdcl cn0 cz elfzelz uzid 3syl cfn wrdfin hashcl uzaddcl fzoss2 sseldd - ccatlen wa fzass4 bicomi simplbi pfxlen oveq1d oveq2d eleqtrrd ccatval1 - eqtrd syl3anc pfxfv 3eqtrd ) AGDFECUAUBMZNGDFUCMZCUDMZDEDONZUEUFMZUDMZN - ZGWFNZGDNZAGWDWIADBUGZPZFQEUHMZPZEQWGUHMZPZCWMPZWDWIRHIJKCDEFWMWOWQWMUI - UJUKAWFWMPZWHWMPZGQWFONZSMZPWJWKRAWEWMPZWSWTAWNXDHBDFULUMZKBWECUNTAWNXA - HBDEWGURUMAGQFCONZUOMZSMZXCAQFSMZXHGAXGFUPNZPZXIXHUQAFXJPZXFUSPZXKAWPFU - TPXLIFQEVAFVBVCAWSCVDPXMKBCVECVFVCXFFFVGTFQXGVHUMLVIAXBXGQSAXBWEONZXFUO - MZXGAXDWSXBXORXEKBWECVJTAXNFXFUOAWNFWQPZXNFRHAWPWRXPIJWPWRVKZXPEFWGUHMP - ZXPXRVKXQQFEWGVLVMVNTZBDFVOTZVPVTVQVRBWFWHGVSWAAWKGWENZWLAXDWSGQXNSMZPW - KYARXEKAGXIYBLAXNFQSXTVQVRBWECGVSWAAWNXPGXIPYAWLRHXSLGFBDWBWAVTWC $. + swrdcl cn0 elfzelz uzid 3syl lencl uzaddcl fzoss2 sseldd ccatlen fzass4 + cz wa biimpri simpld pfxlen oveq1d eqtrd oveq2d eleqtrrd ccatval1 pfxfv + syl3anc 3eqtrd ) AGDFECUAUBMZNGDFUCMZCUDMZDEDONZUEUFMZUDMZNZGWDNZGDNZAG + WBWGADBUGZPZFQEUHMZPZEQWEUHMZPZCWKPZWBWGRHIJKCDEFWKWMWOWKUIUJUKAWDWKPZW + FWKPZGQWDONZSMZPWHWIRAWCWKPZWQWRAWLXBHBDFULUMZKBWCCUNTAWLWSHBDEWEURUMAG + QFCONZUOMZSMZXAAQFSMZXFGAXEFUPNZPZXGXFUQAFXHPZXDUSPZXIAWNFVIPXJIFQEUTFV + AVBAWQXKKBCVCUMXDFFVDTFQXEVEUMLVFAWTXEQSAWTWCONZXDUOMZXEAXBWQWTXMRXCKBB + WCCVGTAXLFXDUOAWLFWOPZXLFRHAWNWPXNIJWNWPVJZXNEFWEUHMPZXNXPVJXOQFEWEVHVK + VLTZBDFVMTZVNVOVPVQBWDWFGVRVTAWIGWCNZWJAXBWQGQXLSMZPWIXSRXCKAGXGXTLAXLF + QSXRVPVQBWCCGVRVTAWLXNGXGPXSWJRHXQLGFBDVSVTVOWA $. $} ${ @@ -152413,20 +152423,20 @@ Splicing words (substring replacement) splfv2a $p |- ( ph -> ( ( S splice <. F , T , R >. ) ` ( F + X ) ) = ( R ` X ) ) $= ( caddc co cfv wcel cc0 wceq cn0 syl cotp csplice chash cconcat csubstr - cpfx cop cword cfz splval syl13anc elfznn0 nn0cnd cfzo cz elfzoelz zcnd - addcomd cuz nn0uz syl6eleq elfzuz3 uztrn syl2anc sylanbrc pfxlen oveq2d - elfzuzb eqtr4d fveq12d pfxcl ccatcl swrdcl 0nn0 nn0addcl sylancr fzoss1 - wss eleq2s ccatlen oveq1d wrdfin hashcl 3eqtrd sseqtr4d fzoaddel sseldd - cfn nn0zd eqeltrd ccatval1 syl3anc ccatval3 ) AFGMNZDFECUAUBNZOGDFUFNZU - COZMNZWPCUDNZDEDUCOZUGUENZUDNZOZWRWSOZGCOZAWNWRWOXBADBUHZPZFQEUINZPZEQW - TUINZPZCXFPZWOXBRHIJKCDEFXFXHXJXFUJUKAWNGFMNZWRAFGAFAXIFSPZIFEULTZUMZAG - AGQCUCOZUNNPZGUOPLGQXQUPTUQURAWQFGMAXGFXJPZWQFRHAFQUSOZPWTFUSOZPZXSAFSX - TXOUTVAAWTEUSOPZEYAPZYBAXKYCJEQWTVBTAXIYDIFQEVBTEWTFVCVDFQWTVHVEBDFVFVD - ZVGZVIVJAWSXFPZXAXFPZWRQWSUCOZUNNZPXCXDRAWPXFPZXLYGAXGYKHBDFVKTZKBWPCVL - VDAXGYHHBDEWTVMTAWRXMYJYFAQFMNZXQFMNZUNNZYJXMAYOQYNUNNZYJAYMSPZYOYPVRZA - QSPXNYQVNXOQFVOVPYRYMXTSYMQYNVQUTVSTAYIYNQUNAYIWQXQMNZFXQMNYNAYKXLYIYSR - YLKBWPCVTVDAWQFXQMYEWAAFXQXPAXQACWHPZXQSPAXLYTKBCWBTCWCTUMURWDVGWEAXRFU - OPXMYOPLAFXOWIGQXQFWFVDWGWJBWSXAWRWKWLAYKXLXRXDXERYLKLBWPCGWMWLWD $. + cpfx cop cword cfz splval syl13anc elfznn0 nn0cnd cfzo elfzonn0 addcomd + cuz nn0uz syl6eleq elfzuz3 uztrn syl2anc elfzuzb sylanbrc pfxlen oveq2d + eqtr4d fveq12d pfxcl ccatcl swrdcl 0nn0 nn0addcl sylancr fzoss1 ccatlen + wss eleq2s oveq1d lencl 3eqtrd sseqtr4d nn0zd fzoaddel eqeltrd ccatval1 + cz sseldd syl3anc ccatval3 ) AFGMNZDFECUAUBNZOGDFUFNZUCOZMNZWMCUDNZDEDU + COZUGUENZUDNZOZWOWPOZGCOZAWKWOWLWSADBUHZPZFQEUINZPZEQWQUINZPZCXCPZWLWSR + HIJKCDEFXCXEXGXCUJUKAWKGFMNZWOAFGAFAXFFSPZIFEULTZUMZAGAGQCUCOZUNNPZGSPL + GXNUOTUMUPAWNFGMAXDFXGPZWNFRHAFQUQOZPWQFUQOZPZXPAFSXQXLURUSAWQEUQOPZEXR + PZXSAXHXTJEQWQUTTAXFYAIFQEUTTEWQFVAVBFQWQVCVDBDFVEVBZVFZVGVHAWPXCPZWRXC + PZWOQWPUCOZUNNZPWTXARAWMXCPZXIYDAXDYHHBDFVITZKBWMCVJVBAXDYEHBDEWQVKTAWO + XJYGYCAQFMNZXNFMNZUNNZYGXJAYLQYKUNNZYGAYJSPZYLYMVQZAQSPXKYNVLXLQFVMVNYO + YJXQSYJQYKVOURVRTAYFYKQUNAYFWNXNMNZFXNMNYKAYHXIYFYPRYIKBBWMCVPVBAWNFXNM + YBVSAFXNXMAXNAXIXNSPKBCVTTUMUPWAVFWBAXOFWGPXJYLPLAFXLWCGQXNFWDVBWHWEBWP + WRWOWFWIAYHXIXOXAXBRYIKLBWMCGWJWIWA $. $} $} @@ -152445,24 +152455,24 @@ Splicing words (substring replacement) ( S splice <. F , T , R >. ) = ( ( A ++ R ) ++ C ) ) $= ( co wcel cn0 wceq cotp csplice cpfx cconcat chash cfv cop csubstr ccatcl cword syl2anc eqeltrd lencl syl caddc nn0addcld splval syl13anc cc0 nn0uz - wa cfz cuz syl6eleq cz nn0zd uzid uzaddcl elfzuzb sylanbrc ccatlen fveq2d - oveq1d 3eqtr4d ccatpfx syl3anc eluzfz2 pfxid 3eqtrd wb pfxcl swrdcl eqtrd - pfxlen ccatopth syl221anc mpbid simpld uztrn simprd oveq12d ) AFHGEUAUBQZ - FHUCQZEUDQZFGFUEUFZUGUHQZUDQZBEUDQZDUDQAFIUJZRZHSRGSREWSRWLWQTAFBCUDQZDUD - QZWSNAXAWSRZDWSRZXBWSRABWSRZCWSRZXCJKIBCUIUKZLIXADUIUKULZAHBUEUFZSOAXEXIS - RJIBUMUNULZAGHCUEUFZUOQZSPAHXKXJAXFXKSRZKICUMUNZUPULZMEFGHWSSSWSUQURAWNWR - WPDUDAWMBEUDAWMBTZFHGUGUHQZCTZAWMXQUDQZXATZXPXRVAZAXSFGUCQZXAAWTHUSGVBQRZ - GUSWOVBQZRZXSYBTXHAHUSVCUFZRZGHVCUFZRZYCAHSYFXJUTVDZAGXLYHPAHYHRZXMXLYHRA - HVERYKAHXJVFHVGUNXNXKHHVHUKULZHUSGVIVJAGYFRWOGVCUFZRZYEAGSYFXOUTVDAWOGDUE - UFZUOQZYMAXBUEUFZXAUEUFZYOUOQZWOYPAXCXDYQYSTXGLIXADVKUKAFXBUENVLAGYRYOUOA - XLXIXKUOQZGYRAHXIXKUOOVMPAXEXFYRYTTJKIBCVKUKVNZVMVNAGYMRZYOSRZYPYMRAGVERU - UBAGXOVFGVGUNAXDUUCLIDUMUNYOGGVHUKULZGUSWOVIVJZIFHGVOVPAYBXATZWPDTZAYBWPU - DQZXBTZUUFUUGVAZAUUHFWOUCQZFXBAWTYEWOYDRZUUHUUKTXHUUEAWOYFRUULAWOSYFAWTWO - SRXHIFUMUNUTVDUSWOVQUNIFGWOVOVPAWTUUKFTXHIFVRUNNVSAYBWSRZWPWSRZXCXDYBUEUF - ZYRTUUIUUJVTAWTUUMXHIFGWAUNAWTUUNXHIFGWOWBUNXGLAUUOGYRAWTYEUUOGTXHUUEIFGW - DUKUUAWCYBWPXADIWEWFWGZWHWCAWMWSRZXQWSRZXEXFWMUEUFZXITXTYAVTAWTUUQXHIFHWA - UNAWTUURXHIFHGWBUNJKAUUSHXIAWTHYDRZUUSHTXHAYGWOYHRZUUTYJAYNYIUVAUUDYLGWOH - WIUKHUSWOVIVJIFHWDUKOWCWMXQBCIWEWFWGWHVMAUUFUUGUUPWJWKWC $. + wa cfz cuz syl6eleq nn0zd uzaddcl elfzuzb sylanbrc ccatlen fveq2d 3eqtr4d + uzidd oveq1d ccatpfx syl3anc pfxid 3eqtrd wb pfxcl swrdcl pfxlen ccatopth + eluzfz2 eqtrd syl221anc mpbid simpld uztrn simprd oveq12d ) AFHGEUAUBQZFH + UCQZEUDQZFGFUEUFZUGUHQZUDQZBEUDQZDUDQAFIUJZRZHSRGSREWRRWKWPTAFBCUDQZDUDQZ + WRNAWTWRRZDWRRZXAWRRABWRRZCWRRZXBJKIBCUIUKZLIWTDUIUKULZAHBUEUFZSOAXDXHSRJ + IBUMUNULZAGHCUEUFZUOQZSPAHXJXIAXEXJSRZKICUMUNZUPULZMEFGHWRSSWRUQURAWMWQWO + DUDAWLBEUDAWLBTZFHGUGUHQZCTZAWLXPUDQZWTTZXOXQVAZAXRFGUCQZWTAWSHUSGVBQRZGU + SWNVBQZRZXRYATXGAHUSVCUFZRZGHVCUFZRZYBAHSYEXIUTVDZAGXKYGPAHYGRXLXKYGRAHAH + XIVEVLXMXJHHVFUKULZHUSGVGVHAGYERWNGVCUFZRZYDAGSYEXNUTVDAWNGDUEUFZUOQZYKAX + AUEUFZWTUEUFZYMUOQZWNYNAXBXCYOYQTXFLIIWTDVIUKAFXAUENVJAGYPYMUOAXKXHXJUOQZ + GYPAHXHXJUOOVMPAXDXEYPYRTJKIIBCVIUKVKZVMVKAGYKRYMSRZYNYKRAGAGXNVEVLAXCYTL + IDUMUNYMGGVFUKULZGUSWNVGVHZIFHGVNVOAYAWTTZWODTZAYAWOUDQZXATZUUCUUDVAZAUUE + FWNUCQZFXAAWSYDWNYCRZUUEUUHTXGUUBAWNYERUUIAWNSYEAWSWNSRXGIFUMUNUTVDUSWNWC + UNIFGWNVNVOAWSUUHFTXGIFVPUNNVQAYAWRRZWOWRRZXBXCYAUEUFZYPTUUFUUGVRAWSUUJXG + IFGVSUNAWSUUKXGIFGWNVTUNXFLAUULGYPAWSYDUULGTXGUUBIFGWAUKYSWDYAWOWTDIWBWEW + FZWGWDAWLWRRZXPWRRZXDXEWLUEUFZXHTXSXTVRAWSUUNXGIFHVSUNAWSUUOXGIFHGVTUNJKA + UUPHXHAWSHYCRZUUPHTXGAYFWNYGRZUUQYIAYLYHUURUUAYJGWNHWHUKHUSWNVGVHIFHWAUKO + WDWLXPBCIWBWEWFWGVMAUUCUUDUUMWIWJWD $. $} @@ -152553,49 +152563,48 @@ Splicing words (substring replacement) revccat $p |- ( ( S e. Word A /\ T e. Word A ) -> ( reverse ` ( S ++ T ) ) = ( ( reverse ` T ) ++ ( reverse ` S ) ) ) $= - ( wcel cc0 chash cfv caddc co cfzo wceq syl cc eqtrd oveq2d syl2anr cz c1 - cmin adantr vx cword wa cconcat creverse wfn ccatcl revcl wrdf ffn revlen - wf 4syl ccatlen lencl nn0cnd addcom syl2an mpbid 3syl oveqan12rd cv wo id - fneq2d nn0zd adantl fzospliti simpll simplr fzoval eleq2d biimpa eleqtrrd - cfz fznn0sub2 ccatval3 syl3anc oveq1d 1cnd addsubd peano2zm zcnd ad2antlr - ad2antrr elfzoelz fveq2d adantll 3eqtr4d cuz wss cn0 uzid uzaddcl syl2anc - revfv eqeltrd fzoss2 sselda syl2an2r biimpar eqtr4d zaddcl fzrev2i subidd - ccatval1 subsub3d addcl sub32d pncan2 oveq12d 3eltr4d fzoss1 nn0uz eleq2s - fzosubel3 sseqtr4d ccatval2 jaodan syldan eqfnfvd ) BAUBZDZCYBDZUCZUAECFG - ZBFGZHIZJIZBCUDIZUEGZCUEGZBUEGZUDIZYEYKEYKFGZJIZUFZYKYIUFYEYJYBDZYKYBDYPA - YKULYQABCUGZAYJUHAYKUIYPAYKUJUMYEYPYIYKYEYOYHEJYEYOYJFGZYHYEYRYOYTKYSAYJU - KLYEYTYGYFHIZYHABCUNZYCYGMDZYFMDZUUAYHKYDYCYGABUOZUPZYDYFACUOZUPZYGYFUQUR - NZNOVEUSYEYNEYNFGZJIZUFZYNYIUFYEYNYBDZUUKAYNULUULYDYLYBDZYMYBDZUUMYCACUHZ - ABUHZAYLYMUGPAYNUIUUKAYNUJUTYEUUKYIYNYEUUJYHEJYEUUJYLFGZYMFGZHIZYHYDUUNUU - OUUJUUTKYCUUPUUQAYLYMUNPYDYCUURYFUUSYGHACUKZABUKVAZNOVEUSYEUAVBZYIDZUVCEY - FJIZDZUVCYFYHJIZDZVCZUVCYKGZUVCYNGZKZUVDUVDYFQDZUVIYEUVDVDYDUVMYCYDYFUUGV - FZVGZUVCEYHYFVHPYEUVFUVLUVHYEUVFUCZYTRSIZUVCSIZYJGZUVCYLGZUVJUVKUVPYFRSIZ - UVCSIZYGHIZYJGZUWBCGZUVSUVTUVPYCYDUWBUVEDUWDUWEKYCYDUVFVIYCYDUVFVJUVPUWBE - UWAVOIZUVEUVPUVCUWFDZUWBUWFDYEUVFUWGYEUVEUWFUVCYEUVMUVEUWFKZUVOEYFVKLZVLV - MUVCUWAVPLYEUWHUVFUWITVNABCUWBVQVRUVPUVRUWCYJUVPUVRUWAYGHIZUVCSIZUWCYEUVR - UWKKUVFYEUVQUWJUVCSYEUVQYHRSIZUWJYEYTYHRSUUIVSZYEYFYGRYDUUDYCUUHVGZYCUUCY - DUUFTZYEVTZWANVSTUVPUWAYGUVCYDUWAMDYCUVFYDUWAYDUVMUWAQDUVNYFWBLWCWDYCUUCY - DUVFUUFWEUVFUVCMDZYEUVFUVCUVCEYFWFWCVGWANWGYDUVFUVTUWEKYCACUVCWPWHWIYEYRU - VFUVCEYTJIZDZUVJUVSKZYSYEUVEUWRUVCYEYTYFWJGZDUVEUWRWKYEYTYHUXAUUIYEYFUXAD - ZYGWLDZYHUXADYEUVMUXBUVOYFWMLYCUXCYDUUETYGYFYFWNWOWQYFEYTWRLWSAYJUVCWPZWT - UVPUUNUUOUVCEUURJIZDZUVKUVTKYDUUNYCUVFUUPWDYCUUOYDUVFUUQWEYEUXFUVFYEUXEUV - EUVCYEUURYFEJYDUURYFKZYCUVAVGZOVLXAAYLYMUVCXFVRWIYEUVHUCZUVSUVCUURSIZYMGZ - UVJUVKUXIUVRBGZYGRSIZUXJSIZBGZUVSUXKUXIUVRUXNBUXIUVRUXMUVCYFSIZSIZUXNUXIU - VRUXMYFHIZUVCSIZUXQYEUVRUXSKUVHYEUVQUXRUVCSYEUVQUUARSIUXRYEYTUUARSUUBVSYE - YGYFRUWOUWNUWPWANVSTUXIUXMUVCYFYCUXMMDYDUVHYCUXMYCYGQDZUXMQDYCYGUUEVFZYGW - BLWCWEUVHUWQYEUVHUVCUVCYFYHWFWCVGYDUUDYCUVHUUHWDXGXBYEUXNUXQKUVHYEUXJUXPU - XMSYEUURYFUVCSUXHOOTXBWGUXIYCYDUVREYGJIZDUVSUXLKYCYDUVHVIZYCYDUVHVJUXIUWL - UVCSIZUWLUWLSIZUWLYFSIZVOIZUVRUYBYEUWLQDZUVHUVCYFUWLVOIZDZUYDUYGDYEYHQDZU - YHYDUVMUXTUYKYCUVNUYAYFYGXCPZYHWBLZYEUVHUYJYEUVGUYIUVCYEUYKUVGUYIKUYLYFYH - VKLVLVMUWLUVCYFUWLXDWTYEUVRUYDKUVHYEUVQUWLUVCSUWMVSTYEUYBUYGKUVHYEUYBEUXM - VOIZUYGYEUXTUYBUYNKYCUXTYDUYATZEYGVKLYEUYEEUYFUXMVOYEUWLYEUWLUYMWCXEYEUYF - YHYFSIZRSIUXMYEYHRYFYDUUDUUCYHMDYCUUHUUFYFYGXHPUWPUWNXIYEUYPYGRSYDUUDUUCU - YPYGKYCUUHUUFYFYGXJPVSNXKXBTXLABCUVRXFVRUXIYCUXJUYBDUXKUXOKUYCUXIUXJUXPUY - BUXIUURYFUVCSYDUXGYCUVHUVAWDOUVHUVHUXTUXPUYBDYEUVHVDUYOUVCYFYGXPPWQABUXJW - PWOWIYEYRUVHUWSUWTYSYEUVGUWRUVCYEUVGYIUWRYEYFWLDZUVGYIWKZYDUYQYCUUGVGUYRY - FEWJGWLYFEYHXMXNXOLYEYTYHEJUUIOXQWSUXDWTUXIUUNUUOUVCUURUUTJIZDZUVKUXKKYDU - UNYCUVHUUPWDYCUUOYDUVHUUQWEYEUYTUVHYEUYSUVGUVCYEUURYFUUTYHJUXHUVBXKVLXAAY - LYMUVCXRVRWIXSXTYA $. + ( wcel cc0 chash cfv caddc co cfzo wceq cc oveq2d syl2anr eqtrd adantl c1 + syl cmin ad2antlr vx cword cconcat creverse wfn ccatcl revcl wrdfn revlen + wa 3syl ccatlen lencl nn0cnd addcom syl2an 3eqtrd fneq2d mpbid oveqan12rd + cv wo cz nn0zd fzospliti simpll simplr cfz fzoval eleq2d biimpa fznn0sub2 + id eleqtrrd ccatval3 syl3anc oveq1d adantr 1cnd addsubd peano2zm ad2antrr + zcnd elfzoelz fveq2d revfv adantll 3eqtr4d cuz wss uzaddcl eqeltrd fzoss2 + cn0 uzidd sselda syl2an2r biimpar ccatval1 subsub3d zaddcl fzrev2i subidd + 3eqtr4rd addcl sub32d pncan2 oveq12d eqtr4d 3eltr4d simpl fzosubel3 nn0uz + fzoss1 eleq2s sseqtr4d ccatval2 jaodan syldan eqfnfvd ) BAUBZDZCYADZUJZUA + ECFGZBFGZHIZJIZBCUCIZUDGZCUDGZBUDGZUCIZYDYJEYJFGZJIZUEZYJYHUEYDYIYADZYJYA + DYPABCUFZAYIUGAYJUHUKYDYOYHYJYDYNYGEJYDYNYIFGZYFYEHIZYGYDYQYNYSKYRAYIUIRA + ABCULZYBYFLDZYELDZYTYGKYCYBYFABUMZUNZYCYEACUMZUNZYFYEUOUPZUQMURUSYDYMEYMF + GZJIZUEZYMYHUEYDYMYADZUUKYCYKYADZYLYADZUULYBACUGZABUGZAYKYLUFNAYMUHRYDUUJ + YHYMYDUUIYGEJYDUUIYKFGZYLFGZHIZYGYCUUMUUNUUIUUSKYBUUOUUPAAYKYLULNYCYBUUQY + EUURYFHACUIZABUIUTZOMURUSYDUAVAZYHDZUVBEYEJIZDZUVBYEYGJIZDZVBZUVBYJGZUVBY + MGZKZUVCUVCYEVCDZUVHYDUVCVMYCUVLYBYCYEUUFVDZPUVBEYGYEVENYDUVEUVKUVGYDUVEU + JZYSQSIZUVBSIZYIGZUVBYKGZUVIUVJUVNYEQSIZUVBSIZYFHIZYIGZUVTCGZUVQUVRUVNYBY + CUVTUVDDUWBUWCKYBYCUVEVFYBYCUVEVGUVNUVTEUVSVHIZUVDUVNUVBUWDDZUVTUWDDYDUVE + UWEYDUVDUWDUVBYCUVDUWDKZYBYCUVLUWFUVMEYEVIRZPVJVKUVBUVSVLRYCUWFYBUVEUWGTV + NABCUVTVOVPUVNUVPUWAYIUVNUVPUVSYFHIZUVBSIZUWAYDUVPUWIKUVEYDUVOUWHUVBSYDUV + OYGQSIZUWHYDYSYGQSYDYSYTYGUUAUUHOZVQZYDYEYFQYCUUCYBUUGPZYBUUBYCUUEVRZYDVS + ZVTOVQVRUVNUVSYFUVBYCUVSLDYBUVEYCUVSYCUVLUVSVCDUVMYEWARWCTYBUUBYCUVEUUEWB + UVEUVBLDZYDUVEUVBUVBEYEWDWCPVTOWEYCUVEUVRUWCKYBACUVBWFWGWHYDYQUVEUVBEYSJI + ZDZUVIUVQKZYRYDUVDUWQUVBYDYSYEWIGZDUVDUWQWJYDYSYGUWTUWKYCYEUWTDYFWNDYGUWT + DYBYCYEUVMWOUUDYFYEYEWKNWLYEEYSWMRWPAYIUVBWFZWQUVNUUMUUNUVBEUUQJIZDZUVJUV + RKYCUUMYBUVEUUOTYBUUNYCUVEUUPWBYDUXCUVEYDUXBUVDUVBYDUUQYEEJYCUUQYEKYBUUTP + ZMVJWRAYKYLUVBWSVPWHYDUVGUJZUVQUVBUUQSIZYLGZUVIUVJUXEUVPBGZYFQSIZUXFSIZBG + ZUVQUXGUXEUVPUXJBUXEUXIUVBYESIZSIZUXIYEHIZUVBSIZUXJUVPUXEUXIUVBYEYBUXILDY + CUVGYBUXIYBYFVCDZUXIVCDYBYFUUDVDZYFWARWCWBUVGUWPYDUVGUVBUVBYEYGWDWCPYCUUC + YBUVGUUGTWTYCUXJUXMKYBUVGYCUXFUXLUXISYCUUQYEUVBSUUTMZMTYDUVPUXOKUVGYDUVOU + XNUVBSYDUVOYTQSIUXNYDYSYTQSUUAVQYDYFYEQUWNUWMUWOVTOVQVRXDWEUXEYBYCUVPEYFJ + IZDUVQUXHKYBYCUVGVFYBYCUVGVGUXEUWJUVBSIZUWJUWJSIZUWJYESIZVHIZUVPUXSYDUWJV + CDZUVGUVBYEUWJVHIZDZUXTUYCDYDYGVCDZUYDYCUVLUXPUYGYBUVMUXQYEYFXANZYGWARZYD + UVGUYFYDUVFUYEUVBYDUYGUVFUYEKUYHYEYGVIRVJVKUWJUVBYEUWJXBWQYDUVPUXTKUVGYDU + VOUWJUVBSUWLVQVRYDUXSUYCKUVGYDUXSEUXIVHIZUYCYDUXPUXSUYJKYBUXPYCUXQVRZEYFV + IRYDUYAEUYBUXIVHYDUWJYDUWJUYIWCXCYDUYBYGYESIZQSIUXIYDYGQYEYCUUCUUBYGLDYBU + UGUUEYEYFXENUWOUWMXFYDUYLYFQSYCUUCUUBUYLYFKYBUUGUUEYEYFXGNVQOXHXIVRXJABCU + VPWSVPYDYBUVGUXFUXSDUXGUXKKYBYCXKUXEUXFUXLUXSYCUXFUXLKYBUVGUXRTUVGUVGUXPU + XLUXSDYDUVGVMUYKUVBYEYFXLNWLABUXFWFWQWHYDYQUVGUWRUWSYRYDUVFUWQUVBYDUVFYHU + WQYCUVFYHWJZYBYCYEWNDUYMUUFUYMYEEWIGWNYEEYGXNXMXORPYDYSYGEJUWKMXPWPUXAWQU + XEUUMUUNUVBUUQUUSJIZDZUVJUXGKYCUUMYBUVGUUOTYBUUNYCUVGUUPWBYDUYOUVGYDUYNUV + FUVBYDUUQYEUUSYGJUXDUVAXHVJWRAYKYLUVBXQVPWHXRXSXT $. $( Reversal is an involution on words. (Contributed by Mario Carneiro, 1-Oct-2015.) $) @@ -153095,19 +153104,18 @@ because the border cases ( ` M = N ` , ` -. ( M ..^ N ) C_ ( 0 ..^ L ) ` 16-Oct-2022.) $) cshwlen $p |- ( ( W e. Word V /\ N e. ZZ ) -> ( # ` ( W cyclShift N ) ) = ( # ` W ) ) $= - ( wcel wa ccsh co chash cfv wi c0 3eqtrd fveq2d caddc adantr syl2anc ex syl - wceq cc cword oveq1 0csh0 a1i eqcom biimpi a1d wne cmo csubstr cpfx cconcat - cz cop cshword swrdcl pfxcl ccatlen lennncl pm3.21 com24 pm2.43i imp31 cmin - cc0 cfz simpl pm3.22 adantl zmodfzp1 cn0 lencl nn0fz0 sylib swrdlen syl3anc - cn pfxlen oveq12d nn0cnd zmodcl ancoms npcan syl2an eqtrd expcom pm2.61ine - ) CBUAZDZAUMDZEZCAFGZHIZCHIZSZJCKCKSZWOWKWPWLCHWPWLKAFGZKCCKAFUBWQKSWPAUCUD - WPKCSCKUEUFLMUGWKCKUHZWOWKWREZWMCAWNUIGZWNUNUJGZCWTUKGZULGZHIZXAHIZXBHIZNGZ - WNWKWMXDSWRWKWLXCHABCUOMOWKXDXGSZWRWIXHWJWIXAWHDXBWHDXHBCWTWNUPBCWTUQBXAXBU - RPOOWSWIWNVQDZWJEZEZXGWNSWIWJWRXKWIWJWRXKJJWIWRWJWIXKWIWRWJWIXKJZJZWIWREXIX - MBCUSXIWJXLXJWIUTQRQVAVBVCXKXGWNWTVDGZWTNGZWNXKXEXNXFWTNXKWIWTVEWNVFGZDZWNX - PDZXEXNSWIXJVGZXKWJXIEZXQXJXTWIXIWJVHVIAWNVJRZWIXRXJWIWNVKDXRBCVLZWNVMVNOBC - WTWNVOVPXKWIXQXFWTSXSYABCWTVRPVSWIWNTDWTTDZXOWNSXJWIWNYBVTWJXIYCXTWTAWNWAVT - WBWNWTWCWDWERLWFWG $. + ( wcel wa ccsh co chash cfv wceq wi c0 fveq2d caddc adantr ex syl ancoms cc + nn0cnd cword cz 0csh0 oveq1 id 3eqtr4a a1d wne cmo cop csubstr cpfx cconcat + cshword swrdcl pfxcl ccatlen syl2anc ad2antrr cn lennncl pm3.21 com24 imp31 + pm2.43i cmin cc0 cfz simpl zmodfzp1 adantl cn0 lencl nn0fz0 swrdlen syl3anc + sylib pfxlen sylan2 oveq12d zmodcl npcan syl2an 3eqtrd expcom pm2.61ine + eqtrd ) CBUAZDZAUBDZEZCAFGZHIZCHIZJZKCLCLJZWOWKWPWLCHWPLAFGLWLCAUCCLAFUDWPU + EUFMUGWKCLUHZWOWKWQEZWMCAWNUIGZWNUJUKGZCWSULGZUMGZHIZWTHIZXAHIZNGZWNWKWMXCJ + WQWKWLXBHABCUNMOWIXCXFJZWJWQWIWTWHDXAWHDXGBCWSWNUOBCWSUPBBWTXAUQURUSWRWIWNU + TDZWJEZEZXFWNJWIWJWQXJWIWJWQXJKKWIWQWJWIXJWIWQWJWIXJKZKZWIWQEXHXLBCVAXHWJXK + XIWIVBPQPVCVEVDXJXFWNWSVFGZWSNGZWNXJXDXMXEWSNXJWIWSVGWNVHGZDZWNXODZXDXMJWIX + IVIXIXPWIWJXHXPAWNVJRZVKWIXQXIWIWNVLDXQBCVMZWNVNVQOBCWSWNVOVPXIWIXPXEWSJXRB + CWSVRVSVTWIWNSDWSSDZXNWNJXIWIWNXSTWJXHXTWJXHEWSAWNWATRWNWSWBWCWGQWDWEWF $. $( A cyclically shifted word is a function from a half-open range of integers of the same length as the word as domain to the set of symbols for the @@ -154080,7 +154088,7 @@ the symbol at any position is repeated at multiples of L (modulo the cats1len $p |- ( # ` T ) = N $= ( chash cfv cs1 cconcat co fveq2i c1 caddc cvv wcel eqtri cword ccatlen wceq s1cli mp2an s1len oveq12i ) BJKAELZMNZJKZDBUIJFOUJCPQNZDUJAJKZUHJK - ZQNZUKARUAZSUHUOSUJUNUCGEUDRAUHUBUEULCUMPQHEUFUGTITT $. + ZQNZUKARUAZSUHUOSUJUNUCGEUDRRAUHUBUEULCUMPQHEUFUGTITT $. $} ${ @@ -159421,7 +159429,7 @@ reflection about the origin (under the map ` x |-> -u x ` ). (Contributed A. k e. ( ZZ>= ` j ) ( abs ` ( ( F ` k ) - ( F ` j ) ) ) < x ) -> E. y e. RR A. k e. Z ( abs ` ( F ` k ) ) < y ) $= ( vz cv cfv wcel wral clt wbr wrex cr wa wi cz vw cc cmin co cabs cuz crp - abscl ralimi r19.29uz ralimdv caubnd2 syl6 cfz wss fzssuz sseqtr4i ssralv + abscl ralimi r19.29uz ralimdv caubnd2 syl6 cfz wss fzssuz sseqtrri ssralv ex ax-mp cle cfn fzfi fimaxre3 mpan c1 caddc peano2re adantl ltp1 mpd3an3 lelttr mpan2d expcom impcom ralim syl brralrspcev syl6an rexlimdva mpd wo cif max1 3adant3 simp3 simp1 ifcl ancoms ltletr syl3anc max2 simp2 3expia @@ -165732,7 +165740,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ isumclim3 $p |- ( ph -> F ~~> sum_ k e. Z A ) $= ( vm vx cli cfv csu wcel wbr cv cdm climdm sylib caddc cmpt cseq sumfc wa eqidd cc fmpttd ffvelrnda isum syl5eqr cio cvv seqex a1i cfz co cres wceq - wss cuz fzssuz sseqtr4i resmpt ax-mp fveq1i fvres syl5reqr eqtri syl6eleq + wss cuz fzssuz sseqtrri resmpt ax-mp fveq1i fvres syl5reqr eqtri syl6eleq sumeq2i simpr simpl elfzuz syl6eleqr syl2an fsumser eqtr2d iotabidv df-fv climeq 3eqtr4g eqtrd breqtrrd ) AEEOPZGBDQZOAEOUAZREWHOSJEUBUCAWIUDDGBUEZ FUFZOPZWHAWIGMTZWKPZMQWMGBMDUGAWOMWKFGHIAWNGRZUHWOUIAGUJWNWKADGBUJKUKULZU @@ -171856,7 +171864,7 @@ seq n ( x. , ( k e. Z |-> A ) ) ~~> y ) ) $. iprodclim3 $p |- ( ph -> F ~~> prod_ k e. Z A ) $= ( vm vx cli cfv wcel cprod cdm wbr climdm sylib cmul cmpt cv prodfc eqidd cseq wa cc fmpttd ffvelrnda iprod syl5eqr cio cvv seqex a1i cfz cres wceq - wss cuz fzssuz sseqtr4i resmpt ax-mp fveq1i fvres syl5reqr prodeq2i eqtri + wss cuz fzssuz sseqtrri resmpt ax-mp fveq1i fvres syl5reqr prodeq2i eqtri co simpr syl6eleq elfzuz syl6eleqr sylan2 fprodser eqtr2d climeq iotabidv adantlr df-fv 3eqtr4g eqtrd breqtrrd ) AGGRSZICEUAZRAGRUBZTGWKRUCMGUDUEAW LUFEICUGZHUKZRSZWKAWLIPUHZWNSZPUAWPICPEUIABWRPFWNHIJKLAWQITZULWRUJAIUMWQW @@ -172889,7 +172897,7 @@ seq m ( x. , G ) ~~> z ) ) $. 0nn0 3ne0 div12d 3t2e6 divmuli mpbir mulcomd syl5eq 3nn0 2ne0 expcl mpan2 eqnetri subcld subsubd addcld adddid subdid recidi mulassd 3eqtr3a add12d 6cn addassd 3eqtr2d 3eqtrd lelttrd posdifd mpbid addgt0d gt0ne0d 1eluzge0 - 0lt1 4bc3eq4 3p1e4 fzssp1 sseqtr4i 4bc2eq6 2p2e4 nn0uz wss 3nn nnuz fzss2 + 0lt1 4bc3eq4 3p1e4 fzssp1 sseqtrri 4bc2eq6 2p2e4 nn0uz wss 3nn nnuz fzss2 cn 3sstr4i bcn1 df-4 sylbi 5pos bcn0 subid1i 4p1e5 fsum1 bpoly0 1nn0 mp1i 0z nn0cn 4ne0 divcan2d bpoly1 syl5eqr 2nn0 bpoly2 4d2e2 bpoly3 sqcl deccl nn0cni dfdec10 10re recni mulcli addid1i 10pos mulne0i 6pos divcli mulid2 @@ -173422,7 +173430,7 @@ seq m ( x. , G ) ~~> z ) ) $. wa wral ralrimiva eleq1d rspc impcom syl2an fvmpts eqtr4d expcom w3a cmul nfel1 simp3 peano2uzs mpan9 3adant3 oveq12d elfzuz adantlr fprodp1 fsump1 fzfid fsumcl efadd eqtrd 3eqtr4d 3exp com12 a2d eqcomi eleq2s uzind4 cres - mpcom wss fzssuz sseqtr4i resmpt ax-mp fveq1i fvres syl5reqr prodfc eqtri + mpcom wss fzssuz sseqtrri resmpt ax-mp fveq1i fvres syl5reqr prodfc eqtri prodeq2i sumeq2i sumfc fveq2i 3eqtr3g ) ADEKLZJUCZCFBMNZUGZNZJOZYMYNCFBUG ZNZJPZMNZYMYOCOZYMBCPZMNEDUDNZQAYRUUBRZAEFUUEHGUEADUAUCZKLZYQJOZUUHYTJPZM NZRZUFADDKLZYQJOZUUMYTJPZMNZRZUFADUBUCZKLZYQJOZUUSYTJPZMNZRZUFZADUURUHUIL @@ -175089,7 +175097,7 @@ The circle constant (tau = 2 pi) znnen $p |- ZZ ~~ NN $= ( cz cn cdom wbr cen cxp ccrd cdm wcel cmin wfo com mp2an wss cc subf ax-mp nnsscn mp2 cvv cres con0 omelon nnenom ensymi isnumi xpnum cima wfun xpss12 - wf ffun fdmi sseqtr4i fores wceq wb foeq3 mpbir fodomnum xpnnen domentr zex + wf ffun fdmi sseqtrri fores wceq wb foeq3 mpbir fodomnum xpnnen domentr zex dfz2 nnssz ssdomg sbth ) ABCDZBACDZABEDABBFZCDZVJBEDVHVJGHZIZVJAJVJUAZKZVKB VLIZVPVMLUBILBEDVPUCBLUDUELBUFMZVQBBUGMVOVJJVJUHZVNKZJUIZVJJHZNVSOOFZOJUKVT PWBOJULQVJWBWABONZWCVJWBNRRBOBOUJMWBOJPUMUNVJJUOMAVRUPVOVSUQVDAVRVJVNURQUSV @@ -182934,7 +182942,7 @@ being prime ( ` Prime = { p e. NN | ... ` ), but even if ` p e. NN0 ` was pm4.38 df-ne nesym anbi12i ioran bitr4i syl6bb syl6 syld eluzelz caddc 1z imp zltp1le mpan df-2 breq1i syl6bbr zltlem1 anbi12d peano2zm elfz mp3an2 bitr4d bitr3d anasss expcom pm5.32d fzssuz 2eluzge1 uzss ax-mp sstri nnuz - 2z sseqtr4i sseli pm4.71ri notbid syl5bb pm5.74da bi2.04 3bitr3g ralbidv2 + 2z sseqtrri sseli pm4.71ri notbid syl5bb pm5.74da bi2.04 3bitr3g ralbidv2 wss con2b pm5.32i bitri ) BUACBDEFZCZAUBZBUCGZXQHIZXQBIZUDZJZAKUEZLXPXRMZ ADBHUFNZUGNZUEZLABUHXPYCYGXPYBYDAKYFXPXRXQKCZYAJZJXRXQYFCZMZJYHYBJYJYDJXP XRYIYKYIYHYAMZLZMXPXRLZYKYHYAUIYNYMYJYNYMYHYJLYJYNYHYLYJYHYNYLYJOZYHXPXRY @@ -184237,7 +184245,7 @@ reduced fraction representation (no common factors, denominator wn simprd wi dvdslegcd syl31anc mp2and simprbi wb nnle1eq1 mpbid sylanbrc breqtrd eqtrd dvdsmul2 dvdstr syl3anc opelxpd eqeltrd ralrimiva wfun wf1o cdm wfn crth f1ofn fnfun ssrab3 fndm sseqtrrid syl2an cmpt eqtri eqeltrrd - eqtr3d sylib dfphi2 fveq2i syl6eqr funimass4 mpbird xpss12 sseqtr4i sseli + eqtr3d sylib dfphi2 fveq2i syl6eqr funimass4 mpbird xpss12 sseqtrri sseli ccnv f1ocnvfv2 wf f1ocnv f1of ffvelrn cbvmptv opelxp funfvima2 imp syldan rpmul eqelssd wf1 f1of1 elexi f1imaen sylancl eqbrtrrd xpfi hashen sylibr syl5reqr rabeqi oveq12d 3eqtr4d ) AKUCUDZFUCUDZJUCUDZUEUFZHIUEUFZUGUDZHUG @@ -193303,7 +193311,7 @@ class of all (base sets of) groups is proper. (Contributed by Mario isstruct2 $p |- ( F Struct X <-> ( X e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( F \ { (/) } ) /\ dom F C_ ( ... ` X ) ) ) $= ( vx vf cstr cvv wcel wa cle cdif wfun cdm cfz cfv wss w3a cun cfn wceq - cn wbr cxp cin c0 csn brstruct brrelex12i ssun1 undif1 sseqtr4i wfn simp2 + cn wbr cxp cin c0 csn brstruct brrelex12i ssun1 undif1 sseqtrri wfn simp2 funfnd c1st c2nd cop elinel2 1st2nd2 syl 3ad2ant1 fveq2d co fzfi eqeltrri df-ov difss dmss ax-mp simp3 sstrid ssfid fnfi syl2anc p0ex unexg sylancl syl6eqel ssexg sylancr elex jca simpr eleq1d simpl difeq1d funeqd sseq12d @@ -194415,13 +194423,13 @@ C_ dom ( S sSet <. I , E >. ) ) $= $( The base set of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) $) 2strbas $p |- ( B e. V -> B = ( Base ` G ) ) $= - ( cbs c1 cop 2strstr baseid cnx cfv csn cpr snsspr1 sseqtr4i strfv ) ADKF + ( cbs c1 cop 2strstr baseid cnx cfv csn cpr snsspr1 sseqtrri strfv ) ADKF LEMABCDEGHIJNOPKQAMZRUCPCQBMZSDUCUDTGUAUB $. $( The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) $) 2strop $p |- ( .+ e. V -> .+ = ( E ` G ) ) $= - ( c1 cop 2strstr ndxid cnx cfv csn cbs cpr snsspr2 sseqtr4i strfv ) BDCFK + ( c1 cop 2strstr ndxid cnx cfv csn cbs cpr snsspr2 sseqtrri strfv ) BDCFK ELABCDEGHIJMCEHJNOCPBLZQORPALZUCSDUDUCTGUAUB $. $} @@ -194441,7 +194449,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= not depending on the hard-coded index value of the base set. (Contributed by AV, 22-Sep-2020.) $) 2strbas1 $p |- ( B e. V -> B = ( Base ` G ) ) $= - ( cbs cnx cfv cop 2strstr1 baseid csn cpr snsspr1 sseqtr4i strfv ) ACIEJI + ( cbs cnx cfv cop 2strstr1 baseid csn cpr snsspr1 sseqtrri strfv ) ACIEJI KZDLABCDFGHMNTALZOUADBLZPCUAUBQFRS $. 2str1.e $e |- E = Slot N $. @@ -194549,20 +194557,20 @@ C_ dom ( S sSet <. I , E >. ) ) $= $( The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) $) rngbase $p |- ( B e. V -> B = ( Base ` R ) ) $= - ( cbs c1 cop rngstr baseid cnx cfv csn cplusg cmulr ctp snsstp1 sseqtr4i + ( cbs c1 cop rngstr baseid cnx cfv csn cplusg cmulr ctp snsstp1 sseqtrri c3 strfv ) ACGEHTIABCDFJKLGMAIZNUBLOMBIZLPMDIZQCUBUCUDRFSUA $. $( The additive operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) $) rngplusg $p |- ( .+ e. V -> .+ = ( +g ` R ) ) $= - ( cplusg c1 cop rngstr plusgid cnx cfv csn cbs cmulr ctp snsstp2 sseqtr4i + ( cplusg c1 cop rngstr plusgid cnx cfv csn cbs cmulr ctp snsstp2 sseqtrri c3 strfv ) BCGEHTIABCDFJKLGMBIZNLOMAIZUBLPMDIZQCUCUBUDRFSUA $. $( The multiplicative operation of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) $) rngmulr $p |- ( .x. e. V -> .x. = ( .r ` R ) ) $= - ( cmulr c1 cop rngstr mulrid cnx cfv csn cbs cplusg ctp snsstp3 sseqtr4i + ( cmulr c1 cop rngstr mulrid cnx cfv csn cbs cplusg ctp snsstp3 sseqtrri c3 strfv ) DCGEHTIABCDFJKLGMDIZNLOMAIZLPMBIZUBQCUCUDUBRFSUA $. $} @@ -194610,28 +194618,28 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 6-May-2015.) $) srngbase $p |- ( B e. X -> B = ( Base ` R ) ) $= ( cbs c1 c4 cop srngstr baseid cnx cfv csn cplusg cmulr ctp snsstp1 ssun1 - cstv cun sseqtr4i sstri strfv ) ACHFIJKABCDEGLMNHOAKZPUGNQOBKZNRODKZSZCUG + cstv cun sseqtrri sstri strfv ) ACHFIJKABCDEGLMNHOAKZPUGNQOBKZNRODKZSZCUG UHUITUJUJNUBOEKPZUCCUJUKUAGUDUEUF $. $( The addition operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) $) srngplusg $p |- ( .+ e. X -> .+ = ( +g ` R ) ) $= ( cplusg c1 c4 cop srngstr plusgid cnx cfv csn cbs cmulr ctp snsstp2 cstv - cun ssun1 sseqtr4i sstri strfv ) BCHFIJKABCDEGLMNHOBKZPNQOAKZUGNRODKZSZCU + cun ssun1 sseqtrri sstri strfv ) BCHFIJKABCDEGLMNHOBKZPNQOAKZUGNRODKZSZCU HUGUITUJUJNUAOEKPZUBCUJUKUCGUDUEUF $. $( The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) $) srngmulr $p |- ( .x. e. X -> .x. = ( .r ` R ) ) $= ( cmulr c1 c4 cop srngstr mulrid cnx cfv csn cbs cplusg ctp snsstp3 ssun1 - cstv cun sseqtr4i sstri strfv ) DCHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZCUH + cstv cun sseqtrri sstri strfv ) DCHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZCUH UIUGTUJUJNUBOEKPZUCCUJUKUAGUDUEUF $. $( The involution function of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015.) $) srnginvl $p |- ( .* e. X -> .* = ( *r ` R ) ) $= ( cstv c1 c4 cop srngstr starvid cnx cfv csn cbs cplusg cmulr ctp ssun2 - cun sseqtr4i strfv ) ECHFIJKABCDEGLMNHOEKPZNQOAKNROBKNSODKTZUEUBCUEUFUAGU + cun sseqtrri strfv ) ECHFIJKABCDEGLMNHOEKPZNQOAKNROBKNSODKTZUEUBCUEUFUAGU CUD $. $} @@ -194672,7 +194680,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) $) lmodbase $p |- ( B e. X -> B = ( Base ` W ) ) $= ( cbs c1 c6 cop lmodstr baseid cnx cfv csn cplusg csca ctp snsstp1 cvsca - cun ssun1 sseqtr4i sstri strfv ) AEHFIJKABCDEGLMNHOAKZPUGNQOBKZNRODKZSZEU + cun ssun1 sseqtrri sstri strfv ) AEHFIJKABCDEGLMNHOAKZPUGNQOBKZNRODKZSZEU GUHUITUJUJNUAOCKPZUBEUJUKUCGUDUEUF $. $( The additive operation of a constructed left vector space. (Contributed @@ -194680,7 +194688,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= 29-Aug-2015.) $) lmodplusg $p |- ( .+ e. X -> .+ = ( +g ` W ) ) $= ( cplusg c1 c6 cop lmodstr plusgid cnx cfv csn cbs csca ctp snsstp2 cvsca - cun ssun1 sseqtr4i sstri strfv ) BEHFIJKABCDEGLMNHOBKZPNQOAKZUGNRODKZSZEU + cun ssun1 sseqtrri sstri strfv ) BEHFIJKABCDEGLMNHOBKZPNQOAKZUGNRODKZSZEU HUGUITUJUJNUAOCKPZUBEUJUKUCGUDUEUF $. $( The set of scalars of a constructed left vector space. (Contributed by @@ -194688,14 +194696,14 @@ C_ dom ( S sSet <. I , E >. ) ) $= 29-Aug-2015.) $) lmodsca $p |- ( F e. X -> F = ( Scalar ` W ) ) $= ( csca c1 c6 cop lmodstr scaid cnx cfv csn cbs cplusg ctp snsstp3 cvsca - cun ssun1 sseqtr4i sstri strfv ) DEHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZEU + cun ssun1 sseqtrri sstri strfv ) DEHFIJKABCDEGLMNHODKZPNQOAKZNROBKZUGSZEU HUIUGTUJUJNUAOCKPZUBEUJUKUCGUDUEUF $. $( The scalar product operation of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) $) lmodvsca $p |- ( .x. e. X -> .x. = ( .s ` W ) ) $= - ( cvsca c1 c6 cop lmodstr vscaid cnx cfv csn cbs cplusg csca ctp sseqtr4i + ( cvsca c1 c6 cop lmodstr vscaid cnx cfv csn cbs cplusg csca ctp sseqtrri cun ssun2 strfv ) CEHFIJKABCDEGLMNHOCKPZNQOAKNROBKNSODKTZUEUBEUEUFUCGUAUD $. $} @@ -194729,7 +194737,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsbase $p |- ( B e. V -> B = ( Base ` A ) ) $= ( cbs c1 c8 cop ipsstr baseid cnx cfv csn cplusg ctp cmulr csca cvsca cip - snsstp1 cun ssun1 sseqtr4i sstri strfv ) BAJHKLMABCDEFGINOPJQBMZRUKPSQCMZ + snsstp1 cun ssun1 sseqtrri sstri strfv ) BAJHKLMABCDEFGINOPJQBMZRUKPSQCMZ PUAQFMZTZAUKULUMUEUNUNPUBQDMPUCQEMPUDQGMTZUFAUNUOUGIUHUIUJ $. $( The additive operation of a constructed inner product space. @@ -194737,7 +194745,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsaddg $p |- ( .+ e. V -> .+ = ( +g ` A ) ) $= ( cplusg c1 c8 cop ipsstr plusgid cnx cfv csn cbs ctp cmulr snsstp2 cvsca - csca cip cun ssun1 sseqtr4i sstri strfv ) CAJHKLMABCDEFGINOPJQCMZRPSQBMZU + csca cip cun ssun1 sseqtrri sstri strfv ) CAJHKLMABCDEFGINOPJQCMZRPSQBMZU KPUAQFMZTZAULUKUMUBUNUNPUDQDMPUCQEMPUEQGMTZUFAUNUOUGIUHUIUJ $. $( The multiplicative operation of a constructed inner product space. @@ -194745,7 +194753,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsmulr $p |- ( .X. e. V -> .X. = ( .r ` A ) ) $= ( cmulr c1 c8 cop ipsstr mulrid cnx cfv csn cbs ctp cplusg csca cvsca cip - snsstp3 cun ssun1 sseqtr4i sstri strfv ) FAJHKLMABCDEFGINOPJQFMZRPSQBMZPU + snsstp3 cun ssun1 sseqtrri sstri strfv ) FAJHKLMABCDEFGINOPJQFMZRPSQBMZPU AQCMZUKTZAULUMUKUEUNUNPUBQDMPUCQEMPUDQGMTZUFAUNUOUGIUHUIUJ $. $( The set of scalars of a constructed inner product space. (Contributed @@ -194753,7 +194761,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipssca $p |- ( S e. V -> S = ( Scalar ` A ) ) $= ( csca c1 c8 cop ipsstr scaid cnx cfv csn cvsca ctp cip snsstp1 cbs cmulr - cplusg cun ssun2 sseqtr4i sstri strfv ) DAJHKLMABCDEFGINOPJQDMZRUKPSQEMZP + cplusg cun ssun2 sseqtrri sstri strfv ) DAJHKLMABCDEFGINOPJQDMZRUKPSQEMZP UAQGMZTZAUKULUMUBUNPUCQBMPUEQCMPUDQFMTZUNUFAUNUOUGIUHUIUJ $. $( The scalar product operation of a constructed inner product space. @@ -194761,7 +194769,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsvsca $p |- ( .x. e. V -> .x. = ( .s ` A ) ) $= ( cvsca c1 c8 cop ipsstr vscaid cnx cfv csn csca ctp snsstp2 cplusg cmulr - cip cbs cun ssun2 sseqtr4i sstri strfv ) EAJHKLMABCDEFGINOPJQEMZRPSQDMZUK + cip cbs cun ssun2 sseqtrri sstri strfv ) EAJHKLMABCDEFGINOPJQEMZRPSQDMZUK PUDQGMZTZAULUKUMUAUNPUEQBMPUBQCMPUCQFMTZUNUFAUNUOUGIUHUIUJ $. $( The multiplicative operation of a constructed inner product space. @@ -194769,7 +194777,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 29-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) $) ipsip $p |- ( I e. V -> I = ( .i ` A ) ) $= ( cip c1 c8 cop ipsstr ipid cnx cfv csn csca ctp cvsca snsstp3 cbs cplusg - cmulr cun ssun2 sseqtr4i sstri strfv ) GAJHKLMABCDEFGINOPJQGMZRPSQDMZPUAQ + cmulr cun ssun2 sseqtrri sstri strfv ) GAJHKLMABCDEFGINOPJQGMZRPSQDMZPUAQ EMZUKTZAULUMUKUBUNPUCQBMPUDQCMPUEQFMTZUNUFAUNUOUGIUHUIUJ $. $} @@ -194818,14 +194826,14 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) $) phlbase $p |- ( B e. X -> B = ( Base ` H ) ) $= ( cbs c1 c8 cop phlstr baseid cnx cfv csn cplusg csca ctp snsstp1 cip cpr - cvsca cun ssun1 sseqtr4i sstri strfv ) AEIGJKLABCDEFHMNOIPALZQUJORPBLZOSP + cvsca cun ssun1 sseqtrri sstri strfv ) AEIGJKLABCDEFHMNOIPALZQUJORPBLZOSP CLZTZEUJUKULUAUMUMOUDPDLOUBPFLUCZUEEUMUNUFHUGUHUI $. $( The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) $) phlplusg $p |- ( .+ e. X -> .+ = ( +g ` H ) ) $= - ( cplusg c1 c8 cop phlstr plusgid cnx cfv csn cbs csca ctp cvsca sseqtr4i + ( cplusg c1 c8 cop phlstr plusgid cnx cfv csn cbs csca ctp cvsca sseqtrri snsstp2 cip cpr cun ssun1 sstri strfv ) BEIGJKLABCDEFHMNOIPBLZQORPALZUJOS PCLZTZEUKUJULUCUMUMOUAPDLOUDPFLUEZUFEUMUNUGHUBUHUI $. @@ -194833,7 +194841,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Mario Carneiro, 6-Oct-2013.) (Revised by Mario Carneiro, 29-Aug-2015.) $) phlsca $p |- ( T e. X -> T = ( Scalar ` H ) ) $= - ( csca c1 c8 cop phlstr scaid cnx cfv csn cbs cplusg ctp snsstp3 sseqtr4i + ( csca c1 c8 cop phlstr scaid cnx cfv csn cbs cplusg ctp snsstp3 sseqtrri cvsca cip cpr cun ssun1 sstri strfv ) CEIGJKLABCDEFHMNOIPCLZQORPALZOSPBLZ UJTZEUKULUJUAUMUMOUCPDLOUDPFLUEZUFEUMUNUGHUBUHUI $. @@ -194842,7 +194850,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 29-Aug-2015.) $) phlvsca $p |- ( .x. e. X -> .x. = ( .s ` H ) ) $= ( cvsca c1 c8 cop phlstr vscaid cnx cfv csn cip cpr snsspr1 cbs ctp ssun2 - cplusg csca cun sseqtr4i sstri strfv ) DEIGJKLABCDEFHMNOIPDLZQUJORPFLZSZE + cplusg csca cun sseqtrri sstri strfv ) DEIGJKLABCDEFHMNOIPDLZQUJORPFLZSZE UJUKTULOUAPALOUDPBLOUEPCLUBZULUFEULUMUCHUGUHUI $. $( The inner product (Hermitian form) operation of a constructed @@ -194850,7 +194858,7 @@ C_ dom ( S sSet <. I , E >. ) ) $= (Revised by Mario Carneiro, 29-Aug-2015.) $) phlip $p |- ( ., e. X -> ., = ( .i ` H ) ) $= ( cip c1 c8 cop phlstr ipid cnx cfv csn cvsca cpr snsspr2 cbs cplusg csca - ctp cun ssun2 sseqtr4i sstri strfv ) FEIGJKLABCDEFHMNOIPFLZQORPDLZUJSZEUK + ctp cun ssun2 sseqtrri sstri strfv ) FEIGJKLABCDEFHMNOIPFLZQORPDLZUJSZEUK UJTULOUAPALOUBPBLOUCPCLUDZULUEEULUMUFHUGUHUI $. $} @@ -194877,19 +194885,19 @@ C_ dom ( S sSet <. I , E >. ) ) $= $( The base set of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) $) topgrpbas $p |- ( B e. X -> B = ( Base ` W ) ) $= - ( cbs c1 cop topgrpstr baseid cnx cfv csn cplusg cts ctp snsstp1 sseqtr4i + ( cbs c1 cop topgrpstr baseid cnx cfv csn cplusg cts ctp snsstp1 sseqtrri c9 strfv ) ADGEHTIABCDFJKLGMAIZNUBLOMBIZLPMCIZQDUBUCUDRFSUA $. $( The additive operation of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) $) topgrpplusg $p |- ( .+ e. X -> .+ = ( +g ` W ) ) $= - ( cplusg c1 c9 cop topgrpstr plusgid cnx cfv csn cbs cts snsstp2 sseqtr4i + ( cplusg c1 c9 cop topgrpstr plusgid cnx cfv csn cbs cts snsstp2 sseqtrri ctp strfv ) BDGEHIJABCDFKLMGNBJZOMPNAJZUBMQNCJZTDUCUBUDRFSUA $. $( The topology of a constructed topological group. (Contributed by Mario Carneiro, 29-Aug-2015.) $) topgrptset $p |- ( J e. X -> J = ( TopSet ` W ) ) $= - ( cts c1 cop topgrpstr tsetid cnx cfv csn cbs cplusg ctp snsstp3 sseqtr4i + ( cts c1 cop topgrpstr tsetid cnx cfv csn cbs cplusg ctp snsstp3 sseqtrri c9 strfv ) CDGEHTIABCDFJKLGMCIZNLOMAIZLPMBIZUBQDUCUDUBRFSUA $. $} @@ -194926,19 +194934,19 @@ C_ dom ( S sSet <. I , E >. ) ) $= Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) $) otpsbas $p |- ( B e. V -> B = ( Base ` K ) ) $= ( cbs c1 cc0 cdc cop otpsstr baseid cnx cfv csn cts cple ctp snsstp1 - sseqtr4i strfv ) ACGEHHIJKABCDFLMNGOAKZPUCNQOBKZNRODKZSCUCUDUETFUAUB $. + sseqtrri strfv ) ACGEHHIJKABCDFLMNGOAKZPUCNQOBKZNRODKZSCUCUDUETFUAUB $. $( The open sets of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) $) otpstset $p |- ( J e. V -> J = ( TopSet ` K ) ) $= ( cts c1 cc0 cdc cop otpsstr tsetid cnx cfv csn cbs cple ctp snsstp2 - sseqtr4i strfv ) BCGEHHIJKABCDFLMNGOBKZPNQOAKZUCNRODKZSCUDUCUETFUAUB $. + sseqtrri strfv ) BCGEHHIJKABCDFLMNGOBKZPNQOAKZUCNRODKZSCUDUCUETFUAUB $. $( The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015.) (Revised by AV, 9-Sep-2021.) $) otpsle $p |- ( .<_ e. V -> .<_ = ( le ` K ) ) $= ( cple c1 cc0 cdc cop otpsstr pleid cnx cfv csn cbs cts ctp snsstp3 strfv - sseqtr4i ) DCGEHHIJKABCDFLMNGODKZPNQOAKZNROBKZUCSCUDUEUCTFUBUA $. + sseqtrri ) DCGEHHIJKABCDFLMNGODKZPNQOAKZNROBKZUCSCUDUEUCTFUBUA $. $} ${ @@ -194998,42 +195006,42 @@ C_ dom ( S sSet <. I , E >. ) ) $= 20-Aug-2015.) $) odrngbas $p |- ( B e. V -> B = ( Base ` W ) ) $= ( cbs c1 c2 cdc cop odrngstr baseid cnx cfv csn ctp cplusg cmulr cts cple - snsstp1 cds cun ssun1 sseqtr4i sstri strfv ) AHJGKKLMNABCDEFHIOPQJRANZSUL + snsstp1 cds cun ssun1 sseqtrri sstri strfv ) AHJGKKLMNABCDEFHIOPQJRANZSUL QUARCNZQUBRDNZTZHULUMUNUEUOUOQUCRENQUDRFNQUFRBNTZUGHUOUPUHIUIUJUK $. $( The addition operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngplusg $p |- ( .+ e. V -> .+ = ( +g ` W ) ) $= ( cplusg c1 c2 cdc cop odrngstr plusgid cnx cfv csn ctp cbs cmulr snsstp2 - cts cple cds cun ssun1 sseqtr4i sstri strfv ) CHJGKKLMNABCDEFHIOPQJRCNZSQ + cts cple cds cun ssun1 sseqtrri sstri strfv ) CHJGKKLMNABCDEFHIOPQJRCNZSQ UARANZULQUBRDNZTZHUMULUNUCUOUOQUDRENQUERFNQUFRBNTZUGHUOUPUHIUIUJUK $. $( The multiplication operation of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngmulr $p |- ( .x. e. V -> .x. = ( .r ` W ) ) $= ( cmulr c1 c2 cdc cop odrngstr mulrid cnx cfv csn ctp cplusg snsstp3 cple - cbs cts cds cun ssun1 sseqtr4i sstri strfv ) DHJGKKLMNABCDEFHIOPQJRDNZSQU + cbs cts cds cun ssun1 sseqtrri sstri strfv ) DHJGKKLMNABCDEFHIOPQJRDNZSQU DRANZQUARCNZULTZHUMUNULUBUOUOQUERENQUCRFNQUFRBNTZUGHUOUPUHIUIUJUK $. $( The open sets of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngtset $p |- ( J e. V -> J = ( TopSet ` W ) ) $= ( cts c1 c2 cdc cop odrngstr tsetid cnx cfv csn ctp cds snsstp1 cbs cmulr - cple cplusg cun ssun2 sseqtr4i sstri strfv ) EHJGKKLMNABCDEFHIOPQJRENZSUL + cple cplusg cun ssun2 sseqtrri sstri strfv ) EHJGKKLMNABCDEFHIOPQJRENZSUL QUERFNZQUARBNZTZHULUMUNUBUOQUCRANQUFRCNQUDRDNTZUOUGHUOUPUHIUIUJUK $. $( The order of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngle $p |- ( .<_ e. V -> .<_ = ( le ` W ) ) $= ( cple c1 c2 cdc cop odrngstr pleid cnx cfv csn ctp cts cds snsstp2 cmulr - cbs cplusg cun ssun2 sseqtr4i sstri strfv ) FHJGKKLMNABCDEFHIOPQJRFNZSQUA + cbs cplusg cun ssun2 sseqtrri sstri strfv ) FHJGKKLMNABCDEFHIOPQJRFNZSQUA RENZULQUBRBNZTZHUMULUNUCUOQUERANQUFRCNQUDRDNTZUOUGHUOUPUHIUIUJUK $. $( The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015.) $) odrngds $p |- ( D e. V -> D = ( dist ` W ) ) $= ( cds c1 c2 cdc cop odrngstr dsid cnx cfv csn ctp cts cple snsstp3 cplusg - cbs cmulr cun ssun2 sseqtr4i sstri strfv ) BHJGKKLMNABCDEFHIOPQJRBNZSQUAR + cbs cmulr cun ssun2 sseqtrri sstri strfv ) BHJGKKLMNABCDEFHIOPQJRBNZSQUAR ENZQUBRFNZULTZHUMUNULUCUOQUERANQUDRCNQUFRDNTZUOUGHUOUPUHIUIUJUK $. $} @@ -205196,7 +205204,7 @@ is always a subcategory (and it is full, meaning that all morphisms of $( A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017.) $) homarw $p |- ( X H Y ) C_ A $= - ( co crn cuni ovssunirn arwval sseqtr4i ) DECHCIJACDEKABCFGLM $. + ( co crn cuni ovssunirn arwval sseqtrri ) DECHCIJACDEKABCFGLM $. $} ${ @@ -213413,7 +213421,7 @@ net proof size (existence part)? $) U. { y e. F | ( y i^i x ) = (/) } ) >. } ) ) $= ( vf vo wcel cvv cnx cfv cop cpr cv wceq wa opeq2d cbs cts cple coc cin cordt c0 crab cuni cmpt cun elex cipo wss copab csb cxp xpex simpl prss - vex sylibr ssopab2i df-xp sseqtr4i ssexi a1i sseq2 anbi1d syl6eqr simpr + vex sylibr ssopab2i df-xp sseqtrri ssexi a1i sseq2 anbi1d syl6eqr simpr opabbidv fveq2d preq12d id rabeq unieqd mpteq12dv adantr uneq12d df-ipo csbied2 prex unex fvmpt syl5eq syl ) CFKCLKZDMUANZCOZMUBNZEUFNZOZPZMUCN ZEOZMUDNZACBQZAQZUEUGRZBCUHZUIZUJZOZPZUKZRCFULWHDCUMNXFGICJWSWRPZIQZUNZ @@ -213439,7 +213447,7 @@ net proof size (existence part)? $) ipolerval $p |- ( F e. V -> { <. x , y >. | ( { x , y } C_ F /\ x C_ y ) } = ( le ` I ) ) $= ( wcel cv cpr wss wa copab cnx cfv cop cple wceq cvv vex c1 cbs cts cordt - coc cin c0 crab cuni cmpt cun simpl prss sylibr ssopab2i sseqtr4i sqxpexg + coc cin c0 crab cuni cmpt cun simpl prss sylibr ssopab2i sseqtrri sqxpexg cxp df-xp ssexg sylancr cdc ipostr pleid csn snsspr1 ssun2 sstri syl eqid strfv ipoval fveq2d eqtr4d ) CEGZAHZBHZICJZVOVPJZKZABLZMUANCOMUBNVTUCNZOI ZMPNVTOZMUDNACVPVOUEUFQBCUGUHUIZOZIZUJZPNZDPNVNVTRGZVTWHQVNVTCCUQZJWJRGWI @@ -217419,18 +217427,18 @@ proposition to be be proved (the first four hypotheses tell its values FAEVCVJVEZYKVHSZYSYJYJVIVKYIYJYKMYIYJUUTVLZYIYKUVCVLZYIVMZVNYIUUEYJVAYIYJ VTHMVTHUUEYJQUVEVOYJMVPVQZWAVRYIYQROVAJZYIUUSYQRHUUTYJVHSVSWBZYIOYMWCLZAU UHYNYIOYNUHJZWDLZAYNTZUVKAYNTYIYNYBHZUVNYEYAUVOYHADEWEWFAYNWGSYIUVMUVKAYN - YIUVMOYLWDLZUVKYIUVLYLOWDYEYAUVLYLQYHADEWHWFWIYIYLWJHUVPUVKQYIYJYKUVAYIYK - UVCVFZWLOYLWKSWMWNWOZXBWPYIUUDYRUUGYTBYIBUAYNDOYQUVJYIUUHOYQWCLZHZIYCYDUU - HOYJWDLZHZUUHYNJUUHDJQYCYDYAYHUVTWQYCYDYAYHUVTWRYIUWBUVTYIUWAUVSUUHYIYJWJ - HUWAUVSQUVAOYJWKSZWSWTADEUUHXCXAXDYIUUGYSYJKLZBYNOYJKLZPZJYTYIYMUWDUUFUWF - YIUUEUWEBYNYIUUEYJUWEUVHYIYJUVEXEXFXGYIYMYKYJKLZMNLUWDYIYLUWGMNYIYJYKUVEU - VFXHXKYIYKYJMUVFUVEUVGXIWMXJYIBUAEYNYJOYSYIYSRUVIUVDVSWBZUVAYIUUHOYSWCLZH - ZIZUUHYJKLYNJZUUHEJZUWKYCYDUUHOYKWDLZHZUWLUWMQYCYDYAYHUWJWQYCYDYAYHUWJWRY - IUWOUWJYIUWNUWIUUHYIYKWJHUWNUWIQUVQOYKWKSZWSWTADEUUHXLXAXMXNXFXOWMYIABYNC - OYMUCFGUUNYIYMRUVIYIYLVBHYMRHYIYJYKUUTUVCXPYLVHSVSWBUVRXQYIUUBYRUUCYTBYIA - BDCOYQUCFGUUNUVJYIUWAADTZUVSADTYIYCUWQYAYCYDYHXRADWGSYIUWAUVSADUWCWNWOXQY - IABECOYSUCFGUUNUWHYIUWNAETZUWIAETYIYDUWRYAYCYDYHXTAEWGSYIUWNUWIAEUWPWNWOX - QXOXS $. + YIUVMOYLWDLZUVKYIUVLYLOWDYEYAUVLYLQYHAADEWHWFWIYIYLWJHUVPUVKQYIYJYKUVAYIY + KUVCVFZWLOYLWKSWMWNWOZXBWPYIUUDYRUUGYTBYIBUAYNDOYQUVJYIUUHOYQWCLZHZIYCYDU + UHOYJWDLZHZUUHYNJUUHDJQYCYDYAYHUVTWQYCYDYAYHUVTWRYIUWBUVTYIUWAUVSUUHYIYJW + JHUWAUVSQUVAOYJWKSZWSWTADEUUHXCXAXDYIUUGYSYJKLZBYNOYJKLZPZJYTYIYMUWDUUFUW + FYIUUEUWEBYNYIUUEYJUWEUVHYIYJUVEXEXFXGYIYMYKYJKLZMNLUWDYIYLUWGMNYIYJYKUVE + UVFXHXKYIYKYJMUVFUVEUVGXIWMXJYIBUAEYNYJOYSYIYSRUVIUVDVSWBZUVAYIUUHOYSWCLZ + HZIZUUHYJKLYNJZUUHEJZUWKYCYDUUHOYKWDLZHZUWLUWMQYCYDYAYHUWJWQYCYDYAYHUWJWR + YIUWOUWJYIUWNUWIUUHYIYKWJHUWNUWIQUVQOYKWKSZWSWTADEUUHXLXAXMXNXFXOWMYIABYN + COYMUCFGUUNYIYMRUVIYIYLVBHYMRHYIYJYKUUTUVCXPYLVHSVSWBUVRXQYIUUBYRUUCYTBYI + ABDCOYQUCFGUUNUVJYIUWAADTZUVSADTYIYCUWQYAYCYDYHXRADWGSYIUWAUVSADUWCWNWOXQ + YIABECOYSUCFGUUNUWHYIUWNAETZUWIAETYIYDUWRYAYCYDYHXTAEWGSYIUWNUWIAEUWPWNWO + XQXOXS $. $( Obsolete version of ~ gsumccat as of 13-Jan-2024. Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) 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(Contributed by Mario ( vg wcel cvv cv cfv wa cbs vs cpr wss copab wceq elex fvexi ssex cminusg co cplusg simpl fveq2d syl6eqr sseq2d fveq1d eqidd oveq123d simpr eleq12d cqg anbi12d opabbidv df-eqg cxp xpex vex prss sylibr ssopab2i df-xp ssexi - sseqtr4i ovmpoa syl5eq syl2an ) FHOFPOZEPOZDAQZBQZUBZIUCZVSGRZVTCUJZEOZSZ + sseqtrri ovmpoa syl5eq syl2an ) FHOFPOZEPOZDAQZBQZUBZIUCZVSGRZVTCUJZEOZSZ ABUDZUEEIUCFHUFEIIFTJUGZUHVQVRSDFEVAUJWGMNUAFEPPWANQZTRZUCZVSWIUIRZRZVTWI UKRZUJZUAQZOZSZABUDWGVAWIFUEZWPEUEZSZWRWFABXAWKWBWQWEXAWJIWAXAWJFTRIXAWIF TWSWTULZUMJUNUOXAWOWDWPEXAWMWCVTVTWNCXAWNFUKRCXAWIFUKXBUMLUNXAVSWLGXAWLFU @@ -221970,7 +221978,7 @@ by a normal subgroup (resp. two-sided ideal). (Contributed by Mario cycsubg $p |- ( ( G e. Grp /\ A e. X ) -> ran F = |^| { s e. ( SubGrp ` G ) | A e. s } ) $= ( cgrp wcel wa crn cv csubg cfv crab cint wss wi ssintab cycsubgss mpgbir - cab df-rab inteqi sseqtr4i a1i cycsubgcl eleq2 elrab sylibr intss1 eqssd + cab df-rab inteqi sseqtrri a1i cycsubgcl eleq2 elrab sylibr intss1 eqssd syl ) EKLBFLMZDNZBGOZLZGEPQZRZSZURVCTUQURUSVALUTMZGUEZSZVCURVFTVDURUSTUAG VDGURUBABUSCDEFHIJUCUDVBVEUTGVAUFUGUHUIUQURVBLZVCURTUQURVALBURLZMVGABCDEF HIJUJUTVHGURVAUSURBUKULUMURVBUNUPUO $. @@ -226176,22 +226184,22 @@ operation is a permutation group (group consisting of permutations), see 27-Aug-2015.) $) psgnuni $p |- ( ph -> ( -u 1 ^ ( # ` W ) ) = ( -u 1 ^ ( # ` X ) ) ) $= ( cfv cexp co wcel syl wceq syl2anc c1 cneg chash cword cn0 lencl m1expcl - cz nn0zd zcnd cc cc0 wne neg1cn neg1ne0 expne0i mp3an12i cdiv m1expaddsub - caddc cmin wa expsub mpanl12 eqtr3d creverse cconcat revcl ccatlen revlen - oveq2d eqtrd ccatcl cgsu cplusg c0g cid cres fveq2d eqid symgtrinv eqtr2d - cminusg cgrp cbs symggrp cmnd grpmnd symgtrf sswrd sseldi gsumwcl grprinv - wss ax-mp gsumccat syl3anc symgid 3eqtr4d psgnunilem4 diveq1d ) AUAUBZFUC - NZOPZXBGUCNZOPZAXDAXCUHQZXDUHQAXCAFCUDZQZXCUEQKCFUFRUIZXCUGRUJAXFAXEUHQZX - FUHQAXEAGXHQZXEUEQLCGUFRUIZXEUGRUJXBUKQZXBULUMZAXKXFULUMUNUOXMXBXEUPUQAXB - XCXEUTPZOPZXDXFURPZUAAXBXCXEVAPOPZXQXRAXGXKXSXQSXJXMXCXEUSTAXGXKXSXRSZXJX - MXNXOXGXKVBXTUNUOXBXCXEVCVDTVEAXBFGVFNZVGPZUCNZOPXQUAAYCXPXBOAYCXCYAUCNZU - TPZXPAXIYAXHQZYCYESKAXLYFLCGVHRZCFYAVITAYDXEXCUTAXLYDXESLCGVJRVKVLVKABCDE - YBHIJAXIYFYBXHQKYGCFYAVMTADFVNPZDYAVNPZDVONZPZDVPNZDYBVNPZVQBVRZAYKYHYHDW - CNZNZYJPZYLAYIYPYHYJAYPDGVNPZYONZYIAYHYRYOMVSABEQZXLYSYISJLBCDYOEGIHYOVTZ - WATWBVKADWDQZYHDWENZQZYQYLSAYTUUBJBDEHWFRZADWGQZFUUCUDZQZUUDAUUBUUFUUEDWH - RZAXHUUGFCUUCWNXHUUGWNUUCBCDIHUUCVTZWICUUCWJWOZKWKZUUCDFUUJWLTUUCYJDYOYHY - LUUJYJVTZYLVTUUAWMTVLAUUFUUHYAUUGQYMYKSUUIUULAXHUUGYAUUKYGWKUUCYJDFYAUUJU - UMWPWQAYTYNYLSJBDEHWRRWSWTVEVEXA $. + cz nn0zd zcnd cc cc0 wne neg1cn neg1ne0 expne0i mp3an12i cmin m1expaddsub + caddc cdiv wa expsub mpanl12 creverse cconcat revcl ccatlen revlen oveq2d + eqtr2d ccatcl cgsu cplusg c0g cid cres cminusg fveq2d eqid symgtrinv cgrp + cbs symggrp cmnd grpmnd 3syl wss symgtrf sswrd ax-mp sseldi gsumwcl eqtrd + grprinv gsumccat syl3anc symgid 3eqtr4d psgnunilem4 3eqtr3d diveq1d ) AUA + UBZFUCNZOPZXCGUCNZOPZAXEAXDUHQZXEUHQAXDAFCUDZQZXDUEQKCFUFRUIZXDUGRUJAXGAX + FUHQZXGUHQAXFAGXIQZXFUEQLCGUFRUIZXFUGRUJXCUKQZXCULUMZAXLXGULUMUNUOXNXCXFU + PUQAXCXDXFURPOPZXCXDXFUTPZOPZXEXGVAPZUAAXHXLXQXSSXKXNXDXFUSTAXHXLXQXTSZXK + XNXOXPXHXLVBYAUNUOXCXDXFVCVDTAXSXCFGVENZVFPZUCNZOPUAAXRYDXCOAYDXDYBUCNZUT + PZXRAXJYBXIQZYDYFSKAXMYGLCGVGRZCCFYBVHTAYEXFXDUTAXMYEXFSLCGVIRVJVKVJABCDE + YCHIJAXJYGYCXIQKYHCFYBVLTADFVMPZDYBVMPZDVNNZPZDVONZDYCVMPZVPBVQZAYLYIYIDV + RNZNZYKPZYMAYJYQYIYKAYQDGVMPZYPNZYJAYIYSYPMVSABEQZXMYTYJSJLBCDYPEGIHYPVTZ + WATVKVJADWBQZYIDWCNZQZYRYMSAUUAUUCJBDEHWDZRADWEQZFUUDUDZQZUUEAUUAUUCUUGJU + UFDWFWGZAXIUUHFCUUDWHXIUUHWHUUDBCDIHUUDVTZWICUUDWJWKZKWLZUUDDFUUKWMTUUDYK + DYPYIYMUUKYKVTZYMVTUUBWOTWNAUUGUUIYBUUHQYNYLSUUJUUMAXIUUHYBUULYHWLUUDYKDF + YBUUKUUNWPWQAUUAYOYMSJBDEHWRRWSWTWNXAXB $. $} ${ @@ -230061,45 +230069,45 @@ U C_ ( T .(+) U ) ) $= ( A ++ ( M o. ( reverse ` A ) ) ) .~ (/) ) $= ( wcel cfv cconcat co c0 vc va vb vm vu c2o cxp cword creverse ccom fviss wbr cid eqsstri sseli cv wi wceq id fveq2 rev0 syl6eq coeq2d co02 oveq12d - cs1 breq1d imbi2d ccatidid wer efger a1i wrd0 cvv efgrcl simprd eleqtrrid - erref eqbrtrid wa crn simprl wf revcl ad2antrl efgmf wrdco sylancl ccatcl - syl2anc adantr eleqtrrd chash cs2 cotp csplice cc0 cfz cuz lencl syl6eleq - cn0 nn0uz caddc ccatlen cz nn0zd uzid syl uzaddcl eqeltrd sylanbrc simprr - elfzuzb efgtval syl3anc ffvelrni s2cld ccatrid eqcomd oveq1d eqidd oveq2i - hash0 nn0cnd addid1d syl5req s1cld revs1 oveq1i ccatco s1co 3eqtrd oveq2d - splval2 revccat ccatass df-s2 syl6eqr 3eqtr2rd wfn cmpo efgtf 3syl fnovrn - ffn eqeltrrd efgi2 ersym ertr mpand expcom a2d wrdind mpcom ) EIUFUGZUHZP - EKPZEJEUIQZUJZRSZTFULZKUUQEKUUQUMQUUQLUUQUKUNUOUURUAUPZJUVCUIQZUJZRSZTFUL - ZUQUURTTRSZTFULZUQUURUBUPZJUVJUIQZUJZRSZTFULZUQUURUVJUCUPZVFZRSZJUVQUIQZU - JZRSZTFULZUQUURUVBUQUAUBUCEUUPUVCTURZUVGUVIUURUWBUVFUVHTFUWBUVCTUVETRUWBU - 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(This provides an explicit expression for the @@ -230189,16 +230197,16 @@ U C_ ( T .(+) U ) ) $= (Contributed by Mario Carneiro, 1-Oct-2015.) $) efgsval2 $p |- ( ( A e. Word W /\ B e. W /\ ( A ++ <" B "> ) e. dom S ) -> ( S ` ( A ++ <" B "> ) ) = B ) $= - ( cword wcel cs1 cconcat co cdm w3a cfv chash cmin cc0 caddc wceq efgsval - c1 3ad2ant3 cc cn0 lencl 3ad2ant1 nn0cnd ax-1cn pncan sylancl simp1 simp2 - s1cld ccatlen syl2anc oveq2i syl6eq oveq1d addid2d 3eqtr4d fveq2d cfzo cn - s1len 1nn eqeltri a1i lbfzo0 sylibr ccatval3 syl3anc s1fv 3ad2ant2 3eqtrd - eqtrd ) GRUEZUFZHRUFZGHUGZUHUIZKUJUFZUKZWRKULZWRUMULZUSUNUIZWRULZUOGUMULZ - UPUIZWRULZHWSWOXAXDUQWPABCDEFIJKLMNOWRPQRSTUAUBUCUDURUTWTXCXFWRWTXEUSUPUI - ZUSUNUIZXEXCXFWTXEVAUFUSVAUFXIXEUQWTXEWOWPXEVBUFWSRGVCVDVEZVFXEUSVGVHWTXB - XHUSUNWTXBXEWQUMULZUPUIZXHWTWOWQWNUFZXBXLUQWOWPWSVIZWTHRWOWPWSVJVKZRGWQVL - VMXKUSXEUPHWBZVNVOVPWTXEXJVQVRVSWTXGUOWQULZHWTWOXMUOUOXKVTUIUFZXGXQUQXNXO - WTXKWAUFZXRXSWTXKUSWAXPWCWDWEXKWFWGRGWQUOWHWIWPWOXQHUQWSHRWJWKWMWL $. + ( cword wcel cs1 cconcat co cdm cfv wceq wa chash cmin efgsval caddc s1cl + c1 ccatlen sylan2 s1len oveq2i syl6eq oveq1d cc lencl nn0cnd ax-1cn pncan + cc0 sylancl addid2d eqtr4d adantr eqtrd fveq2d cfzo adantl cn 1nn eqeltri + simpl lbfzo0 mpbir a1i ccatval3 syl3anc s1fv 3eqtrd sylan9eqr 3impa ) GRU + EZUFZHRUFZGHUGZUHUIZKUJUFZWQKUKZHULWRWNWOUMZWSWQUNUKZUSUOUIZWQUKZHABCDEFI + JKLMNOWQPQRSTUAUBUCUDUPWTXCVKGUNUKZUQUIZWQUKZVKWPUKZHWTXBXEWQWTXBXDUSUQUI + ZUSUOUIZXEWTXAXHUSUOWTXAXDWPUNUKZUQUIZXHWOWNWPWMUFZXAXKULHRURZRRGWPUTVAXJ + USXDUQHVBZVCVDVEWNXIXEULWOWNXIXDXEWNXDVFUFUSVFUFXIXDULWNXDRGVGVHZVIXDUSVJ + VLWNXDXOVMVNVOVPVQWTWNXLVKVKXJVRUIUFZXFXGULWNWOWCWOXLWNXMVSXPWTXPXJVTUFXJ + USVTXNWAWBXJWDWEWFRGWPVKWGWHWOXGHULWNHRWIVSWJWKWL $. $( The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (Contributed @@ -230273,45 +230281,44 @@ U C_ ( T .(+) U ) ) $= efgsp1 $p |- ( ( F e. dom S /\ A e. ran ( T ` ( S ` F ) ) ) -> ( F ++ <" A "> ) e. dom S ) $= ( vi va cdm wcel cfv crn wa cs1 cconcat co cword c0 csn cdif cc0 cv chash - cmin cfzo wral wne efgsdm simp1bi adantr eldifad cfz c2o cxp cotp csplice - c1 cs2 cmpo wceq wf crab efgsf fdmi feq2i mpbir ffvelrni efgtf syl simprd - frnd simpr sseldd s1cld ccatcl syl2anc cn caddc ccatlen oveq2i syl6eq cn0 - s1len lencl nn0p1nn eqeltrd cfn wb wrdfin hashnncl mpbid eldifsn sylanbrc - 3syl eldifsni mpbird lbfzo0 sylibr ccatval1 syl3anc simp2bi simp3bi sseli - cun w3a fzo0ss1 syl3an3 cuz wss cz cle elfzoel2 peano2zm zred lem1d eluz2 - wbr syl3anbrc fzoss2 fveq2d rneqd eleq12d elfzo1elm1fzo0 ralbidva addid2d - nn0cnd 1nn eqeltri a1i ccatval3 eqtr3d s1fv adantl fzo0end efgsval eqtr4d - 3expa 3eltr4d fveq2 fvoveq1 ralsn ralunb oveq2d syl6eleq fzosplitsn eqtrd - fvex nnuz raleqdv ) OJUGZUHZGOJUIZKUIZUJZUHZUKZOGULZUMUNZRUOZUPUQZURZUHZU - SUVPUIZHUHUEUTZUVPUIZUWBVOVBUNZUVPUIZKUIZUJZUHZUEVOUVPVAUIZVCUNZVDZUVPUVH - 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(Contributed by Mario Carneiro, 1-Oct-2015.) 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efgsval is uniquely determined, given the ending representation. @@ -230670,89 +230677,89 @@ an extension of the previous (inserting an element and its inverse at efgredlemc $p |- ( ph -> ( P e. 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KTUEUHUWGWNXOXIVIUYPUEWXFNUWHUWIXSWXDWXGUYPVLZWXIYGZYGZBCDEFGHIWXGJKL + MNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMAUFWPZNVKZVTVKVWGVTVKUWJUWKWXRUGWPZ + NVKVMVQWXQVKVQWXSVKVMUWLUWLUGUYPWKUFUYPWKWXAWXOUNYTAUYQWXAWXOUOYTAVVB + WXAWXOUPYTAVWGVWHVMWXAWXOUQYTAUXJUWOWXAWXOURYTUSUTAUYHWXAWXOVAYTAVURW + XAWXOVBYTAVWLWXAWXOVCYTAVWTWXAWXOVDYTAVWGVWIVMWXAWXOVEYTAVWHVWQVMWXAW + XOVFYTAUYBUWOWXAWXOVGYTAWXAWXOUWMWXDWXNWXIUWPWXPWXFWXHWXDWXNWXIUWQUWR + UWSUWTUXAUXBUXCUXDUXCUXD $. $( The reduced word that forms the base of the sequence in ~ efgsval is uniquely determined, given the ending representation. (Contributed @@ -231097,28 +231104,28 @@ an extension of the previous (inserting an element and its inverse at ZUQZVWIVXEVYOWSVXBVWKVXDXFYIUTZUQZVYPVXDYIUTZUQVYRVWKVXCVYSUQVVOYJUQZVY TVWKVXCYJVYSVWKVVBVXCYJUQVYAVURGYKYBZYLYMVWHWUBVUQVWIVVOVVSUVEWCZVVOXFV XCYNWQVWKVYPUYSXGUTZHXGUTZXTVLZWUAVWKVUTVVAVYPWUGWSVUQVUTVWJVVGXSVYGVUR - UYSHYOWQVWKWUEWUAUQWUFYJUQZWUGWUAUQVWKWUEVXCVVSXTVLZWUAVWKVVBVVCWUEWUIW - SVYAVYBVURGUXQYOWQVWKVVSVXCXTVLZVVOVXCXTVLZYIUTZWUIWUAVWKVVSVVOYIUTUQZV - XCUVFUQWUJWULUQVWHWUMVUQVWIVVOXFVVSUVGWCVWKVXCWUCUVJVXCVVOVVSUVHWQVWKVV - SVXCVWKVVSVWKVVCVVSYJUQVYBVURUXQYKYBZYPVWKVXCWUCYPZYQVWKWUKVXDYIVWKVVOV - XCVWKVVOWUDYPWUOYQUVKUVLYRVWKVVAWUHVYGVURHYKYBWUFVXDWUEYNWQYRVXDXFVYPUV - MUVNZVWRBCDEVVPJLQRTVXDUAUYTUBUCUDUEYEYDVWKVXGYSVXNVXIUYTVXDVXDVURVYDYS - VUSUQVWKVURUWHVOVWKVXQVVAVXNVUSUQVYFVYGVURVXKHWPWQVYEVWKVXGYSVKVLZVXNVK - VLVXGVXNVKVLZVXGVXKVKVLZHVKVLZUYTVWKWUQVXGVXNVKVWKVXRWUQVXGWSVYDVURVXGU - VOYBYFVWKVXRVXQVVAWUTWURWSVYDVYFVYGVURVXGVXKHYCYDVWKWUSUYSHVKVWKWUSGVXF - VXKVKVLZVKVLZUYSVWKVVBVXTVXQWUSWVBWSVYAVYCVYFVURGVXFVXKYCYDVWKWVAUXQGVK - VWKWVAUXQVVSYAVLZUXQVWKVVCVWHVVSVVTUQZWVAWVCWSVYBVWQVWKVVSVYSUQWVDVWKVV - SYJVYSWUNYLYMXFVVSUVPYBVURUXQVVOVVSUVQYDVWKVVCWVCUXQWSVYBVURUXQUVRYBYGY - HYGYFUVSVWKVXGXGUTZVXCVXFXGUTZXTVLZVXDVWKVVBVXTWVEWVGWSVYAVYCVURGVXFYOW - QVWKWVFVVOVXCXTVWKVVCVWHWVFVVOWSVYBVWQVURUXQVVOUVTWQYHUWAVWKVXDYSXGUTZX - TVLVXDXFXTVLVXDWVHXFVXDXTUWIUWBVWKVXDVWKVXDVWKVXCVVOWUCWUDUWCYPUWDUWEUW - FYGUWGVWKVWTVYQVURWGZXIZVYRVWIVXEVXAUQVWKVWSWVIUAVWTXMZWVJVXBVWSVWTUMUP - VYQVURUYTVWFXLVLXOWSWVKBCDEJLQRTUAUYTUMUPUBUCUDUEXPXBWVIUAVWTXQXRWUPVWR - VYQVURVXDVVPVWTUWJYDYRBCDEUYTVWNJLQRTUAUBUCUDUEUWKWQABCDEFGHIJKLMNOPQRS - TUAUXQVVQUBUCUDUEUFUGUHVPWBVWBUXQVVMSVMVVRVWLVVMUXQSUMYTUJYTUWRVVMVVQUX - QSUWLUWMUWNUWOUWPUWQUWSUXNSWTUQZUYFUYGUYJURZXNUXNUYGUAWTUQWVLVVLUAVUSWI - UBUWTUASWTUXAUXBUYCWVMUISWTUXOSWSZUXPUYGUYBUYJUAUXOSUXCWVNUYAUYIUJUAWVN - UXTUYHUXSUXOSUXQUXHUXDUXEUXFUXIYBUXGSUYDUXJYBUXK $. + VURUYSHYOWQVWKWUEWUAUQWUFYJUQZWUGWUAUQVWKWUEVXCVVSXTVLZWUAVWKVVBVVCWUEW + UIWSVYAVYBVURVURGUXQYOWQVWKVVSVXCXTVLZVVOVXCXTVLZYIUTZWUIWUAVWKVVSVVOYI + UTUQZVXCUVFUQWUJWULUQVWHWUMVUQVWIVVOXFVVSUVGWCVWKVXCWUCUVJVXCVVOVVSUVHW + QVWKVVSVXCVWKVVSVWKVVCVVSYJUQVYBVURUXQYKYBZYPVWKVXCWUCYPZYQVWKWUKVXDYIV + WKVVOVXCVWKVVOWUDYPWUOYQUVKUVLYRVWKVVAWUHVYGVURHYKYBWUFVXDWUEYNWQYRVXDX + FVYPUVMUVNZVWRBCDEVVPJLQRTVXDUAUYTUBUCUDUEYEYDVWKVXGYSVXNVXIUYTVXDVXDVU + RVYDYSVUSUQVWKVURUWHVOVWKVXQVVAVXNVUSUQVYFVYGVURVXKHWPWQVYEVWKVXGYSVKVL + ZVXNVKVLVXGVXNVKVLZVXGVXKVKVLZHVKVLZUYTVWKWUQVXGVXNVKVWKVXRWUQVXGWSVYDV + URVXGUVOYBYFVWKVXRVXQVVAWUTWURWSVYDVYFVYGVURVXGVXKHYCYDVWKWUSUYSHVKVWKW + USGVXFVXKVKVLZVKVLZUYSVWKVVBVXTVXQWUSWVBWSVYAVYCVYFVURGVXFVXKYCYDVWKWVA + UXQGVKVWKWVAUXQVVSYAVLZUXQVWKVVCVWHVVSVVTUQZWVAWVCWSVYBVWQVWKVVSVYSUQWV + DVWKVVSYJVYSWUNYLYMXFVVSUVPYBVURUXQVVOVVSUVQYDVWKVVCWVCUXQWSVYBVURUXQUV + RYBYGYHYGYFUVSVWKVXGXGUTZVXCVXFXGUTZXTVLZVXDVWKVVBVXTWVEWVGWSVYAVYCVURV + URGVXFYOWQVWKWVFVVOVXCXTVWKVVCVWHWVFVVOWSVYBVWQVURUXQVVOUVTWQYHUWAVWKVX + DYSXGUTZXTVLVXDXFXTVLVXDWVHXFVXDXTUWIUWBVWKVXDVWKVXDVWKVXCVVOWUCWUDUWCY + PUWDUWEUWFYGUWGVWKVWTVYQVURWGZXIZVYRVWIVXEVXAUQVWKVWSWVIUAVWTXMZWVJVXBV + WSVWTUMUPVYQVURUYTVWFXLVLXOWSWVKBCDEJLQRTUAUYTUMUPUBUCUDUEXPXBWVIUAVWTX + QXRWUPVWRVYQVURVXDVVPVWTUWJYDYRBCDEUYTVWNJLQRTUAUBUCUDUEUWKWQABCDEFGHIJ + KLMNOPQRSTUAUXQVVQUBUCUDUEUFUGUHVPWBVWBUXQVVMSVMVVRVWLVVMUXQSUMYTUJYTUW + RVVMVVQUXQSUWLUWMUWNUWOUWPUWQUWSUXNSWTUQZUYFUYGUYJURZXNUXNUYGUAWTUQWVLV + VLUAVUSWIUBUWTUASWTUXAUXBUYCWVMUISWTUXOSWSZUXPUYGUYBUYJUAUXOSUXCWVNUYAU + YIUJUAWVNUXTUYHUXSUXOSUXQUXHUXDUXEUXFUXIYBUXGSUYDUXJYBUXK $. $} $( Two extension sequences have related endpoints iff they have the same @@ -238235,55 +238242,54 @@ factorization into prime power factors (even if the exponents are pgpfaclem1 $p |- ( ph -> E. s e. Word C ( G dom DProd s /\ ( G DProd s ) = U ) ) $= ( cword wcel cdprd cdm wbr co wceq cv wrex csn cfv csubg cress ccyg - cpgp crn cin wss cbs cmre cgrp cacs subggrp syl eqid subgacs acsmre - wa 3syl subgbas eleqtrd mrcsncl syl2anc subsubg mpbid simpld oveq1i - wb simprd ressabs syl5eq cycsubgcyg2 eqeltrrd cprime pgpprm subgpgp - brelrng syl3anc elind oveq2 eleq1d elrab2 sylanbrc cc0 chash c0g wf - cfzo wrdf sylancl c1 caddc cn0 cun syl6eq oveq2d eqtrd cres cop wfn - ffn breqtrrd cvv fvex cn 3eqtrd sneqd dprdsubg ccntz ssrab3 fzodisj - cats1cld fss c0 cz lencl nn0zd fzosn ineq2d syl5reqr cconcat fveq2i - cs1 s1cld ccatlen s1len oveq2i cuz nn0uz fzosplitsn cats1un reseq1d - syl6eleq wn fzonel fsnunres dprdsn sylancr ssun2 snss mpbir fnressn - eleqtrrid fveq1i nn0cnd addid2d eqcomd fveq2d 1nn a1i lbfzo0 sylibr - eqeltrid ccatval3 s1fv mp1i opeq2d ablcntzd ineq12d syl6eqr 3eqtr4d - incom subg0 dmdprdsplit2 clsm dprdsplit cabl lsmcom subgss sseqtr4d - oveq12d subglsm breq2 eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHV - EVFZVGZKHVEVHZIVIZKRVJZUXLVGZKUXPVEVHZIVIZWJZRUXJVKADGHPVLZMVMZVBUS - AUYBKVNVMZVDZKUYBVOVHZVPVQVRZVSZVDZUYBDVDZAUYDUYBIVTZAUYBLVNVMZVDZU - YDUYJWJZAUYKLWAVMZWBVMVDZPUYNVDZUYLALWCVDZUYKUYNWDVMVDUYOAIUYCVDZUY - QUEIKLUGWEWFZUYNLUYNWGZWHUYKUYNWIWKAPIUYNUNAUYRIUYNVIUEIKLUGWLWFZWM - ZUYKPMUYNUHWNWOAUYRUYLUYMWTUEUYBIKLUGWPWFWQZWRZAVPUYFUYEALUYBVOVHZU - YEVPAVUEKIVOVHZUYBVOVHZUYELVUFUYBVOUGWSAUYRUYJVUGUYEVIUEAUYDUYJVUCX - AZIUYBKUYCXBWOXCAUYQUYPVUEVPVDUYSVUBPUYNLMUYTUHXDWOXEZAEXFVDZUYEVPV - DEUYEVQVGZUYEUYFVDAEKVQVGZVUJUCEKXGWFVUIAVULUYDVUKUCVUDEUYBKXHWOEUY - EVQXFVPXIXJXKKSVJZVOVHZUYGVDZUYHSUYBUYCDVUMUYBVIVUNUYEUYGVUMUYBKVOX - LXMUAXNXOZUUDZAXPGXQVMZXTVHZVURVLZHKXPHXQVMZXTVHZKXRVMZKUUAVMZAVVBD - HXSZDUYCVTVVBUYCHXSAUXKVVEVUQDHYAZWFVUOSUYCDUAUUBVVBDUYCHUUEYBZAUUF - VUSVURVURYCYDVHZXTVHZVSVUSVUTVSXPVURVVHUUCAVVIVUTVUSAVURUUGVDVVIVUT - VIAVURAGUXJVDZVURYEVDUSDGUUHWFZUUIVURUUJWFUUKUULZAVVBXPVVHXTVHZVUSV - UTYFZAVVAVVHXPXTAVVAVURUYBUUOZXQVMZYDVHZVVHAVVAGVVOUUMVHZXQVMZVVQHV - VRXQVBUUNAVVJVVOUXJVDZVVSVVQVIUSAUYBDVUPUUPZDGVVOUUQWOXCVVPYCVURYDU - YBUURZUUSYGYHAVURXPUUTVMZVDVVMVVNVIAVURYEVWCVVKUVAUVEXPVURUVBWFYIZV - VDWGZVVCWGZAKGHVUSYJZUXLUTAVWGGVURUYBYKZVLZYFZVUSYJZGAHVWJVUSAHVVRV - WJVBAVVJUYIVVRVWJVIUSVUPGUYBDUVCWOXCUVDAGVUSYLZVURVUSVDUVFVWKGVIAVV - JVUSDGXSVWLUSDGYAVUSDGYMWKXPVURUVGVUSGVURUYBUVHYBYIZYNZAKVWIHVUTYJZ - UXLAKVWIUXLVGZKVWIVEVHZUYBVIZAVURYOVDUYDVWPVWRWJGXQYPZVUDVURUYBKYOU - VIUVJZWRAVWOVURVURHVMZYKZVLZVWIAHVVBYLZVURVVBVDVWOVXCVIAUXKVVEVXDVU - QVVFVVBDHYMWKAVURVVNVVBVURVVNVDVUTVVNVTVUTVUSUVKVURVVNVWSUVLUVMVWDU - VOVVBVURHUVNWOAVXBVWHAVXAUYBVURAVXAXPVURYDVHZVVRVMZXPVVOVMZUYBAVXAV - URVVRVMVXFVURHVVRVBUVPAVURVXEVVRAVXEVURAVURAVURVVKUVQUVRUVSUVTXCAVV - JVVTXPXPVVPXTVHVDZVXFVXGVIUSVWAAVVPYQVDVXHAVVPYCYQVWBYCYQVDAUWAUWBU - WEVVPUWCUWDDGVVOXPUWFXJUYBYOVDVXGUYBVIAUYAMYPUYBYOUWGUWHYRUWIYSYIZY - NZAKVWGVEVHZKVWOVEVHZKVVDVWEUBAKVWGUXLVGVXKUYCVDVWNVWGKYTWFZAKVWOUX - LVGVXLUYCVDVXJVWOKYTWFUWJAUYBOVSZQVLVXKVXLVSZVVCVLUQAVXOOUYBVSVXNAV - XKOVXLUYBAVXKKGVEVHOAVWGGKVEVWMYHVAYIZAVXLVWQUYBAVWOVWIKVEVXIYHAVWP - VWRVWTXAYIZUWKOUYBUWNYGAVVCQAVVCLXRVMZQAUYRVVCVXRVIUEIKLVVCUGVWFUWO - WFUKUWLYSUWMUWPZAUXNUYBOKUWQVMZVHZUYBOFVHZIAUXNVXKVXLVXTVHOUYBVXTVH - ZVYAAVUSVUTVXTHKVVBVVGVVLVWDVXTWGZVXSUWRAVXKOVXLUYBVXTVXPVXQUXCAKUW - SVDOUYCVDUYDVYCVYAVIUBAVXKOUYCVXPVXMXEVUDVXTOUYBKVYDUWTXJYRAUYRUYJO - IVTVYAVYBVIUEVUHAOUYNIAOUYKVDOUYNVTUPUYNOLUYTUXAWFVUAUXBFVXTIUYBOKL - UGVYDULUXDXJURYRUXTUXMUXOWJRHUXJUXPHVIZUXQUXMUXSUXOUXPHKUXLUXEVYEUX - RUXNIUXPHKVEXLUXFUXGUXHUXI $. + wa cpgp crn cin wss cbs cmre cgrp cacs subggrp eqid subgacs acsmred + syl subgbas eleqtrd mrcsncl syl2anc wb subsubg simpld oveq1i simprd + mpbid ressabs syl5eq eqeltrrd cprime pgpprm subgpgp brelrng syl3anc + cycsubgcyg2 elind oveq2 eleq1d sylanbrc cats1cld cc0 chash cfzo c0g + elrab2 wf sylancl c1 caddc cn0 cun syl6eq oveq2d eqtrd cres cop wfn + wrdfn breqtrrd cvv fvex mpbir cn 3eqtrd sneqd dprdsubg ccntz ssrab3 + wrdf fss c0 fzodisj cz lencl nn0zd fzosn ineq2d syl5reqr cs1 fveq2i + cconcat s1cld ccatlen s1len oveq2i nn0uz fzosplitsn cats1un reseq1d + cuz syl6eleq wn fzonel fsnunres dprdsn sylancr ssun2 snss eleqtrrid + fnressn nn0cnd addid2d fveq2d fveq1i syl6reqr eqeltri ccatval3 s1fv + 1nn lbfzo0 mp1i opeq2d ablcntzd ineq12d incom subg0 syl6eqr 3eqtr4d + a1i dmdprdsplit2 clsm dprdsplit oveq12d cabl lsmcom subgss sseqtr4d + subglsm breq2 eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHVEVFZVGZK + HVEVHZIVIZKRVJZUXJVGZKUXNVEVHZIVIZVQZRUXHVKADGHPVLZMVMZVBUSAUXTKVNV + MZVDZKUXTVOVHZVPVRVSZVTZVDZUXTDVDZAUYBUXTIWAZAUXTLVNVMZVDZUYBUYHVQZ + AUYILWBVMZWCVMVDPUYLVDZUYJAUYIUYLALWDVDZUYIUYLWEVMVDAIUYAVDZUYNUEIK + LUGWFWJZUYLLUYLWGZWHWJWIAPIUYLUNAUYOIUYLVIUEIKLUGWKWJZWLZUYIPMUYLUH + WMWNAUYOUYJUYKWOUEUXTIKLUGWPWJWTZWQZAVPUYDUYCALUXTVOVHZUYCVPAVUBKIV + OVHZUXTVOVHZUYCLVUCUXTVOUGWRAUYOUYHVUDUYCVIUEAUYBUYHUYTWSZIUXTKUYAX + AWNXBAUYNUYMVUBVPVDUYPUYSPUYLLMUYQUHXIWNXCZAEXDVDZUYCVPVDEUYCVRVGZU + YCUYDVDAEKVRVGZVUGUCEKXEWJVUFAVUIUYBVUHUCVUAEUXTKXFWNEUYCVRXDVPXGXH + XJKSVJZVOVHZUYEVDZUYFSUXTUYADVUJUXTVIVUKUYCUYEVUJUXTKVOXKXLUAXSXMZX + NZAXOGXPVMZXQVHZVUOVLZHKXOHXPVMZXQVHZKXRVMZKUUAVMZAVUSDHXTZDUYAWAVU + SUYAHXTAUXIVVBVUNDHUUCWJVULSUYADUAUUBVUSDUYAHUUDYAZAUUEVUPVUOVUOYBY + CVHZXQVHZVTVUPVUQVTXOVUOVVDUUFAVVEVUQVUPAVUOUUGVDVVEVUQVIAVUOAGUXHV + DZVUOYDVDUSDGUUHWJZUUIVUOUUJWJUUKUULZAVUSXOVVDXQVHZVUPVUQYEZAVURVVD + XOXQAVURVUOUXTUUMZXPVMZYCVHZVVDAVURGVVKUUOVHZXPVMZVVMHVVNXPVBUUNAVV + FVVKUXHVDZVVOVVMVIUSAUXTDVUMUUPZDDGVVKUUQWNXBVVLYBVUOYCUXTUURZUUSYF + YGAVUOXOUVDVMZVDVVIVVJVIAVUOYDVVSVVGUUTUVEXOVUOUVAWJYHZVVAWGZVUTWGZ + AKGHVUPYIZUXJUTAVWCGVUOUXTYJZVLZYEZVUPYIZGAHVWFVUPAHVVNVWFVBAVVFUYG + VVNVWFVIUSVUMGUXTDUVBWNXBUVCAGVUPYKZVUOVUPVDUVFVWGGVIAVVFVWHUSDGYLW + JXOVUOUVGVUPGVUOUXTUVHYAYHZYMZAKVWEHVUQYIZUXJAKVWEUXJVGZKVWEVEVHZUX + TVIZAVUOYNVDUYBVWLVWNVQGXPYOZVUAVUOUXTKYNUVIUVJZWQAVWKVUOVUOHVMZYJZ + VLZVWEAHVUSYKZVUOVUSVDVWKVWSVIAUXIVWTVUNDHYLWJAVUOVVJVUSVUOVVJVDVUQ + VVJWAVUQVUPUVKVUOVVJVWOUVLYPVVTUVMVUSVUOHUVNWNAVWRVWDAVWQUXTVUOAVWQ + XOVUOYCVHZVVNVMZXOVVKVMZUXTAVXBVUOVVNVMVWQAVXAVUOVVNAVUOAVUOVVGUVOU + VPUVQVUOHVVNVBUVRUVSAVVFVVPXOXOVVLXQVHVDZVXBVXCVIUSVVQVXDAVXDVVLYQV + DVVLYBYQVVRUWCUVTVVLUWDYPUWMDGVVKXOUWAXHUXTYNVDVXCUXTVIAUXSMYOUXTYN + UWBUWEYRUWFYSYHZYMZAKVWCVEVHZKVWKVEVHZKVVAVWAUBAKVWCUXJVGVXGUYAVDVW + JVWCKYTWJZAKVWKUXJVGVXHUYAVDVXFVWKKYTWJUWGAUXTOVTZQVLVXGVXHVTZVUTVL + UQAVXKOUXTVTVXJAVXGOVXHUXTAVXGKGVEVHOAVWCGKVEVWIYGVAYHZAVXHVWMUXTAV + WKVWEKVEVXEYGAVWLVWNVWPWSYHZUWHOUXTUWIYFAVUTQAVUTLXRVMZQAUYOVUTVXNV + IUEIKLVUTUGVWBUWJWJUKUWKYSUWLUWNZAUXLUXTOKUWOVMZVHZUXTOFVHZIAUXLVXG + VXHVXPVHOUXTVXPVHZVXQAVUPVUQVXPHKVUSVVCVVHVVTVXPWGZVXOUWPAVXGOVXHUX + TVXPVXLVXMUWQAKUWRVDOUYAVDUYBVXSVXQVIUBAVXGOUYAVXLVXIXCVUAVXPOUXTKV + XTUWSXHYRAUYOUYHOIWAVXQVXRVIUEVUEAOUYLIAOUYIVDOUYLWAUPUYLOLUYQUWTWJ + UYRUXAFVXPIUXTOKLUGVXTULUXBXHURYRUXRUXKUXMVQRHUXHUXNHVIZUXOUXKUXQUX + MUXNHKUXJUXCVYAUXPUXLIUXNHKVEXKUXDUXEUXFUXG $. $} $( Lemma for ~ pgpfac . (Contributed by Mario Carneiro, 27-Apr-2016.) @@ -239222,7 +239228,7 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). issrg.p $e |- .+ = ( +g ` R ) $. issrg.t $e |- .x. = ( .r ` R ) $. issrg.0 $e |- .0. = ( 0g ` R ) $. - $( The predicate "is a semiring." (Contributed by Thierry Arnoux, + $( The predicate "is a semiring". (Contributed by Thierry Arnoux, 21-Mar-2018.) $) issrg $p |- ( R e. SRing <-> ( R e. CMnd /\ G e. Mnd /\ A. x e. B ( A. y e. B A. z e. B @@ -239679,8 +239685,8 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). srg1expzeq1.g $e |- G = ( mulGrp ` R ) $. srg1expzeq1.t $e |- .x. = ( .g ` G ) $. srg1expzeq1.1 $e |- .1. = ( 1r ` R ) $. - $( The exponentiation (by a nonnegative integer) of the unity element of a - (semi)ring, analogous to ~ mulgnn0z . (Contributed by AV, + $( The exponentiation (by a nonnegative integer) of the multiplicative + identity of a semiring, analogous to ~ mulgnn0z . (Contributed by AV, 25-Nov-2019.) $) srg1expzeq1 $p |- ( ( R e. SRing /\ N e. NN0 ) -> ( N .x. .1. ) = .1. ) $= ( csrg wcel cmnd cn0 co wceq srgmgp cbs cfv eqid ringidval mulgnn0z sylan @@ -239692,20 +239698,22 @@ elements and the subgroup containing only the identity ( ~ simpgnsgbid ). -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- The binomial theorem for semirings -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- -In this section, we prove the binomial theorem for semirings, ~ srgbinom , -which is a generalization of the binomial theorem for complex numbers, -~ binom : ` ( A + B ) ^ N ` is the sum from ` k = 0 ` to ` N ` of -` ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) `. -Notice that the binomial theorem would also hold in the non-unital case (that -is, in a "rg") and actually, the additive unit is not needed in its proof -either. Therefore, it could be proven for even more general cases. An example -would be the integrable nonnegative (resp. positive) bounded functions on -` RR `. + In this section, we prove the binomial theorem for semirings, ~ srgbinom , + which is a generalization of the binomial theorem for complex numbers, + ~ binom : ` ( A + B ) ^ N ` is the sum from ` k = 0 ` to ` N ` of + ` ( N _C k ) x. ( ( A ^ k ) x. ( B ^ ( N - k ) ) `. + + Note that the binomial theorem also holds in the non-unital case (that is, in + a "rg") and actually, the additive unit is not needed in its proof either. + Therefore, it can be proven in even more general cases. An example is the + "rg" (resp. "rg without a zero") of integrable nonnegative (resp. positive) + functions on ` RR `. + + Special cases of the binomial theorem are ~ csrgbinom (binomial theorem for + commutative semirings) and ~ crngbinom (binomial theorem for commutative + rings). -Special cases of the binomial theorem are ~ csrgbinom (binomial theorem for -commutative semirings) and ~ crngbinom (binomial theorem for commutative -rings). $) ${ @@ -250038,7 +250046,7 @@ U C_ ( N ` { X } ) ) -> ( U = ( N ` { X } ) \/ U = { .0. } ) ) $= jca alrimiv 3jca csn cdif wral simpr1 simpr2 simplr1 ssdifssd fvexi ssexg wn cbs sylancl simplr3 difssd simpr neldifsn nelne1 necomd psseq1 psseq1d fveq2 imbi12d spcgv syl3c simprbi simplr2 clss ad2antrr adantrr lspcl cun - dfpss3 ssun1 undif1 sseqtr4i lspssid simprr snssd sstrid syl3anc eqsstrrd + dfpss3 ssun1 undif1 sseqtrri lspssid simprr snssd sstrid syl3anc eqsstrrd unssd lspssp expr mtod ralrimiva wb islbs2 adantr mpbir3and impbida ) EUA KZABKZADLZACMZDUBZFUCZANZYGCMZDNZUDZFUFZOZYBYCPZYDYFYLYCYDYBABDEGHUEZUGYC YFYBABCDEGHIUHUGYNYKFYBYCYHYJYBYCYHOZYIDLZYIDQZYJYPEUIKZYGDLZYQYBYCYSYHEU @@ -260865,7 +260873,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldbas $p |- CC = ( Base ` CCfld ) $= ( cc cvv wcel ccnfld cbs cfv wceq c1 c3 cdc cop cnfldstr baseid cnx csn ctp cnex cun ssun1 sstri cplusg caddc cmulr cmul snsstp1 cstv ccj cts cabs cmin - ccom cmopn cple cle cds cunif cmetu df-cnfld sseqtr4i strfv ax-mp ) ABCADEF + ccom cmopn cple cle cds cunif cmetu df-cnfld sseqtrri strfv ax-mp ) ABCADEF GQADEBHHIJKLMNEFAKZOVBNUAFUBKZNUCFUDKZPZDVBVCVDUEVEVENUFFUGKOZRZDVEVFSVGVGN UHFUIUJUKZULFKNUMFUNKNUOFVHKPNUPFVHUQFKORZRDVGVISURUSTTUTVA $. @@ -260875,7 +260883,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldadd $p |- + = ( +g ` CCfld ) $= ( caddc cvv wcel ccnfld cplusg cfv addex c1 c3 cdc cop cnfldstr plusgid cnx wceq csn ctp cun ssun1 sstri cbs cmulr cmul snsstp2 cstv ccj cabs cmin ccom - cc cts cmopn cple cle cds cunif cmetu df-cnfld sseqtr4i strfv ax-mp ) ABCAD + cc cts cmopn cple cle cds cunif cmetu df-cnfld sseqtrri strfv ax-mp ) ABCAD EFOGADEBHHIJKLMNEFAKZPNUAFUJKZVBNUBFUCKZQZDVCVBVDUDVEVENUEFUFKPZRZDVEVFSVGV GNUKFUGUHUIZULFKNUMFUNKNUOFVHKQNUPFVHUQFKPRZRDVGVISURUSTTUTVA $. @@ -260885,7 +260893,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldmul $p |- x. = ( .r ` CCfld ) $= ( cmul cvv wcel ccnfld cmulr cfv wceq mulex cdc cop cnfldstr mulrid cnx csn c1 c3 ctp cun ssun1 sstri cbs cc cplusg snsstp3 cstv ccj cts cabs cmin ccom - caddc cmopn cple cle cds cunif cmetu df-cnfld sseqtr4i strfv ax-mp ) ABCADE + caddc cmopn cple cle cds cunif cmetu df-cnfld sseqtrri strfv ax-mp ) ABCADE FGHADEBOOPIJKLMEFAJZNMUAFUBJZMUCFUKJZVBQZDVCVDVBUDVEVEMUEFUFJNZRZDVEVFSVGVG MUGFUHUIUJZULFJMUMFUNJMUOFVHJQMUPFVHUQFJNRZRDVGVISURUSTTUTVA $. @@ -260895,7 +260903,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldcj $p |- * = ( *r ` CCfld ) $= ( ccj cvv wcel ccnfld cstv cfv wceq cc wf cjf cnex fex2 mp3an c1 c3 cop cnx csn ctp cun cdc cnfldstr starvid cbs cplusg caddc cmulr cmul ssun2 cts cabs - cmin ccom cmopn cple cunif cmetu ssun1 df-cnfld sseqtr4i sstri strfv ax-mp + cmin ccom cmopn cple cunif cmetu ssun1 df-cnfld sseqtrri sstri strfv ax-mp cle cds ) ABCZADEFGHHAIHBCZVGVFJKKHHABBLMADEBNNOUAPUBUCQEFAPRZQUDFHPQUEFUFP QUGFUHPSZVHTZDVHVIUIVJVJQUJFUKULUMZUNFPQUOFVDPQVEFVKPSQUPFVKUQFPRTZTDVJVLUR USUTVAVBVC $. @@ -260906,7 +260914,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldtset $p |- ( MetOpen ` ( abs o. - ) ) = ( TopSet ` CCfld ) $= ( cabs cmin ccom cmopn cfv cvv wcel ccnfld cts wceq fvex c1 cdc cop cnx csn c3 ctp cun sstri cnfldstr tsetid cple cle cds snsstp1 cunif cmetu ssun1 cbs - cc cplusg caddc cmulr cmul cstv ccj ssun2 df-cnfld sseqtr4i strfv ax-mp ) A + cc cplusg caddc cmulr cmul cstv ccj ssun2 df-cnfld sseqtrri strfv ax-mp ) A BCZDEZFGVDHIEJVCDKVDHIFLLQMNUAUBOIEVDNZPVEOUCEUDNZOUEEVCNZRZHVEVFVGUFVHVHOU GEVCUHENPZSZHVHVIUIVJOUJEUKNOULEUMNOUNEUONROUPEUQNPSZVJSHVJVKURUSUTTTVAVB $. @@ -260920,7 +260928,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldle $p |- <_ = ( le ` CCfld ) $= ( cle ctsr wcel ccnfld cple cfv wceq letsr c1 c3 cdc cop cnfldstr pleid cnx csn cts ctp cun sstri cabs cmin ccom cmopn cds snsstp2 cunif cmetu ssun1 cc - cbs cplusg caddc cmulr cmul cstv ccj ssun2 df-cnfld sseqtr4i strfv ax-mp ) + cbs cplusg caddc cmulr cmul cstv ccj ssun2 df-cnfld sseqtrri strfv ax-mp ) ABCADEFGHADEBIIJKLMNOEFALZPOQFUAUBUCZUDFLZVCOUEFVDLZRZDVEVCVFUFVGVGOUGFVDUH FLPZSZDVGVHUIVIOUKFUJLOULFUMLOUNFUOLROUPFUQLPSZVISDVIVJURUSUTTTVAVB $. @@ -260931,7 +260939,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring ( cabs cmin ccom cvv wcel ccnfld cds cfv wceq cc cr wf cnex cop cnx csn ctp c1 cun sstri cxp absf subf fco mp2an xpex reex fex2 mp3an cdc cnfldstr dsid c3 cts cmopn cple cle snsstp3 cunif cmetu ssun1 cbs cplusg caddc cmulr cmul - cstv ccj ssun2 df-cnfld sseqtr4i strfv ax-mp ) ABCZDEZVNFGHIJJUAZKVNLZVPDEK + cstv ccj ssun2 df-cnfld sseqtrri strfv ax-mp ) ABCZDEZVNFGHIJJUAZKVNLZVPDEK DEVOJKALVPJBLVQUBUCVPJKABUDUEJJMMUFUGVPKVNDDUHUIVNFGDRRUMUJNUKULOGHVNNZPOUN HVNUOHNZOUPHUQNZVRQZFVSVTVRURWAWAOUSHVNUTHNPZSZFWAWBVAWCOVBHJNOVCHVDNOVEHVF NQOVGHVHNPSZWCSFWCWDVIVJVKTTVLVM $. @@ -260941,7 +260949,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring cnfldunif $p |- ( metUnif ` ( abs o. - ) ) = ( UnifSet ` CCfld ) $= ( cabs cmin ccom cmetu cfv cvv wcel ccnfld cunif wceq c1 c3 cdc cop cnx csn fvex ctp cun ssun2 cnfldstr unifid cts cmopn cle cds cbs cplusg caddc cmulr - cple cc cmul cstv ccj df-cnfld sseqtr4i sstri strfv ax-mp ) ABCZDEZFGVBHIEJ + cple cc cmul cstv ccj df-cnfld sseqtrri sstri strfv ax-mp ) ABCZDEZFGVBHIEJ VADQVBHIFKKLMNUAUBOIEVBNPZOUCEVAUDENOUKEUENOUFEVANRZVCSZHVCVDTVEOUGEULNOUHE UINOUJEUMNROUNEUONPSZVESHVEVFTUPUQURUSUT $. @@ -263308,30 +263316,30 @@ According to Wikipedia ("Integer", 25-May-2019, cneg cid cdif cfn crab wfn eqid psgnfn fndm ssrab3 ressbas2 cnmsgnbas cvv fvexi ressplusg prex cmgp cnfldmul mgpplusg csubg cgrp psgndmsubg subggrp ccnfld syl cnmsgngrp a1i wral wfun fnfun funfn mpbi cgsu chash cexp cpmtr - wf crn cword wrex psgnvali wi cz lencl nn0zd m1expcl2 prcom adantl eleq1a - syl6eleq adantld rexlimdva syl5 ralrimiv ffnfv sylanbrc cconcat psgnvalii - ccatcl sylan2 cmnd symggrp grpmnd symgtrf sswrd sseli syl3an 3expb fveq2d - gsumccat caddc ccatlen oveq2d cc neg1cn ad2antll ad2antrl expaddd 3eqtr3d - cn0 eqtrd wb oveq12 eqeqan12d syl5ibrcom rexlimdvva anim12i reeanv sylibr - an4s impel isghmd ) AFNZKUABUBOZPDCEEUCZUDUDUIZUEZUUCBUFOZUGUUCDUFOQKUHZU - JUKUCULNZKUUFUUCEUUHKUUFUMZUNZUUCUUIQUUFAUUIBEKGUUFUOZUUIUOHUPZUUIEUQRURU - UCUUFDBIUUKUSRZCJUTUUCVANUUBDUBOQUUCDUFUUMVBUUCUUBBDVAIUUBUOZVCRUUEVANPCU - BOQUDUUDVDUUEPVLVEOZCVAJVLPUUOUUOUOVFVGVCRUUAUUCBVHONDVINABEFGHVJUUCBDIVK - VMCVINUUACJVNVOUUAEUUCUNZUUGEOZUUENZKUUCVPUUCUUEEWEUUPUUAEVQZUUPUUJUUSUUL - UUIEVRREVSVTVOUUAUURKUUCUUGUUCNZUUGBLUHZWASZQZUUQUUDUVAWBOZWCSZQZTZLAWDOW - FZWGZWHZUUAUURLAUUGUVHBEGUVHUOZHWIZUUAUVGUURLUVIUUAUVAUVINZTZUVFUURUVCUVN - UVEUUENZUVFUURWJUVMUVOUUAUVMUVDWKNZUVOUVMUVDUVHUVAWLZWMUVPUVEUUDUDUEUUEUV - DWNUUDUDWOWRVMWPUVEUUEUUQWQVMWSWTXAXBKUUCUUEEXCXDUUAUVGUAUHZBMUHZWASZQZUV - REOZUUDUVSWBOZWCSZQZTZTZMUVIWHLUVIWHZUUGUVRUUBSZEOZUUQUWBPSZQZUUTUVRUUCNZ - TZUUAUWGUWLLMUVIUVIUUAUVMUVSUVINZTZTZUWLUWGUVBUVTUUBSZEOZUVEUWDPSZQZUWQBU - VAUVSXESZWASZEOZUUDUXBWBOZWCSZUWSUWTUWPUUAUXBUVINUXDUXFQUVHUVAUVSXGAUVHBE - FUXBGUVKHXFXHUWQUXCUWREUUAUVMUWOUXCUWRQZUUABXINZUVMUVAUUFWGZNUWOUVSUXINUX - GUUABVINUXHABFGXJBXKVMUVIUXIUVAUVHUUFUGUVIUXIUGUUFAUVHBUVKGUUKXLUVHUUFXMR - ZXNUVIUXIUVSUXJXNUUFUUBBUVAUVSUUKUUNXRXOXPXQUWQUXFUUDUVDUWCXSSZWCSUWTUWQU - XEUXKUUDWCUWPUXEUXKQUUAUVHUVAUVSXTWPYAUWQUUDUVDUWCUUDYBNUWQYCVOUWOUWCYHNU - UAUVMUVHUVSWLYDUVMUVDYHNUUAUWOUVQYEYFYIYGUVCUWAUVFUWEUWLUXAYJUVCUWATZUVFU - WETUWJUWSUWKUWTUXLUWIUWREUUGUVBUVRUVTUUBYKXQUUQUVEUWBUWDPYKYLYRYMYNUWNUVJ - UWFMUVIWHZTUWHUUTUVJUWMUXMUVLMAUVRUVHBEGUVKHWIYOUVGUWFLMUVIUVIYPYQYSYT $. + wf crn cword wrex psgnvali wi cz lencl nn0zd m1expcl2 prcom syl6eleq 3syl + eleq1a adantld rexlimiv ralrimiv sylanbrc cconcat ccatcl psgnvalii sylan2 + syl5 ffnfv cmnd symggrp grpmnd symgtrf sswrd sseli gsumccat syl3an fveq2d + 3expb caddc ccatlen adantl oveq2d cc neg1cn cn0 ad2antll ad2antrl expaddd + 3eqtr3d oveq12 eqeqan12d an4s syl5ibrcom rexlimdvva anim12i reeanv sylibr + eqtrd wb impel isghmd ) AFNZKUABUBOZPDCEEUCZUDUDUIZUEZUUDBUFOZUGUUDDUFOQK + UHZUJUKUCULNZKUUGUUDEUUIKUUGUMZUNZUUDUUJQUUGAUUJBEKGUUGUOZUUJUOHUPZUUJEUQ + RURUUDUUGDBIUULUSRZCJUTUUDVANUUCDUBOQUUDDUFUUNVBUUDUUCBDVAIUUCUOZVCRUUFVA + NPCUBOQUDUUEVDUUFPVLVEOZCVAJVLPUUPUUPUOVFVGVCRUUBUUDBVHONDVINABEFGHVJUUDB + DIVKVMCVINUUBCJVNVOUUBEUUDUNZUUHEOZUUFNZKUUDVPUUDUUFEWEUUQUUBEVQZUUQUUKUU + TUUMUUJEVRREVSVTVOUUBUUSKUUDUUHUUDNZUUHBLUHZWASZQZUURUUEUVBWBOZWCSZQZTZLA + WDOWFZWGZWHZUUBUUSLAUUHUVIBEGUVIUOZHWIZUVKUUSWJUUBUVHUUSLUVJUVBUVJNZUVGUU + SUVDUVNUVEWKNZUVFUUFNUVGUUSWJUVNUVEUVIUVBWLZWMUVOUVFUUEUDUEUUFUVEWNUUEUDW + OWPUVFUUFUURWRWQWSWTVOXGXAKUUDUUFEXHXBUUBUVHUAUHZBMUHZWASZQZUVQEOZUUEUVRW + BOZWCSZQZTZTZMUVJWHLUVJWHZUUHUVQUUCSZEOZUURUWAPSZQZUVAUVQUUDNZTZUUBUWFUWK + LMUVJUVJUUBUVNUVRUVJNZTZTZUWKUWFUVCUVSUUCSZEOZUVFUWCPSZQZUWPBUVBUVRXCSZWA + SZEOZUUEUXAWBOZWCSZUWRUWSUWOUUBUXAUVJNUXCUXEQUVIUVBUVRXDAUVIBEFUXAGUVLHXE + XFUWPUXBUWQEUUBUVNUWNUXBUWQQZUUBBXINZUVNUVBUUGWGZNUWNUVRUXHNUXFUUBBVINUXG + ABFGXJBXKVMUVJUXHUVBUVIUUGUGUVJUXHUGUUGAUVIBUVLGUULXLUVIUUGXMRZXNUVJUXHUV + RUXIXNUUGUUCBUVBUVRUULUUOXOXPXRXQUWPUXEUUEUVEUWBXSSZWCSUWSUWPUXDUXJUUEWCU + WOUXDUXJQUUBUVIUVIUVBUVRXTYAYBUWPUUEUVEUWBUUEYCNUWPYDVOUWNUWBYENUUBUVNUVI + UVRWLYFUVNUVEYENUUBUWNUVPYGYHYRYIUVDUVTUVGUWDUWKUWTYSUVDUVTTZUVGUWDTUWIUW + RUWJUWSUXKUWHUWQEUUHUVCUVQUVSUUCYJXQUURUVFUWAUWCPYJYKYLYMYNUWMUVKUWEMUVJW + HZTUWGUVAUVKUWLUXLUVMMAUVQUVIBEGUVLHWIYOUVHUWELMUVJUVJYPYQYTUUA $. $} ${ @@ -281915,7 +281923,7 @@ a topological space (with the topology extractor function coming out the (Contributed by Mario Carneiro, 28-Aug-2015.) $) basdif0 $p |- ( ( B \ { (/) } ) e. TopBases <-> B e. TopBases ) $= ( vx vy c0 cdif ctb wcel cvv cun wss cv cin cuni wral ralbii wceq elinel2 - wb elsni syl csn ssun1 undif1 sseqtr4i snex unexg mpan2 ssexg sylancr cpw + wb elsni syl csn ssun1 undif1 sseqtrri snex unexg mpan2 ssexg sylancr cpw elex indif1 unieqi unidif0 eqtri sseq2i inss2 syl6eqss sstrid rgen ralunb 0ss mpbiran inundif raleqi 3bitr2i inss1 ralrimivw a1i isbasisg pm5.21nii difexg 3bitr4d ) ADUAZEZFGZAHGZAFGZVPAVOVNIZJVSHGZVQAAVNIVSAVNUBAVNUCUDVP @@ -283500,7 +283508,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ctopon cima cmre wrex wfun fnfun fvelima mpan cldmreon topontop wa adantl 0cld uncld ralrimivva eleq1 eleq2 raleqbi1dv 3anbi123d syl5ibcom rexlimiv 3jca cdif cpw simp1 simp2 uneq1 eleq1d uneq2 rspc2v com12 3ad2ant3 3impib - crab wi eqid mretopd simprd simpld cdm wss ssriv sseqtr4i funfvima2 mp2an + crab wi eqid mretopd simprd simpld cdm wss ssriv sseqtrri funfvima2 mp2an fndm eqeltrd impbii ) DHCUBIZUCZJZDCUDIZJZKDJZALZBLZMZDJZBDNZADNZOZWLELZH IZDPZEWJUEZXBHUFZWLXFHQUAZXGRQHUGSZEDWJHUHUIXEXBEWJXCWJJZXDWMJZKXDJZWRXDJ ZBXDNZAXDNZOXEXBXJXKXLXOCXCUJXJXCQJXLCXCUKZXCUNTXJXMABXDXDWPXDJWQXDJULXMX @@ -285206,7 +285214,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on iocpnfordt $p |- ( A (,] +oo ) e. ( ordTop ` <_ ) $= ( vx vy vz cxr wcel cpnf cioc co cle cfv cmpt crn cun ctb ctop eqid letop ax-mp sseldi wa cordt cv cmnf cico ctg wss leordtval eqeltrri tgclb mpbir - cioo bastg sseqtr4i ssun1 wceq wrex oveq1 rspceeqv elrnmpti sylibr adantr + cioo bastg sseqtrri ssun1 wceq wrex oveq1 rspceeqv elrnmpti sylibr adantr mpan2 ovex wn c0 cxp cpw clt df-ioc ixxf fdmi ndmov 0opn syl6eqel pm2.61i ) AEFZGEFZUAZAGHIZJUBKZFZVQWBVRVQBEBUCZGHIZLZMZBEUDWCUEILMZNZULMZNZWAVTWJ WJUFKZWAWJOFZWJWKUGWLWKPFWAWKPBWFWGWIWFQWGQWIQUHZRUIWJUJUKWJOUMSWMUNVQWHW @@ -285219,7 +285227,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on icomnfordt $p |- ( -oo [,) A ) e. ( ordTop ` <_ ) $= ( vx vy vz cmnf cxr wcel cico co cle cfv cmpt crn cun ctb ctop eqid letop ax-mp sseldi wa cordt cv cpnf cioc ctg wss leordtval eqeltrri tgclb mpbir - cioo bastg sseqtr4i ssun1 ssun2 wceq oveq2 rspceeqv mpan2 elrnmpti sylibr + cioo bastg sseqtrri ssun1 ssun2 wceq oveq2 rspceeqv mpan2 elrnmpti sylibr wrex ovex adantl wn c0 cxp cpw clt df-ico ixxf fdmi 0opn syl6eqel pm2.61i ndmov ) EFGZAFGZUAZEAHIZJUBKZGZVSWCVRVSBFBUCZUDUEILMZBFEWDHIZLZMZNZULMZNZ WBWAWKWKUFKZWBWKOGZWKWLUGWMWLPGWBWLPBWEWHWJWEQWHQWJQUHZRUIWKUJUKWKOUMSWNU @@ -285233,7 +285241,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on iooordt $p |- ( A (,) B ) e. ( ordTop ` <_ ) $= ( vx cxr cv cpnf cioc co cmpt crn cmnf cico cun cioo cfv ctb wcel ctop eqid sselii cle cordt ctg leordtval letop eqeltrri tgclb mpbir bastg ax-mp ssun2 - wss sseqtr4i ioorebas ) CDCEZFGHIJZCDKUOLHIJZMZNJZMZUAUBOZABNHZUTUTUCOZVAUT + wss sseqtrri ioorebas ) CDCEZFGHIJZCDKUOLHIJZMZNJZMZUAUBOZABNHZUTUTUCOZVAUT PQZUTVCULVDVCRQVAVCRCUPUQUSUPSUQSUSSUDZUEUFUTUGUHUTPUIUJVEUMUSUTVBUSURUKABU NTT $. @@ -286314,7 +286322,7 @@ converges to zero (in the standard topology on the reals) with this wi jcad adantl toptopon2 resttopon 3adant2 adantr simpr cnf2 syl3anc ccnv jca ex cin vex inex1 wrex simpl1 toponmax syl simpl3 ssexd elrest syl2anc imaeq2 eleq1d ralxfr2d wfun simplrr ffun inpreima cdm cnvimass cnvimarndm - 3syl sseqtr4i simpll2 imass2 sstrid df-ss eqtrd ralbidva simprr biantrurd + 3syl sseqtrri simpll2 imass2 sstrid df-ss eqtrd ralbidva simprr biantrurd cv fssd 3bitrrd bitrd simprl iscn 3bitr4d pm5.21ndd ) DEHIZJZBUAZAKZAEKZU BZCUCJZCUDZABUEZLZBCDUFMJZBCDAUGMZUFMJZXNXSXOXQXSXOUQXNBCDUHNXNXSBXPUIZXL LXQXNXSYBXLXSYBUQXNXSXPDUDZBBCDXPYCXPUJYCUJUKULNXJXLXMUMUNXPABUOUPURXNYAX @@ -287699,7 +287707,7 @@ require the space to be Hausdorff (which would make it the same as T_3), cmpcov2 $p |- ( ( J e. Comp /\ A. x e. X E. y e. J ( x e. y /\ ph ) ) -> E. s e. ( ~P J i^i Fin ) ( X = U. s /\ A. y e. s ph ) ) $= ( wcel cv wa wrex wral cuni wceq cpw cfn cin wss anbi1i an32 dfss3 ralbii - ccmp crab elunirab sylbbr ssrab2 unissi sseqtr4i a1i cmpcov mp3an2 sylan2 + ccmp crab elunirab sylbbr ssrab2 unissi sseqtrri a1i cmpcov mp3an2 sylan2 eqssd ssrab anass 3bitri 3bitr4i elfpw rexbii2 sylib ) DUCHZBIZCIHAJCDKZB ELZJEFIZMNZFACDUDZOPQZKZVGACVFLZJZFDOPQZKVEVBEVHMZNZVJVEEVNEVNRVCVNHZBELV EBEVNUAVPVDBEACVCDUEUBUFVNERVEVNDMEVHDACDUGZUHGUIUJUNVBVHDRVOVJVQVHDEFGUK @@ -288069,7 +288077,7 @@ require the space to be Hausdorff (which would make it the same as T_3), cuni iunxun simprr cmptop restrcl simprd 3syl nfcv nfcsb1v csbeq1a cbviun vex csbeq1 iunxsn eqtri simpl3 nfv nfel1 cbvral sylib ssun2 simprl sylibr nfov snss rspcdva eqeltrid unexg syl2anc resttop eqid restin unieqd inss2 - sseqtr4i restuni sylancl eqtr4d uneq2i ineq1i indir inss1 restabs syl3anc + sseqtrri restuni sylancl eqtr4d uneq2i ineq1i indir inss1 restabs syl3anc sstri a1i eqeltrd eqsstrri uncmp syl22anc exp32 a2d syl5 findcard2 mpcom a2i mpi ) DHIZBUBIZDCJKZLIZABUDZUEZBBMZDABCNZJKZLIZBUFYHYLYMYPOZYGYHYKUGY LGUHZBMZDAYRCNZJKZLIZOZOYLPBMZDPJKZLIZOZOYLUAUHZBMZDAUUHCNZJKZLIZOZOZYLUU @@ -288326,7 +288334,7 @@ require the space to be Hausdorff (which would make it the same as T_3), wb cmptop islp3 3expa notbid ralbidva sylan syl5bbr rexanali nne vex sneq weq difeq2d ineq2d eqeq1d spcev sylbi anim2i reximi sylbir ralimi cmpcov2 ex syl5 adantr sylbid 3adant3 elinel2 sseq2 biimpac infssuni ancoms an42s - anassrs sylanl2 0fin eleq1 mpbiri snfi unfi sylancl ssun1 undif1 sseqtr4i + anassrs sylanl2 0fin eleq1 mpbiri snfi unfi sylancl ssun1 undif1 sseqtrri ss2in mp2an 3sstr4i ssfi exlimiv anim12ci expl reximdva 3adant1 syld mt3i incom undir ) CUAIZBDJZBKILZUBZAMZBCUCUDUDIZADNZBFMZOZKIZFGMZPZYGLZFYHNZQ ZGCUEZKOZNZYLGYNYLLYHYNIZYLYGYJQZFYHNYQFYHYQLFGUFYGUGUHUIYGYJFYHUJUKUHUIY @@ -288419,7 +288427,7 @@ require the space to be Hausdorff (which would make it the same as T_3), (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 14-Aug-2015.) $) indisconn $p |- { (/) , A } e. Conn $= - ( cpr cconn wcel ctop ccld cfv cin cid wss indistop inss1 indislem sseqtr4i + ( cpr cconn wcel ctop ccld cfv cin cid wss indistop inss1 indislem sseqtrri c0 indisuni isconn2 mpbir2an ) OABZCDSEDSSFGZHZOAIGZBZJAKUASUCSTLAMNSUBAPQR $. @@ -289462,15 +289470,15 @@ require the space to be Hausdorff (which would make it the same as T_3), $( Define a space that is locally ` A ` , where ` A ` is a topological property like "compact", "connected", or "path-connected". A topological space is locally ` A ` if every neighborhood of a point - contains an open sub-neighborhood that is ` A ` in the subspace - topology. (Contributed by Mario Carneiro, 2-Mar-2015.) $) + contains an open subneighborhood that is ` A ` in the subspace topology. + (Contributed by Mario Carneiro, 2-Mar-2015.) $) df-lly $a |- Locally A = { j e. Top | A. x e. j A. y e. x E. u e. ( j i^i ~P x ) ( y e. u /\ ( j |`t u ) e. A ) } $. $( Define a space that is n-locally ` A ` , where ` A ` is a topological property like "compact", "connected", or "path-connected". A topological space is n-locally ` A ` if every neighborhood of a point - contains a sub-neighborhood that is ` A ` in the subspace topology. + contains a subneighborhood that is ` A ` in the subspace topology. The terminology "n-locally", where 'n' stands for "neighborhood", is not standard, although this is sometimes called "weakly locally ` A ` ". @@ -290294,7 +290302,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually elrabd elunii syl6eleqr ancoms adantl locfinbas ad3antrrr eleqtrrd simplr eleqtrd eluni2 sylib reximddv expr exlimdv n0 eliun 3imtr4g expimpd ssrdv syl5bi iunfi ssfi expcom sylan9 sylan2b rexlimdva snfi unfi sylancl ssun1 - ex mpd undif1 sseqtr4i jca ctop cmptop finlocfin 3expib syl impbid ) BUAL + ex mpd undif1 sseqtrri jca ctop cmptop finlocfin 3expib syl impbid ) BUAL ZABUBUCLZAMLZCDNZOZXKXLXOXKXLOZXMXNXPAPUDZUEZXQUFZMLZAXSQXMXPXRMLZXQMLXTX PCGRZUGZNZHRZIRZUHZPUIZHAUJZMLZIYBUKZOZGBULMUHZSZYAXLXKJITYJOIBSZJCUKYNXL YOJCAJRZIBCHEUMUNYJJIBCGEUOUPXPYLYAGYMYBYMLXPYBBQZYBMLZOZYLYAUQYBBURXPYSO @@ -293670,7 +293678,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= eqeltrrd imaiun imaeq2d syl5eqr simpl1 3ad2ant2 simp3 r19.21bi ralbidva 3bitr4g mpbird eqsstrrd uniiun syl6eq simpl opelxpd anbi12i coeq1 coeq2 simprll ovmpo ntrss2 sseqtrd imass2 simprlr eqsstrid eqeltrd ralrimivva - cnco sylan2b mpofun xpss12 dmmpo sseqtr4i funimassov sseq1 expr exlimdv + cnco sylan2b mpofun xpss12 dmmpo sseqtrri funimassov sseq1 expr exlimdv sylibr syldan expimpd rexlimdva ) AFDLUKZULUMZUNZUBUOZUNZUPZUYKCUQMUKZU RZUCUOZUSZUDUOZUYRUYPUTZFVAUTZUTZVBZFUYSULUMZVCVDZVEZUDUYKVFZVEZUCVGZVE ZUBUYIURZVHVIZVJZCDVKZBUOZVDZVUNKUQJUOZLUKZMVBZJEGVLUMZVMZUKZVBZVEZBGFV @@ -295907,7 +295915,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario $( Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.) $) hmphdis $p |- ( J ~= ~P A -> J = ~P X ) $= - ( vf vx cpw chmph wbr wss cuni pwuni pweqi sseqtr4i a1i chmeo co cv sylbi + ( vf vx cpw chmph wbr wss cuni pwuni pweqi sseqtrri a1i chmeo co cv sylbi wcel c0 wne hmph wex n0 elpwi wa cima crn imassrn wf unipw eqcomi hmeof1o wf1o f1of frn 3syl adantr sstrid imaex elpw sylibr hmeoopn mpbird ex syl5 vex ssrdv exlimiv eqssd ) BAGZHIZBCGZBVNJVMBBKZGVNBLCVODMNOVMBVLPQZUAUBZV @@ -297795,7 +297803,7 @@ also T_1 (because they are homeomorphic). (Contributed by Mario Carneiro, cif uzin ancoms syl5eq wfn ffn uzssz eqsstri ifcl uzid syl elin2d fnfvima eleqtrrd mp3an12i eqeltrd ralrimiva wb ineq1 eleq1d ralrn eqid fmpt sylib sylibr frnd eqsstrid uztrn2 ex ssrdv df-ss sseli fnfvelrn sylancr elrestr - adantl eqeltrrd wfun cdm ffun fdmi sseqtr4i funimass4 eqssd ) AFGZHIZBUAU + adantl eqeltrrd wfun cdm ffun fdmi sseqtrri funimass4 eqssd ) AFGZHIZBUAU BZHBUCZWTXBDXADJZBKZUDZIZXCXALGZBLGZXBXGMXAFUEZFUFUGFXJHNZXAXJOPFXJHUHQUI ZBAHCUJZDBXALLUKRWTXAXCXFWTXEXCGZDXASZXAXCXFNWTEJZHTZBKZXCGZEFSZXOWTXSEFW TXPFGZULZXRXPAUMUNZAXPUPZHTZXCYBXRXQAHTZKZYEBYFXQCUOYAWTYGYEMXPAUQURUSZHF @@ -298338,7 +298346,7 @@ is called nonprincipal (having empty intersection). Note that examples ufildr $p |- ( F e. ( UFil ` X ) -> ( J u. ( Clsd ` J ) ) = ~P X ) $= ( vx cufil cfv wcel ccld cun cpw wo wss cuni c0 syl5ibr cdif sseld ctop wb cv elssuni csn unieqi uniun 0ex unisn uneq2i un0 3eqtri eqtr2i ufilfil - cfil wceq filunibas syl syl5reqr sseq2d eqid cldss jaod wa ssun1 sseqtr4i + cfil wceq filunibas syl syl5reqr sseq2d eqid cldss jaod wa ssun1 sseqtrri ufilss cconn filconn conntop 3syl eqeltrid biimpa iscld2 syl2an2r difeq1d a1i eleq1d adantr bitr4d sylibrd orim12d mpd ex impbid elun velpw 3bitr4g eqrdv ) ACFGHZEBBIGZJZCKZWHEUAZBHZWLWIHZLZWLCMZWLWJHWLWKHWHWOWPWHWMWPWNWM @@ -298365,7 +298373,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the velpw eldifn sbcieg mpbird 0fin 0ex con2bii mpbi w3a ssfi expcom 3ad2ant3 wi con3d vex 3imtr4g cin eldifi fin1ai sylan 3adant3 inundif incom simprl eqeltrid simprr simpl3 ssdifd ssfid unfi syl2anc eqeltrrid expr orim2d ex - cun mpid anbi12i ioran bitr4i inex1 isfild adantr undif2 sseqtr4i sylancl + cun mpid anbi12i ioran bitr4i inex1 isfild adantr undif2 sseqtrri sylancl ssun2 nsyl ianor sylib elpwi adantl baib syl elpw2g mpbiri syl5bb orbi12d difss ralrimiva isufil sylanbrc csn snfi eleq2s mt2 uffixsn mtoi eq0rdv jca ) BFGHZIZABUBUCIZAUDZJKYFABUEUCIDUFZAIZBYIHZAIZLZDBUGZUHYGYFYIGIZMZDE @@ -302794,7 +302802,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the cv wi wa ccmn eqid tsmsval eleq2d ctopon cfbas wf wb istps sylib tsmsfbas ctps tsmslem1 fmpttd flffbas syl3anc cvv cpw cfn cin inex1g 3syl eqeltrid pwexg adantr rabexg syl ralrimivw wceq imaeq2 sseq1d rexrnmpt ccnv funmpt - wfun cdm ssrab2 ovex dmmpti sseqtr4i funimass3 mp2an sseq2i ss2rab 3bitri + wfun cdm ssrab2 ovex dmmpti sseqtrri funimass3 mp2an sseq2i ss2rab 3bitri mptpreima rexbii syl6bb imbi2d ralbidva anbi2d 3bitrd ) AGJIUBUCZTGBHJIBU PZUDZUEUCZUFZKHCHCUPXLUGZBHUHZUFZUIZUJUCZUKUCULZTZGFTZGDUPZTZXOUAUPZUMZYD UGZUAXSUNZUQZDKUOZURZYCYEXPXNYDTZUQBHUOZCHUNZUQZDKUOZURAXKYAGABCEFHIJKXSU @@ -303988,7 +303996,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed /\ E. w e. u ( w o. w ) C_ v ) ) ) } ) $= ( vx wcel cv cxp cpw wss wel wi wral cid cres w3a cab cvv pwexg ccnv ccom cin wrex cust df-ust id sqxpeqd pweqd sseq2d eleq1d raleqdv reseq2 sseq1d - wceq 3anbi1d 3anbi13d ralbidv 3anbi123d abbidv elex simp1 ss2abi sseqtr4i + wceq 3anbi1d 3anbi13d ralbidv 3anbi123d abbidv elex simp1 ss2abi sseqtrri df-pw sqxpexg 3syl ssexg sylancr fvmptd3 ) EDGZFECHZFHZVMIZJZKZVNVLGZBHZA HZKACLMZAVONZVRVSUCVLGAVLNZOVMPZVRKZVRUAVLGZVSVSUBVRKAVLUDZQZQZBVLNZQZCRV LEEIZJZKZWKVLGZVTAWLNZWBOEPZVRKZWEWFQZQZBVLNZQZCRZSUESFABCUFVMEUOZWJXACXC @@ -312870,7 +312878,7 @@ Normed space homomorphisms (bounded linear operators) qtopbaslem $p |- ( (,) " ( S X. S ) ) e. TopBases $= ( vx vy vz vw vv vu vt cioo wcel cv cin wral wa cxr wceq sseli cfv mp2an cxp cima cvv ctb iooex imaex cle wbr cif anim12i iooin syl2an ifcl ancoms - co cop df-ov opelxpi wfun cdm wss wi cr cpw wf ioof ax-mp xpss12 sseqtr4i + co cop df-ov opelxpi wfun cdm wss wi cr cpw wf ioof ax-mp xpss12 sseqtrri ffun fdmi funfvima2 syl eqeltrid an4s eqeltrd ralrimivva rgen2a wfn ineq1 wb ffn eleq1d ralbidv ralima fveq2 ineq1d ineq2 ineq2d ralxp bitri syl6bb syl6eqr mpbir fiinbas ) JAAUAZUBZUCKCLZDLZMZWQKZDWQNZCWQNZWQUDKJWPUEUFXCE @@ -313349,7 +313357,7 @@ Normed space homomorphisms (bounded linear operators) ( vx vy vz vw vu vv cioo cq cfv wss cxr cv wcel ax-mp wa wrex clt wbr cxp cima ctg crn wceq imassrn wf wfn co wral cr cpw ioof simpll elioo1 biimpa ffn w3a simp1d simp2d qbtwnxr syl3anc simplr simp3d reeanv df-ov 3ad2ant2 - cop opelxpi wfun cdm wi ffun qssre ressxr sstri xpss12 sseqtr4i funfvima2 + cop opelxpi wfun cdm wi ffun qssre ressxr sstri xpss12 sseqtrri funfvima2 mp2an fdmi syl eqeltrid 3ad2ant1 simp3lr simp3rl wb simp2l sseldi syl2anc simp2r mpbir3and simp3ll xrltled iooss1 simp3rr sstrd eleq2 sseq1 anbi12d cle iooss2 rspcev syl12anc rexlimdvv syl5bir mp2and ralrimiva ctb qtopbas @@ -313378,7 +313386,7 @@ Normed space homomorphisms (bounded linear operators) ( cioo ctg cfv cq cxp c2ndc wcel com cdom wbr cen cdm cn qnnen mp2an nnenom entri wss cxr cr crn cima eqid tgqioo ctb qtopbas ccrd cres wfo con0 omelon xpen xpnnen entr2i isnumi wfun wf ioof ffun ax-mp qssre ressxr sstri xpss12 - cpw fdmi sseqtr4i fores fodomnum mp2 domentr 2ndci eqeltri ) AUABCADDEZUBZB + cpw fdmi sseqtrri fores fodomnum mp2 domentr 2ndci eqeltri ) AUABCADDEZUBZB CZFVPVPUCUDVOUEGVOHIJZVPFGUFVOVNIJZVNHKJVQVNUGLGZVNVOAVNUHZUIZVRHUJGHVNKJVS UKVNMHVNMMEZMDMKJZWCVNWBKJNNDMDMULOUMQZPUNHVNUOOAUPZVNALZRWASSEZTVEZAUQWEUR WGWHAUSUTVNWGWFDSRZWIVNWGRDTSVAVBVCZWJDSDSVDOWGWHAURVFVGVNAVHOVNVOVTVIVJVNM @@ -316102,7 +316110,7 @@ Normed space homomorphisms (bounded linear operators) ord xle0neg1 sylanbrc iccssre iccneg negneg1e1 oveq2i syl6eleq xle0neg2 biimpa neeq2 mpbiri necomd ifnefalse id oveq2 1m0e1 oveq12d ax-1cn c0ex imp div0i breq12d cii iccpnfhmeo leisorel mtbid unitssre negnegd eqtr2d - recnd xnegneg iffalse pm2.61dan rnmpt sseqtr4i eqssi dffo2 mpbir2an w3a + recnd xnegneg iffalse pm2.61dan rnmpt sseqtrri eqssi dffo2 mpbir2an w3a mprgbir simpl3 ltled letrd iftrued breqtrrd lt0neg1d eqbrtrrid xlt0neg2 simpl2 mpbird simprbi xrltletrd simpll3 ltnegd xltneg breqan12d sylibrd ltnle 3expia rgen2a soisoi cxp cin elexi inex1 ssid leiso isores1 tsrps @@ -316474,7 +316482,7 @@ topological space to the reals is bounded (above). (Boundedness below bndth $p |- ( ph -> E. x e. RR A. y e. X ( F ` y ) <_ x ) $= ( vw wss cr cioo cmnf wcel clt cxr wa vv vu vz crn cuni cfv cle wral wrex cv wbr csn cxp cpw cfn cin ccn co wf ctg ctopon retopon eqeltri toponunii - cima cnf syl frnd wi wceq unieq imassrn unissi unirnioo sseqtr4i c1 caddc + cima cnf syl frnd wi wceq unieq imassrn unissi unirnioo sseqtrri c1 caddc id ltp1 wb ressxr peano2re sseldi elioomnf mpbir2and cop df-ov mnfxr snid elexi opelxpi sylancr wfun cdm ioof ffun ax-mp snssi mp2an fdmi funfvima2 xpss12 eqeltrid elunii syl2anc ssriv syl6eq sseq2d ineq1d rexeqdv imbi12d @@ -318600,7 +318608,7 @@ a tuple of a topological space (a member of ` TopSp ` , not ` Top ` ) mpbir3and eceq1 oveq1 oveq2d eceq1d opeq12d fmptco phtpcer eqtr2d erref wer csn cxp wbr eqid pcopt2 syl2anc c2 cdiv c4 cmul caddc eqcomd pcoass cle cif pcorev2 ertr2d ertr3d ertr4d pcopt ertrd erthi opeq2d mpteq2dva - pcohtpy rneqd rncoss sseqtr4i syl6eqss ) AIUBZGEUCUDZUEZGUFZLUGUDZUHZHY + pcohtpy rneqd rncoss sseqtrri syl6eqss ) AIUBZGEUCUDZUEZGUFZLUGUDZUHZHY QKLUIUDZUJZYTUJZYRUHZUKZULZFCUEZKFUFZHYTUJZYTUJZULZUMZUNZJAYNFUUFUUIYRU HZUUGYRUHZUKZULZUNUULAFUUNUUMUOUOIUUFQYRUOUPZUUNUOUPAUUGUUFUPZUQZLUGURZ UUGUOYRUSUTUUQUUMUOUPUUSUUTUUIUOYRUSUTVAAUUKUUPAUUKFUUFUUMHUUIKYTUJZYTU @@ -319084,7 +319092,7 @@ and so are its subrings (see ~ subrgcrng ), left modules over such isclmp.v $e |- V = ( Base ` W ) $. isclmp.s $e |- S = ( Scalar ` W ) $. isclmp.k $e |- K = ( Base ` S ) $. - $( The predicate "is a subcomplex module." (Contributed by NM, + $( The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) $) isclmp $p |- ( W e. CMod <-> ( ( W e. Grp /\ S = ( CCfld |`s K ) /\ K e. ( SubRing ` CCfld ) ) @@ -319736,7 +319744,7 @@ vector spaces (see ~ zclmncvs ), because the ring ZZ is not a division ring iscvsp.v $e |- V = ( Base ` W ) $. iscvsp.s $e |- S = ( Scalar ` W ) $. iscvsp.k $e |- K = ( Base ` S ) $. - $( The predicate "is a subcomplex vector space." (Contributed by NM, + $( The predicate "is a subcomplex vector space". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.) $) iscvsp $p |- ( W e. CVec <-> ( ( W e. Grp /\ ( S e. DivRing /\ S = ( CCfld |`s K ) ) @@ -324115,7 +324123,7 @@ can be the countable union of rare closed subsets (where rare means VPVPPUJOSVGVNADVDVEVNANVFVJABVTUKTVDVEVFULSVGVDVFVJAUMHCZVIVMVOUNUOVDVEVFUP VRVGAVBCZWBVDVEWCVFAUQTVJAVTURRVHVJVKAAUSWAUTVAVC $. - $( The predicate "is a subcomplex Hilbert space." A Hilbert space is a + $( The predicate "is a subcomplex Hilbert space". A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) $) @@ -326327,7 +326335,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically wb cin ccom cuni cima rnco2 wa c1st c2nd co ffvelrn elin2d 1st2nd2 fveq2d cop df-ov syl6eqr xp1st xp2nd iccssre syl2anc reex elpw2 sylibr ralrimiva eqsstrd wfn ffn fveq2 eleq1d ralrn mpbird wfun iccf ffun ax-mp rexpssxrxp - cdm frn inss2 sstri fdmi sseqtr4i syl6ss funimass4 sylancr eqsstrid sylib + cdm frn inss2 sstri fdmi sseqtrri syl6ss funimass4 sylancr eqsstrid sylib sspwuni ) DEFFGZUAZAHZIAUBJZFKZLWLUCFLWKWLIAJZUDZWMIAUEWKWOWMLZBMZINZWMOZ BWNPZWKWTCMZANZINZWMOZCDPZWKXDCDWKXADOUFZXCFLXDXFXCXBUGNZXBUHNZIUIZFXFXCX GXHUNZINXIXFXBXJIXFXBWIOZXBXJQXFEWIXBDWJXAAUJUKZXBFFULRUMXGXHIUOUPXFXGFOZ @@ -328162,7 +328170,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically syl3anc cun cv cvol cdm wrex wral cin wi cpw c0 wne 1rp ne0ii r19.2z mpan c1 simprl mblss adantr sstrd rexlimiva rexlimivw inss1 elpwi simpr difssd syl mp3an2i readdcld adantl rpre ad2antlr simprrr ovollecl simprrl sstrid - sslin ovolss syl2anc ssdifssd unssd ovolun syl22anc ssun1 undif1 sseqtr4i + sslin ovolss syl2anc ssdifssd unssd ovolun syl22anc ssun1 undif1 sseqtrri ssdif ax-mp difundir sseqtri difun1 wceq ssequn2 difeq2d syl5eqr sseqtrid sylib uneq1d letrd le2addd mblsplit oveq1d recnd addassd breqtrrd sylancr eqtrd difss leadd2dd rexlimdvaa ralimdva impcom cxr rexrd simprr xralrple @@ -329211,7 +329219,7 @@ Hilbert space (in the algebraic sense, meaning that all algebraically iccf fniunfv iunun ioof wfun cdm wfo fo1st fofn ssv fndm eqimss2 dfimafn2 fnfun 3syl syl2anc fnima eqtr3d rnco2 syl6eq fo2nd 3eqtr3d sseqtrrid cdom syl5eq ovolficcss ssdifssd cen ccrd cres con0 omelon nnenom ensymi isnumi - mp2an fofun fof fdmi sseqtr4i fores ffnd dffn4 sylib fodomnum mpsyl unctb + mp2an fofun fof fdmi sseqtrri fores ffnd dffn4 sylib fodomnum mpsyl unctb foco domentr ctex ssid sseqtrid ssundif ssdomg sylc domtr ovolctb2 jca ) AEBUAZUBUCZFBUAZUBUCZGUUMUUKUDZUEHUFIZAUUKJBUBZUGZKUUPUGZUHZUHZUUKUUMUUKU USUIADLDUJZUULHZUKZDLUVAUUJHZUVAJBUAZHZULZUVAKBUAZHZULZUHZUHZUKZUUMUUTADL @@ -330079,7 +330087,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vz vw va vb vc cfv wcel cicc cv wss wa cr co clt wbr vr vn cioo crn ctg crab cima cuni cvol cdm cpw wral weq fveq2 sseq1d simprr fvex elpw sylibr elrab sylan2b ralrimiva wfun wb cxr cxp wf iccf ffun ax-mp ssrab2 cle cin - cz cn0 dyadf inss2 rexpssxrxp sstri fdmi sseqtr4i funimass4 mp2an sspwuni + cz cn0 dyadf inss2 rexpssxrxp sstri fdmi sseqtrri funimass4 mp2an sspwuni frn sylib cabs cmin ccom cres cbl crp wrex cxmet eqid rexmet cmopn mopni2 tgioo mp3an1 wceq elssuni uniretop syl6sseqr sselda rpre bl2ioo syl2an c1 caddc c2 cexp cdiv cn 2re 1lt2 expnlbnd mp3an23 ad2antrl cfl ad2antrr 2nn @@ -330151,7 +330159,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vx vy cdm wcel cioo cq cxp wss ctb ax-mp cn cdom wbr wral cen com mp2an cxr ioof cvol cima ctg cfv crn cv cuni wceq wa wex wb qtopbas ciun uniiun eltg3 wi ssdomg ccrd cres wfo con0 omelon qnnen xpen xpnnen nnenom entr2i - entri isnumi wfun cr cpw wf ffun qssre ressxr sstri xpss12 sseqtr4i fores + entri isnumi wfun cr cpw wf ffun qssre ressxr sstri xpss12 sseqtrri fores fodomnum mp2 domentr domtr sylancl imassrn wfn co ffn ioombl rgen2w ffnov fdmi mpbir2an frn sstr mpan2 dfss3 sylib iunmbl2 syl2anc eleq1 syl5ibrcom eqeltrid imp exlimiv sylbi eqid tgqioo eleq2s ) AUADZEZAFGGHZUBZUCUDZFUEZ @@ -331381,7 +331389,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by domtr adantr fvco3 sylan eleq1d anbi12d cop fveq2 syl elpreima fco 3syl ffn ismbf mpbid sseldi rsp sylc syl2anc imauni eqtr3d wi cmpo ccrd con0 imaiun ssdomg ensymi cres xpen entri entr2i wfun cpw qssre ressxr sstri - ffun xpss12 fdmi sseqtr4i fores elexi xpdom1 xpdom2 numdom fnmpoi dffn4 + ffun xpss12 fdmi sseqtrri fores elexi xpdom1 xpdom2 numdom fnmpoi dffn4 nnex xpex mpbi sylancl ad2antrl elrnmpo cin elin mbff ffvelrnda opeq12d bitri cnrecnv opex fvmpt biantrurd bitr3d opelxp f1ocnv f1ofn imacnvcnv mp2b bitr3i 3bitr3g bitrd pm5.32da ref imf anandi syl6bbr 3bitr4d eqrdv @@ -331929,7 +331937,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by i1fima $p |- ( F e. dom S.1 -> ( `' F " A ) e. dom vol ) $= ( vy citg1 cdm wcel crn cin ccnv cv csn cima ciun cvol cr wf wceq wss cfn adantr wfun i1ff inpreima iunid imaeq2i imaiun eqtr3i cnvimass cnvimarndm - ffun sseqtr4i df-ss mpbi 3eqtr3g 3syl wral i1frn ssfi sylancl cmbf i1fmbf + ffun sseqtrri df-ss mpbi 3eqtr3g 3syl wral i1frn ssfi sylancl cmbf i1fmbf inss2 wa frnd sstrid sselda mbfimasn syl3anc ralrimiva finiunmbl eqeltrrd syl2anc ) BDEFZCABGZHZBIZCJZKZLZMZVPALZNEZVMOOBPZBUAZVTWAQBUBZOOBUJWDVPVO LZWAVPVNLZHZVTWAAVNBUCVPCVOVRMZLWFVTWIVOVPCVOUDUECVPVOVRUFUGWAWGRWHWAQWAB @@ -331943,7 +331951,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vol ` ( `' F " A ) ) e. RR ) $= ( citg1 cdm wcel cc0 wn cima cvol cfv covol wceq i1fima adantr mblvol syl wa cr wss cin ccnv csn cdif crn wf wfun i1ff inpreima cnvimass cnvimarndm - ffun 3syl sseqtr4i df-ss syl6req c0 elinel1 con3i adantl disjsn sylibr wb + ffun 3syl sseqtrri df-ss syl6req c0 elinel1 con3i adantl disjsn sylibr wb mpbi inss2 frnd sstrid reldisj mpbid imass2 eqsstrd mblss cfn w3a simprbi cmbf isi1f simp3d eqeltrrd ovolsscl syl3anc eqeltrd ) BCDEZFAEZGZQZBUAZAH ZIJZWGKJZRWEWGIDZEZWHWILWBWKWDABMNWGOPWEWGWFRFUBZUCZHZSWNRSZWNKJZREWIREWE @@ -331976,7 +331984,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cfn csn w3a citg1 cv cioo wral wfun ad2antrr ffun funcnvcnv imadif 4syl cdm cpw cxr cxp ioof frn ax-mp sseli elpwid ad2antlr dfss4 sylib eqtr3d imaeq2d fimacnv rembl syl6eqel wi wal cin adantr inpreima iunid imaeq2i - ciun imaiun eqtr3i cnvimass cnvimarndm sseqtr4i df-ss mpbi 3eqtr3g 3syl + ciun imaiun eqtr3i cnvimass cnvimarndm sseqtrri df-ss mpbi 3eqtr3g 3syl inss2 ssfi sylancl simpll c0 elinel1 con3i adantl disjsn sylibr reldisj wb sselda syl2anc ralrimiva finiunmbl eqeltrrd ex alrimiv elndif difexi reex eleq2 notbid imaeq2 eleq1d imbi12d spcv sylc spv adantlr pm2.61dan @@ -340303,7 +340311,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cr wcel cfv co cabs cc va vb cv clt wbr cmin cmul cle wi cicc wral cima cdv csup wss crn imassrn wf absf frn ax-mp sstri a1i c0 wne wfun cdm ffun dvf cres wceq ccncf cncff fdm 3syl ssdmres sylibr rexrd lbicc2 syl3anc wa - cxr funfvima2 imp syl21anc fdmi sseqtr4i mp2an ne0i wrex ax-resscn cncfss + cxr funfvima2 imp syl21anc fdmi sseqtrri mp2an ne0i wrex ax-resscn cncfss ssid sseldi cniccbdd fvelima adantl fveq2d 2fveq3 breq1d rspccva eqbrtrrd mpan fvres adantll fveq2 syl5ibcom rexlimdva impel breq1 syl5 ralrimiv ex reximdva mpd suprcld eqeltrid cdiv simplrr fvresd ad2antrr ffvelrnd recnd @@ -343804,7 +343812,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by deg1n0ima $p |- ( R e. Ring -> ( D " ( B \ { .0. } ) ) C_ NN0 ) $= ( vx crg wcel cv cfv cn0 csn cdif wss adantl cxr wral cima wa wne simpl eldifi eldifsni deg1nn0cl syl3anc ralrimiva wfun cdm wb wf deg1xrf ffun - ax-mp difss fdmi sseqtr4i funimass4 mp2an sylibr ) DKLZJMZBNOLZJAEPZQZU + ax-mp difss fdmi sseqtrri funimass4 mp2an sylibr ) DKLZJMZBNOLZJAEPZQZU AZBVHUBORZVDVFJVHVDVEVHLZUCVDVEALZVEEUDZVFVDVKUEVKVLVDVEAVGUFSVKVMVDVEA EUGSABCDVEEFGHIUHUIUJBUKZVHBULZRVJVIUMATBUNVNABCDFGIUOZATBUPUQVHAVOAVGU RATBVPUSUTJVHOBVAVBVC $. @@ -352965,7 +352973,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the cc0 ax-mp rpssre fss iccssre syl2anc fssres2 sylancr wb syl6ss efcn rescncf mp2an mpisyl cncffvrn mpbird wiso reefiso a1i ioossre eqidd isores3 syl3anc crn ctg cnt ssid ccnfld ctopn eqid tgioo2 dvres mpanl12 resabs1d oveq2d cpr - cdm reelprrecn dvef dmeqi fdmi eqtri sseqtr4i dvres3 mp4an reseq1i reseq12d + cdm reelprrecn dvef dmeqi fdmi eqtri sseqtrri dvres3 mp4an reseq1i reseq12d iccntr 3eqtr3d isoeq1 simpr dvcvx ax-1cn sseldi recnd nncan oveq1d ioossicc eff syl iirev lincmb01cmp syldan eqeltrrd fvresd cxr cle rexrd ltled lbicc2 ubicc2 oveq12d 3brtr3d ) ADEZBDEZABFUAZUBZCUOGUCHZEZUDZCAIHZGCUEHZBIHZJHZKA @@ -355671,7 +355679,7 @@ imaginary part lies in the interval (-pi, pi]. See ( cr crp cres cdv co ce c1 cdiv cfv wceq wtru ccncf wcel wss reeff1o cc a1i cdm eqtri clog ccnv cv cmpt dfrelog oveq2i wf1o f1of ax-mp rpssre fss mp2an wf ax-resscn efcn rescncf mp2 cncffvrn mpbir cpr reelprrecn ssid dvef dmeqi - wb eff fdmi sseqtr4i dvres3 mp4an reseq1i cc0 crn 0nrp rneqi wfo f1ofo forn + wb eff fdmi sseqtrri dvres3 mp4an reseq1i cc0 crn 0nrp rneqi wfo f1ofo forn wn mp2b eleq2i mtbir dvcnvre mptru fveq1i f1ocnvfv2 syl5eq oveq2d mpteq2ia mpan ) BUACDZEFBGBDZUBZEFZACHAUCZIFZUDZWKWMBEUEUFWNACHWOWMJZBWLEFZJZIFZUDZW QWNXBKLAWLBCWLBBMFNZLXCBBWLUMZBCWLUMZCBOXDBCWLUGZXEPBCWLUHUIZUJBCBWLUKULBQO @@ -355911,7 +355919,7 @@ imaginary part lies in the interval (-pi, pi]. See dvloglem $p |- ( log " D ) e. ( TopOpen ` CCfld ) $= ( vx clog cim cpi co cfv wss wcel wb cc cc0 wceq mp2an cr clt wbr wne wa vy vz vw cima ccnv cneg cioo ccnfld ctopn wfun cdm wral csn cdif crn wf1o - cv logf1o f1ofun ax-mp logdmss sseqtr4i funimass4 ellogdm simplbi logdmn0 + cv logf1o f1ofun ax-mp logdmss sseqtrri funimass4 ellogdm simplbi logdmn0 f1odm crp wi logcld imcld cle logimcld simpld pire a1i simprd wn logdmnrp lognegb syl2anc necon3bbid mpbid necomd leneltd cxr renegcli rexri elioo2 w3a syl3anbrc wf wfn imf ffn elpreima mp2b sylanbrc mprgbir df-ioo df-ioc @@ -355957,7 +355965,7 @@ imaginary part lies in the interval (-pi, pi]. See ( `' Im " ( -u _pi (,) _pi ) ) $= ( vx clog cim cpi co cc cc0 wceq wb cfv wcel cr clt wbr pire a1i wa c1 cv cima cres wf1o ccnv cneg cioo csn cdif crn wf1 logf1o f1of1 ax-mp logdmss - wss f1ores mp2an wfun cdm wral f1ofun wf f1of fdmi sseqtr4i funimass4 crp + wss f1ores mp2an wfun cdm wral f1ofun wf f1of fdmi sseqtrri funimass4 crp wi ellogdm simplbi logdmn0 logcld imcld cle logimcld simpld simprd wn wne logdmnrp lognegb syl2anc necon3bbid mpbid necomd leneltd cxr w3a renegcli rexri elioo2 syl3anbrc wfn imf elpreima mp2b sylanbrc mprgbir ce eliooord @@ -356007,7 +356015,7 @@ imaginary part lies in the interval (-pi, pi]. See imf fdmi fssres ffun funcnvres2 mp2b resabs1 3eqtri imaeq1i eqtr4i f1oeq1 reseq2i mpbi ccncf wrel relres dfrel2 f1of mp1i imassrn ssriv sstri logcn eqtr3i logrncn cncffvrn sylibr eqeltrid cin cnt ssid dvres mp4an cnfldtop - dvef dvloglem isopn3i reseq12i eqtri dmeqi dmres sseqtr4i df-ss wne neirr + dvef dvloglem isopn3i reseq12i eqtri dmeqi dmres sseqtrri df-ss wne neirr ctop wn resss eqsstri rnssi eff2 frn sseli eldifsn sylib simprd mto dvcnv wa mptru oveq2i fveq1i f1ocnvfv2 mpan syl5eq oveq2d mpteq2ia 3eqtr3i ) DE FBUAZGZHZITZABUBAUCZUUHJZDUUGITZJZUDTZUEZDFBGZITABUBUUJUDTZUEUUIUUOKLADUU @@ -361953,7 +361961,7 @@ already know is total (except at ` 0 ` ). There are branch points at ( cr c1 co wcel cc wss cmnf cc0 1re sylancr wbr clt a1i wb 0re cv c2 cexp caddc crab wral ax-resscn cioc cdif resqcl readdcl recnd cle wn addgtge0d 0lt1 sqge0 ltnle mpbid cxr w3a mnfxr elioc2 simp3bi nsyl eldifd syl6eleqr - mp2an rgen ssrab mpbir2an sseqtr4i ) FGAUAZUBUCHZUDHZBIZAJUEZCFVQKFJKVPAF + mp2an rgen ssrab mpbir2an sseqtrri ) FGAUAZUBUCHZUDHZBIZAJUEZCFVQKFJKVPAF UFUGVPAFVMFIZVOJLMUHHZUIBVRVOJVSVRVOVRGFIZVNFIVOFIZNVMUJZGVNUKOZULVRVOMUM PZVOVSIZVRMVOQPZWDUNZVRGVNVTVRNRWBMGQPVRUPRVMUQUOVRMFIZWAWFWGSTWCMVOUROUS WEWALVOQPZWDLUTIWHWEWAWIWDVASVBTLMVOVCVHVDVEVFDVGVIVPAJFVJVKEVL $. @@ -366020,7 +366028,7 @@ sum notation (which we used for its unordered summing capabilities) into wss wf1o f1oi f1of fzssz fss mp2an cvv 1ex fdmfifsupp cseq csupp 1z mpbid gsumsubmcl znegcl moddvds ccom fcoi1 fveq1i fvres syl5eq seqfveq cnfldbas ccntz mgpbas cnfldmul mgpplusg crg cmnd cnring ringmgp zsscn cntzcmnf wf1 - f1of1 crn cdm suppssdm dmresi sseqtri sseqtr4i gsumval3 facnn 3eqtr4d cn0 + f1of1 crn cdm suppssdm dmresi sseqtri sseqtrri gsumval3 facnn 3eqtr4d cn0 rnresi nnm1nn0 faccld nncnd ax-1cn subneg sylancl eqtrd breqtrd ) DUBIZDE JKDKLMZUFMZUCZUDMZKUGZLMZUWEUHNZKUIMZUJUWDUWHDOMZUWIDOMZPZDUWJUJUKZUWDUWF CIZUWOUWDUWEUWFIZBULZDUMLMUNMDOMZUWFIZBUWFUOZUWQUWDUWEKUPNZIUWRUWDUWEUQUX @@ -385188,19 +385196,19 @@ The second section ("Tarskian geometry") develops the synthetic treatment of 24-Aug-2017.) $) trkgbas $p |- ( U e. V -> U = ( Base ` W ) ) $= ( cbs c1 c6 cdc cop trkgstr baseid cnx cfv csn cds citv ctp snsstp1 strfv - sseqtr4i ) BEGDHHIJKABCEFLMNGOBKZPUCNQOAKZNROCKZSEUCUDUETFUBUA $. + sseqtrri ) BEGDHHIJKABCEFLMNGOBKZPUCNQOAKZNROCKZSEUCUDUETFUBUA $. $( The measure of a distance in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) $) trkgdist $p |- ( D e. V -> D = ( dist ` W ) ) $= ( cds c1 c6 cdc cop trkgstr dsid cnx cfv csn cbs citv ctp snsstp2 strfv - sseqtr4i ) AEGDHHIJKABCEFLMNGOAKZPNQOBKZUCNROCKZSEUDUCUETFUBUA $. + sseqtrri ) AEGDHHIJKABCEFLMNGOAKZPNQOBKZUCNROCKZSEUDUCUETFUBUA $. $( The congruence relation in a Tarski geometry. (Contributed by Thierry Arnoux, 24-Aug-2017.) $) trkgitv $p |- ( I e. V -> I = ( Itv ` W ) ) $= ( citv c1 c6 cdc cop trkgstr itvid cnx cfv csn cbs cds ctp snsstp3 strfv - sseqtr4i ) CEGDHHIJKABCEFLMNGOCKZPNQOBKZNROAKZUCSEUDUEUCTFUBUA $. + sseqtrri ) CEGDHHIJKABCEFLMNGOCKZPNQOBKZNROAKZUCSEUDUEUCTFUBUA $. $} ${ @@ -393751,9 +393759,9 @@ then any point of the half line ( ` R ` ` A ` ) also lies opposite to isinag.c $e |- ( ph -> C e. P ) $. ${ isinag.g $e |- ( ph -> G e. V ) $. - $( Property for point ` X ` to lie in the angle ` <" A B C "> ` Defnition - 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry Arnoux, - 15-Aug-2020.) $) + $( Property for point ` X ` to lie in the angle ` <" A B C "> ` . + Definition 11.23 of [Schwabhauser] p. 101. (Contributed by Thierry + Arnoux, 15-Aug-2020.) $) isinag $p |- ( ph -> ( X ( inA ` G ) <" A B C "> <-> ( ( A =/= B /\ C =/= B /\ X =/= B ) /\ E. x e. P ( x e. ( A I C ) /\ ( x = B \/ x ( K ` B ) X ) ) ) ) ) $= @@ -408373,19 +408381,17 @@ in common (for j=1, ... , k). In contrast to the definition in Aksoy et $( The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) - (Revised by AV, 2-May-2021.) $) - wlklenvclwlk $p |- ( ( W e. Word ( Vtx ` G ) /\ 1 <_ ( # ` W ) ) + (Revised by AV, 2-May-2021.) (Revised by JJ, 14-Jan-2024.) $) + wlklenvclwlk $p |- ( W e. Word ( Vtx ` G ) -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. ( Walks ` G ) -> ( # ` F ) = ( # ` W ) ) ) $= - ( cc0 cfv cs1 cconcat co wcel c1 chash wbr wa wceq wi caddc cc nn0cn adantr - cn0 cop cwlks cword cle df-br wlklenvp1 wlkcl wrdsymb1 s1cld ccatlen syldan - cvtx s1len a1i oveq2d eqtrd eqeq1d lencl adantl 1cnd addcan2d biimpd syl5bi - eqcom ex com23 syl sylbid com3l sylc sylbir com12 ) ACDCEZFZGHZUABUBEZIZCBU - LEZUCZIZJCKEZUDLZMZAKEZWANZVQAVOVPLZWCWEOZAVOVPUEWFVOKEZWDJPHZNZWDTIZWGVOAB - UFVOABUGWCWJWKWEWCWJWAJPHZWINZWKWEOZWCWHWLWIWCWHWAVNKEZPHZWLVTWBVNVSIWHWPNW - CVMVRVRCUHUIVRCVNUJUKWCWOJWAPWOJNWCVMUMUNUOUPUQVTWMWNOZWBVTWATIZWQVRCURWRWK - WMWEWRWKWMWEOWMWIWLNZWRWKMZWEWLWIVDWTWSWEWTWDWAJWKWDQIWRWDRUSWRWAQIWKWARSWT - UTVAVBVCVEVFVGSVHVIVJVKVL $. + ( cc0 cfv cs1 cconcat co cop cwlks wcel chash cn0 c1 caddc wceq cvtx adantr + wa cc cword wbr df-br wlkcl wlklenvp1 jca sylbir wb ccatws1len eqeq1d eqcom + syl6bb nn0cn adantl lencl nn0cnd 1cnd addcan2d biimpd sylbid expimpd syl5 ) + ACDCEZFGHZIBJEZKZALEZMKZVDLEZVGNOHZPZSZCBQEZUAKZVGCLEZPZVFAVDVEUBZVLAVDVEUC + VQVHVKVDABUDVDABUEUFUGVNVHVKVPVNVHSZVKVJVONOHZPZVPVNVKVTUHVHVNVKVSVJPVTVNVI + VSVJVMCVCUIUJVSVJUKULRVRVTVPVRVGVONVHVGTKVNVGUMUNVNVOTKVHVNVOVMCUOUPRVRUQUR + USUTVAVB $. ${ $d A a b f g p $. $d B a b f g p $. $d G a b f g p $. $d V f g p $. @@ -412163,47 +412169,46 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) ( vi co wcel cfv c1 wa wi c0 cc0 cmin cfzo wceq cwwlksn cpr cconcat caddc clsw cs1 cvv cn0 cword w3a wwlknbp wne cv chash wral wwlknp cop csn simp1 cun simprl cats1un syl2an opex snnz neii intnan df-ne un00 xchbinxr mpbir - wn eqnetrd s1cl ad2antrl ccatcl simplrl simpll adantr fzossfzop1 ad2antlr - a1i sseld imp wb oveq2 eleq2d adantl mpbird ccats1val1 syl3anc fzonn0p1p1 - eqcomd eleqtrrd exp41 impcom eleq1d ralbidva biimpd ex com23 3impia oveq1 - preq12d ad2antll nn0cn ax-1cn pncan sylancl eqtrd fveq2d fzonn0p1 3eqtr4d - cc lsw simpr ccats1val2 biimpcd exp4c com12 3adant3 fvoveq1 ralsng ralunb - fveq2 sylanbrc cfz cuz elnn0uz eluzfz2 sylbi fzelp1 fzosplit 3syl cz nn0z - fzosn syl raleqdv oveq1d uneq2d ccatlen simpl2 s1len nn0cnd 3eqtrd oveq2d - oveq12d peano2nn0 3jca expd cwwlks iswwlksn iswwlks anbi1i syl6bb sylibrd - jca mpcom 3impib ) BEDUAJKZAFKZBUELZAUBZCKZBAUFZUCJZEMUDJZDUAJKZDUGKZEUHK - ZBFUIZKZUJUVAUVBUVENZUVIOZDEFBGUKUVJUVKUVAUVOOUVMUVJUVKNZUVAUVOUVPUVANUVN - UVGPULZUVGUVLKZIUMZUVGLZUVSMUDJZUVGLZUBZCKZIQUVGUNLZMRJZSJZUOZUJZUWEUVHMU - DJZTZNZUVIUVPUVAUVNUWLOZUVKUVAUWMOUVJUVAUVKUWMUVAUVKUVNUWLUVAUVMBUNLZUVHT - ZUVSBLZUWABLZUBZCKZIQESJZUOZUJZUVKUVNNZUWLOICDEFBGHUPUXBUXCUWLUXBUXCNZUWI - UWKUXDUVQUVRUWHUXDUVGBUWNAUQZURZUTZPUXBUVMUVBUVGUXGTUXCUVMUWOUXAUSZUVKUVB - UVEVABAFVBVCUXGPULZUXDUXIBPTZUXFPTZNZVLUXKUXJUXFPUXEUWNAVDVEVFVGUXIUXGPTU - XLUXGPVHBUXFVIVJVKWBVMUXBUVMUVFUVLKZUVRUXCUXHUVBUXMUVKUVEAFVNVOZFBUVFVPVC - UXDUWHUWDIQUVHSJZUOZUXDUXPUWDIUWTEURZUTZUOZUXDUWDIUWTUOZUWDIUXQUOZUXSUXBU - XCUXTUVMUWOUXAUXCUXTOUVMUWONZUXCUXAUXTUYBUXCUXAUXTOUYBUXCNZUXAUXTUYCUWSUW - DIUWTUYCUVSUWTKZNUWRUWCCUYCUYDUWRUWCTZUXCUYBUYDUYEOZUVNUVKUYBUYFOZUVBUVKU - YGOUVEUVBUVKUYBUYDUYEUVBUVKNZUYBNZUYDNZUWPUVTUWQUWBUYJUVTUWPUYJUVMUVBUVSQ - UWNSJZKZUVTUWPTUYHUVMUWOUYDVQZUYIUVBUYDUVBUVKUYBVRZVSZUYJUYLUVSUXOKZUYIUY - DUYPUVKUYDUYPOUVBUYBUVKUWTUXOUVSEVTWCWAWDUYBUYLUYPWEZUYHUYDUWOUYQUVMUWOUY - KUXOUVSUWNUVHQSWFZWGWHWAWIAUVSFBWJWKWMUYJUWBUWQUYJUVMUVBUWAUYKKUWBUWQTUYM - UYOUYJUWAUXOUYKUYDUWAUXOKUYIUVSEWLWHUYBUYKUXOTZUYHUYDUWOUYSUVMUYRWHWAWNAU - WAFBWJWKWMXDWOVSWPWPWDWQWRWSWTXAXBWDUXDUYAEUVGLZUVHUVGLZUBZCKZUXBUXCVUCUV - MUWOUXCVUCOUXAUXCUYBVUCUVNUVKUYBVUCOZUVEUVBUVKVUDOUVEUVBUVKUYBVUCUYIUVEVU - CUYIUVDVUBCUYIUVCUYTAVUAUYIUWNMRJZBLZEBLZUVCUYTUYIVUEEBUYIVUEUVHMRJZEUWOV - UEVUHTUYHUVMUWNUVHMRXCXEUVKVUHETZUVBUYBUVKEXNKMXNKZVUIEXFXGEMXHXIWAXJXKUV - 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(Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) @@ -412222,12 +412227,12 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) XRXOXJXMGXKSLZXNJZXRXORWTXMYCRXAXIDEGUNUOXJXRYCXOXJXRYCXORXJXRKZYCXOYDYBB XNYDYBBMZXFXKSLZBMZXJXRYGXJXRXFQNZXQMZYGXIXRYIUPZXBXGXDYJXHGXFXQQUQUSTXBX GXDYIYGRXHXBXDKZYIBQNZXKMZYGYKYIYLXEQNZPLZXQMYLOPLZXQMYMYKYHYOXQYKXDXEXCJ - ZKZYHYOMXBYQXDXAYQWTAFUTTVAZFBXEVBVCVDYKYOYPXQYKYNOYLPYNOMYKAVEVJVFVDYKYL - XKOXDYLVGJXBXDYLFBVHVITWTXKVGJXAXDWTXKEVKVIUOYKVLVMVNYKYMYGYMYKYFXFYLSLZB - YFYTMXKYLXKYLXFSVOVPYKYRYTBMYSFBXEVQVCVRWKVSVTVSWAXIYEYGUPZXBXRXGXDUUAXHX - GYBYFBGXFXKSWBVDUSWCWDWEWFWKWGWLWHUSWIWJXBXIXOXMRZXAXIUUBRWTXIXAUUBXGXHXA - UUBRZXDXGXHUUCXOXHXAXGXMXOXAXHXGXMRZXOXAXHUUDXOXAXHUHXMXGXFXLJABCDEFHIWMG - XFXLWNWOWPWGWQWAWRWJTWAWS $. + ZKZYHYOMXBYQXDXAYQWTAFUTTVAZFFBXEVBVCVDYKYOYPXQYKYNOYLPYNOMYKAVEVJVFVDYKY + LXKOXDYLVGJXBXDYLFBVHVITWTXKVGJXAXDWTXKEVKVIUOYKVLVMVNYKYMYGYMYKYFXFYLSLZ + BYFYTMXKYLXKYLXFSVOVPYKYRYTBMYSFBXEVQVCVRWKVSVTVSWAXIYEYGUPZXBXRXGXDUUAXH + XGYBYFBGXFXKSWBVDUSWCWDWEWFWKWGWLWHUSWIWJXBXIXOXMRZXAXIUUBRWTXIXAUUBXGXHX + AUUBRZXDXGXHUUCXOXHXAXGXMXOXAXHXGXMRZXOXAXHUUDXOXAXHUHXMXGXFXLJABCDEFHIWM + GXFXLWNWOWPWGWQWAWRWJTWAWS $. $} ${ @@ -414053,62 +414058,60 @@ that the word does not contain the terminating vertex p(n) of the walk, /\ ( A ` 0 ) = ( B ` 0 ) ) -> A. i e. ( 0 ..^ ( ( # ` ( A ++ B ) ) - 1 ) ) { ( ( A ++ B ) ` i ) , ( ( A ++ B ) ` ( i + 1 ) ) } e. ( Edg ` G ) ) $= - ( cfv wcel wa c1 caddc co cc0 cmin cfzo wral wceq wi syl adantr cuz cword - cvtx c0 wne cpr cedg chash clsw w3a cconcat cun csn simpl ad2antrr simplr - cv wss lencl nn0zd fzossrbm1 sselda ccatval1 syl3anc elfzom1elp1fzo sylan - cz preq12d eqcomd eleq1d biimpd ralimdva impancom 3adant3 3ad2ant1 impcom - com12 adantl simprr 3jca ccatval1lsw simpr cc nn0cnd ad2antrl ccatval21sw - npcan1 fveq2d eqtr2d eqtrd com23 expimpd 3adant2 3imp ralunb ovex fvoveq1 - ex fveq2 ralsn anbi2i bitri sylanbrc wb 0z cn lennncl 0p1e1 fveq2i eleq2i - elnnuz bitr4i sylibr fzosplitsnm1 sylancr raleqdv mpbird peano2zm anim1ci - fzosubel3 rspcv 3syl ad3antrrr cn0 nn0addcl syl2an 1nn0 eluzmn addsubassd - sylancl 1cnd eleqtrrd fzoss2 ccatval2 oveq2d eleq2d fzoss1 sylbid syl2anr - sseld imp anim12i zaddcl jca elfzoelz 1z jctir elfzomelpfzo mpbid addsubd - sylibrd ralrimiv exp31 expcom com24 ccatlen oveq1d elnn0uz sylib nnm1nn0 - zcnd fzoun ) AEUBFZUAZGZAUCUDZHZCUPZAFZUVGIJKZAFZUEZEUFFZGZCLAUGFZIMKZNKZ - 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VPVNYTVMXOXP $. + ( cfv wcel wa c1 caddc co cpr cc0 cmin cfzo wral wceq wi adantr cuz cword + cvtx c0 wne cv cedg chash w3a cconcat cun csn simplll simplr wss cz lencl + nn0zd fzossrbm1 syl ad2antrr sselda ccatval1 syl3anc elfzom1elp1fzo sylan + clsw preq12d eleq1d biimprd ralimdva impancom 3adant3 com12 impcom simprl + 3ad2ant1 simpll simprr ccatval1lsw cc nn0cnd npcan1 ad2antrl fveq2d eqtrd + ccatval21sw simpr eqtr4d exbiri com23 expimpd 3adant2 3imp ralunb fvoveq1 + ovex fveq2 ralsn anbi2i bitri sylanbrc wb 0z lennncl fveq2i eleq2i elnnuz + 0p1e1 bitr4i sylibr fzosplitsnm1 sylancr raleqdv mpbird anim1ci fzosubel3 + cn peano2zm rspcv 3syl simp-4l cn0 nn0addcl syl2an 1nn0 eluzmn addsubassd + sylancl 1cnd eleqtrd fzoss2 ccatval2 oveq2d eleq2d ad3antrrr fzoss1 sseld + sylbird imp simpl zaddcl elfzoelz 1z jctir elfzomelpfzo mpbid zcnd adantl + jca addsubd sylibrd ralrimiv exp31 expcom ccatlen oveq1d ad2ant2r elnn0uz + com24 sylib nnm1nn0 fzoun 3ad2antr1 3ad2antl1 ) AEUBFZUAZGZAUCUDZHZCUEZAF + ZUVJIJKZAFZLZEUFFZGZCMAUGFZINKZOKZPZAVFFZMAFZLZUVOGZUHZBUVFGZBUCUDZHZDUEZ + BFZUWIIJKBFZLZUVOGZDMBUGFZINKZOKZPZBVFFMBFZLUVOGZUHZUWBUWRQZUHZUVJABUIKZF + ZUVLUXCFZLZUVOGZCMUXCUGFZINKZOKZPUXGCMUVQOKZUVQUVQUWOJKZOKZUJZPZUXBUXGCUX + KPZUXGCUXMPZUXOUXBUXPUXGCUVSUVRUKZUJZPZUXBUXGCUVSPZUVRUXCFZUVRIJKZUXCFZLZ + UVOGZUXTUWEUWTUYAUXAUWTUWEUYAUWHUWQUWEUYARZUWSUWFUYGUWGUWEUWFUYAUVIUVTUWF + UYARUWDUVIUWFUVTUYAUVIUWFHZUVPUXGCUVSUYHUVJUVSGZHZUXGUVPUYJUXFUVNUVOUYJUX + DUVKUXEUVMUYJUVGUWFUVJUXKGUXDUVKQUVGUVHUWFUYIULZUVIUWFUYIUMZUYHUVSUXKUVJU + VGUVSUXKUNZUVHUWFUVGUVQUOGZUYMUVGUVQUVEAUPZUQZUVQURUSUTVAUVEABUVJVBVCUYJU + VGUWFUVLUXKGZUXEUVMQUYKUYLUYHUYNUYIUYQUVGUYNUVHUWFUYPUTUVJUVQVDVEUVEABUVL + VBVCVGVHVIVJVKVLVMSVPVNVLUWEUWTUXAUYFUVIUWDUWTUXAUYFRZRUVTUWTUVIUWDHZUYRU + WHUWQUYSUYRRUWSUWHUVIUWDUYRUWHUVIHZUXAUWDUYFUYTUXAUYFUWDUYTUXAHZUYEUWCUVO + VUAUYBUWAUYDUWBUYTUYBUWAQZUXAUYTUVGUWFUVHVUBUWHUVGUVHVOZUWFUWGUVIVQZUWHUV + GUVHVRABUVEVSVCSVUAUYDUWRUWBUYTUYDUWRQUXAUYTUYDUVQUXCFZUWRUYTUYCUVQUXCUVG + UYCUVQQZUWHUVHUVGUVQVTGZVUFUVGUVQUYOWAZUVQWBUSWCWDUYTUVGUWFUWGVUEUWRQVUCV + UDUWFUWGUVIUMABUVEWFVCWESUYTUXAWGWHVGVHWIWJWKVPVMWLWMUXTUYAUXGCUXRPZHUYAU + YFHUXGCUVSUXRWNVUIUYFUYAUXGUYFCUVRUVQINWPUVJUVRQZUXFUYEUVOVUJUXDUYBUXEUYD + UVJUVRUXCWQUVJUVRIUXCJWOVGVHWRWSWTXAUWEUWTUXPUXTXBZUXAUVIUVTVUKUWDUVIUXGC + UXKUXSUVIMUOGUVQMIJKZTFZGZUXKUXSQXCUVIUVQXQGZVUNUVEAXDVUNUVQITFZGVUOVUMVU + PUVQVULITXHXEXFUVQXGXIXJMUVQXKXLXMVPVPXNUWEUWTUXAUXQUVIUVTUWTUXAUXQRZRUWD + UWTUVIVUQUWHUWQUVIVUQRZUWSUWHUWQVURUWHUXAUVIUWQUXQUWHUVIUXAUWQUXQRZUVIUWH + UXAVUSRUVIUWHHZUXAUWQUXQVUTUXAHZUWQHUXGCUXMVVAUVJUXMGZUWQUXGVVAVVBHZUWQUV + JUVQNKZBFZVVDIJKZBFZLZUVOGZUXGVVCVVBUWOUOGZHVVDUWPGUWQVVIRVVAVVJVVBVUTVVJ + UXAUWFVVJUVIUWGUWFUWNUOGZVVJUWFUWNUVEBUPZUQZUWNXRUSWCSXOUVJUVQUWOXPUWMVVI + DVVDUWPUWIVVDQZUWLVVHUVOVVNUWJVVEUWKVVGUWIVVDBWQUWIVVDIBJWOVGVHXSXTVVCUXF + VVHUVOVVCUXDVVEUXEVVGVVCUVGUWFUVJUVQUVQUWNJKZOKZGUXDVVEQUVGUVHUWHUXAVVBYA + ZVUTUWFUXAVVBUVIUWFUWGVOUTZVVAUXMVVPUVJVUTUXMVVPUNZUXAVUTVVOUXLTFZGVVSVUT + VVOVVOINKZTFZVVTVUTVVOUOGZIYBGZVVOVWBGUVIUVQYBGZUWNYBGZVWCUWHUVGVWEUVHUYO + SUWFVWFUWGVVLSVWEVWFHVVOUVQUWNYCUQYDYEVVOIYFYHVUTVWAUXLTVUTUVQUWNIUVGVUGU + VHUWHVUHUTZUWFUWNVTGUVIUWGUWFUWNVVLWAWCVUTYIYGZWDYJUXLUVQVVOYKUSSVAUVEABU + VJYLVCVVCUXEUVLUVQNKZBFZVVGVVCUVGUWFUVLVVPGZUXEVWJQVVQVVRVVCUVJUVRVWAOKZG + ZVWKVVAVVBVWMVVAVVBUVJUVQVWAOKZGZVWMVUTVWOVVBXBUXAVUTVWNUXMUVJVUTVWAUXLUV + QOVWHYMYNSVVAVWNVWLUVJVVAUVQUVRTFGZVWNVWLUNUVGVWPUVHUWHUXAUVGUYNVWDVWPUYP + YEUVQIYFYHYOUVQUVRVWAYPUSYQYRYSVVAUYNVWCHZUVJUOGZIUOGZHVWMVWKXBVVBVUTVWQU + XAUVIUYNVVKVWQUWHUVGUYNUVHUYPSUWFVVKUWGVVMSUYNVVKHUYNVWCUYNVVKYTUVQUWNUUA + UUIYDSVVBVWRVWSUVJUVQUXLUUBZUUCUUDUVJIUVQVVOUUEYDUUFUVEABUVLYLVCVVCVWIVVF + BVVCUVJIUVQVVBUVJVTGVVAVVBUVJVWTUUGUUHVVCYIVUTVUGUXAVVBVWGUTUUJWDWEVGVHUU + KVKUULUUMUUNWJUUSYSVLVMVPWMUXGCUXKUXMWNXAUXBUXGCUXJUXNUWEUWTUXJUXNQZUXAUV + IUVTUWTVXAUWDUVIUWQUWHVXAUWSVUTUXJMUXLOKZUXNVUTUXIUXLMOVUTUXIVWAUXLUVGUWF + UXIVWAQUVHUWGUVGUWFHUXHVVOINUVEUVEABUUOUUPUUQVWHWEYMUVIUVQMTFGZUWOYBGZVXB + UXNQUWHUVGVXCUVHUVGVWEVXCUYOUVQUURUUTSUWHUWNXQGVXDUVEBXDUWNUVAUSMUVQUWOUV + BYDWEUVCUVDVLXMXN $. $( The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double @@ -414533,19 +414536,18 @@ that the word does not contain the terminating vertex p(n) of the walk, clwlkclwwlk2 $p |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) ) -> ( E. f f ( ClWalks ` G ) ( P ++ <" ( P ` 0 ) "> ) <-> P e. ( ClWWalks ` G ) ) ) $= - ( wcel c1 chash cfv cle wbr co cpfx wceq wa 3adant1 c2 caddc cuspgr cword - w3a cc0 cs1 cconcat cclwwlk cv cclwlks wex lswccats1fst wrdlenccats1lenm1 - cmin biantrurd adantr oveq2d simpl wrdsymb1 s1cld eqidd pfxccatid syl3anc - eqtr2d eleq1d wb simp1 simp2 ccatcl syl2anc cn0 lencl 1e2m1 a1i breq1d cr - clsw 2re 1red nn0re lesubaddd bitrd syl biimpa s1len oveq2i breqtrrdi jca - ccatlen breqtrrd clwlkclwwlk 3bitr4rd ) DUAHZAEUBZHZIAJKZLMZUCZAUDAKZUEZU - FNZWTJKZIUMNZONZDUGKZHZWTVPKUDWTKPZXEQZAXDHBUHWTDUIKMBUJZWQXFXEWNWPXFWLAE - UKRUNWQAXCXDWNWPAXCPWLWNWPQZXCWTWOONZAXIXBWOWTOWNXBWOPWPWREAULUOUPXIWNWSW - MHZWOWOPXJAPWNWPUQZXIWREEAURUSZXIWOUTAWSWOEVAVBVCRVDWQWLWTWMHZSXALMXHXGVE - WLWNWPVFWQWNXKXNWLWNWPVGWNWPXKWLXMREAWSVHVIWQSWOWSJKZTNZXALWNWPSXPLMWLXIS - WOITNZXPLWNWPSXQLMZWNWOVJHZWPXRVEEAVKXSWPSIUMNZWOLMXRXSIXTWOLIXTPXSVLVMVN - XSSIWOSVOHXSVQVMXSVRWOVSVTWAWBWCXOIWOTWRWDWEWFRWQWNXKQZXAXPPWNWPYAWLXIWNX - KXLXMWGREAWSWHWBWIWTBCDEFGWJVBWK $. + ( wcel c1 chash cfv cle wbr co wceq cpfx c2 wb 3adant1 caddc cuspgr cword + w3a cv cc0 cs1 cconcat cclwlks wex clsw cmin cclwwlk simp1 wrdsymb1 s1cld + wa ccatcl syldan cn0 lencl 1e2m1 breq1i cr 2re a1i nn0re lesubaddd syl5bb + 1red syl biimpa s1len oveq2i breqtrrdi ccatlen breqtrrd wrdlenccats1lenm1 + clwlkclwwlk syl3anc oveq2d adantr simpl pfxccatid eqtr2d eleq1d biantrurd + eqidd lswccats1fst bitr2d bitrd ) DUAHZAEUBZHZIAJKZLMZUCZBUDAUEAKZUFZUGNZ + DUHKMBUIZWSUJKUEWSKOZWSWSJKZIUKNZPNZDULKZHZUPZAXEHZWPWKWSWLHZQXBLMZWTXGRW + KWMWOUMWMWOXIWKWMWOWRWLHZXIWMWOUPZWQEEAUNUOZEAWRUQURSWMWOXJWKXLQWNWRJKZTN + ZXBLXLQWNITNZXOLWMWOQXPLMZWMWNUSHZWOXQREAUTWOQIUKNZWNLMXRXQIXSWNLVAVBXRQI + WNQVCHXRVDVEXRVIWNVFVGVHVJVKXNIWNTWQVLVMVNWMWOXKXBXOOXMEEAWRVOURVPSWSBCDE + FGVRVSWMWOXGXHRWKXLXHXFXGXLAXDXEXLXDWSWNPNZAWMXDXTOWOWMXCWNWSPWQEAVQVTWAX + LWMXKWNWNOXTAOWMWOWBXMXLWNWGAWRWNEWCVSWDWEXLXAXFAEWHWFWISWJ $. $} ${ @@ -415315,51 +415317,50 @@ of fixed length (as words) is also finite. (Contributed by Alexander { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ( Edg ` G ) /\ { ( lastS ` P ) , ( P ` 0 ) } e. ( Edg ` G ) ) ) -> ( P ++ <" ( P ` 0 ) "> ) e. D ) $= - ( wcel cfv wceq wa c1 caddc co cc0 cmin cfzo adantr adantl wb cword chash - cn cvtx cpr cedg wral clsw w3a cs1 cconcat cwwlksn wne ccatws1n0 3ad2ant2 - cv c0 simp2l fstwrdne0 s1cld 3adant3 ccatcl syl2anc csn wi simprl cn0 clt - cz wbr elfzonn0 nnz elfzo0 nn0re nnre peano2rem syl 3jca simpr ltm1d lttr - cr imp syl12anc ex impancom 3adant2 sylbi elfzo0z syl3anbrc adantlr oveq2 - impcom ad2antll ccatval1 syl3anc elfzom1p1elfzo eleqtrrd preq12d ralbidva - eleq2d mpbird eqcomd eleq1d biimpcd expdcom fzo0end fvoveq1 eqcoms eqtr4d - 3imp eqtr2d fveq2 nncn 1cnd npcand fveq2d ccatws1ls 3eqtr3rd ralsn sylibr - lsw cun addsubd oveq2d cuz nnm1nn0 elnn0uz sylib fzosplitsn eqtrd raleqdv - ovex ralunb syl6bb 3ad2ant1 mpbir2and ccatlen a1i eqid s1len oveq1d nnnn0 - id oveq12d cwwlks iswwlksn iswwlks anbi1d bitrd lswccats1 biimpri eqeq12d - lbfzo0 fveq1 elrab2 sylanbrc ) FUCHZCEUDIZUAZHZCUBIZFJZKZDUPZCIZUVELMNZCI - ZUEZEUFIZHZDOFLPNZQNZUGZCUHIZOCIZUEZUVJHZKZUIZCUVPUJZUKNZFEULNZHZUWBUHIZO - UWBIZJZUWBBHUVTUWDUWBUQUMZUWBUUTHZUVEUWBIZUVGUWBIZUEZUVJHZDOUWBUBIZLPNZQN - 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D |-> ( t prefix N ) ) $. @@ -415537,31 +415538,31 @@ the set of closed walks of a fixed length represented by walks (as <-> ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) ) ) $= ( vi wcel c0 wa cc0 cfv co c1 caddc cpr cmin cfzo wral wceq syl adantr cv cword wne cs1 cconcat cedg chash clsw cwwlks cclwwlk simpl fstwrdne s1cld - csn jca ccatlen s1len oveq2d eqtrd oveq1d cc lencl nn0cnd addsubd raleqdv - a1i cun cuz cn0 cn lennncl nnm1nn0 elnn0uz sylib fzosplitsn ralunb syl6bb - 1cnd cz nn0zd elfzom1elfzo sylan ccats1val1 syl3anc elfzom1elp1fzo eleq1d - preq12d ralbidva ovex fveq2 fvoveq1 ralsn fzo0end lsw eqcomd npcan1 eqidd - fveq2d ccats1val2 syl5bb anbi12d 3bitrd ccat0 syl6bi necon3d mpcom cvv wb - adantld wrdv s1cli jctir ccatalpha mpbird w3a eqid iswwlks bitri mpbirand - df-3an isclwwlk 3anass baib 3bitr4rd ) CBUBZFZCGUCZHZEUAZCICJZUDZUEKZJZYI - LMKZYLJZNZAUFJZFZEIYLUGJZLOKZPKZQZYICJZYNCJZNZYQFZEICUGJZLOKZPKZQZCUHJZYJ - NZYQFZHZYLAUIJFZCAUJJFZYHUUBYREIUUHLMKZPKZQZYREUUIQZYREUUHUNZQZHZUUNYHYRE - UUAUURYHYTUUQIPYHYTUUGLMKZLOKUUQYHYSUVDLOYHYSUUGYKUGJZMKZUVDYHYFYKYEFZHZY - SUVFRYHYFUVGYFYGUKZYHYJBBCULZUMUOZBCYKUPSYHUVELUUGMUVELRYHYJUQVFURUSUTYHU - UGLLYFUUGVAFZYGYFUUGBCVBZVCZTYHVRZUVOVDUSURVEYHUUSYREUUIUVAVGZQUVCYHYREUU - RUVPYHUUHIVHJFZUURUVPRYHUUHVIFZUVQYHUUGVJFZUVRBCVKZUUGVLSUUHVMVNIUUHVOSVE - YREUUIUVAVPVQYHUUTUUJUVBUUMYHYRUUFEUUIYHYIUUIFZHZYPUUEYQUWBYMUUCYOUUDUWBY - FYJBFZYIIUUGPKZFZYMUUCRYHYFUWAUVITZYHUWCUWAUVJTZYHUUGVSFZUWAUWEYFUWHYGYFU - UGUVMVTTZYIUUGWAWBYJYIBCWCWDUWBYFUWCYNUWDFZYOUUDRUWFUWGYHUWHUWAUWJUWIYIUU - GWEWBYJYNBCWCWDWGWFWHUVBUUHYLJZUUQYLJZNZYQFZYHUUMYRUWNEUUHUUGLOWIYIUUHRZY - PUWMYQUWOYMUWKYOUWLYIUUHYLWJYIUUHLYLMWKWGWFWLYHUWMUULYQYHUWKUUKUWLYJYHUWK - UUHCJZUUKYHYFUWCUUHUWDFZUWKUWPRUVIUVJYHUVSUWQUVTUUGWMSYJUUHBCWCWDYFUWPUUK - RYGYFUUKUWPCYEWNWOTUSYHUWLUUGYLJZYJYHUUQUUGYLYFUUQUUGRZYGYFUVLUWSUVNUUGWP - STWRYHYFUWCUUGUUGRUWRYJRUVIUVJYHUUGWQYJUUGBCWSWDUSWGWFWTXAXBYHUUOYLGUCZYL - YEFZHZUUBYHUWTUXAUVHYHUWTUVKUVHYGUWTYFUVHYLGCGUVHYLGRCGRZYKGRZHUXCBCYKXCU - XCUXDUKXDXEXIXFYHUXAUVHUVKYHCXGUBZFZYKUXEFZHZUXAUVHXHYFUXHYGYFUXFUXGBCXJY - JXKXLTCYKBXMSXNUOUUOUXBUUBHZXHYHUUOUWTUXAUUBXOUXIEYQABYLDYQXPZXQUWTUXAUUB - XTXRVFXSUUPYHUUNUUPYHUUJUUMXOYHUUNHEYQABCDUXJYAYHUUJUUMYBXRYCYD $. + csn ccatlen s1len oveq2i syl6eq oveq1d cc lencl nn0cnd 1cnd addsubd eqtrd + jca oveq2d raleqdv cun cuz cn0 cn lennncl nnm1nn0 sylib fzosplitsn ralunb + elnn0uz syl6bb nn0zd elfzom1elfzo sylan ccats1val1 syl3anc elfzom1elp1fzo + cz preq12d eleq1d ralbidva ovex fveq2 fvoveq1 ralsn fzo0end eqtr4d npcan1 + fveq2d eqidd ccats1val2 syl5bb anbi12d 3bitrd ccat0 necon3d adantld mpcom + lsw syl6bi wb cvv wrdv s1cli ccatalpha sylancl mpbir2and w3a eqid iswwlks + df-3an bitri a1i mpbirand isclwwlk 3anass baib 3bitr4rd ) CBUBZFZCGUCZHZE + UAZCICJZUDZUEKZJZYKLMKZYNJZNZAUFJZFZEIYNUGJZLOKZPKZQZYKCJZYPCJZNZYSFZEICU + GJZLOKZPKZQZCUHJZYLNZYSFZHZYNAUIJFZCAUJJFZYJUUDYTEIUUJLMKZPKZQZYTEUUKQZYT + EUUJUNZQZHZUUPYJYTEUUCUUTYJUUBUUSIPYJUUBUUILMKZLOKUUSYJUUAUVFLOYJUUAUUIYM + UGJZMKZUVFYJYHYMYGFZHZUUAUVHRYJYHUVIYHYIUKZYJYLBBCULZUMZVFZBBCYMUOSUVGLUU + IMYLUPUQURUSYJUUILLYHUUIUTFZYIYHUUIBCVAZVBZTYJVCZUVRVDVEVGVHYJUVAYTEUUKUV + CVIZQUVEYJYTEUUTUVSYJUUJIVJJFZUUTUVSRYJUUJVKFZUVTYJUUIVLFZUWABCVMZUUIVNSU + UJVRVOIUUJVPSVHYTEUUKUVCVQVSYJUVBUULUVDUUOYJYTUUHEUUKYJYKUUKFZHZYRUUGYSUW + EYOUUEYQUUFUWEYHYLBFZYKIUUIPKZFZYOUUERYJYHUWDUVKTZYJUWFUWDUVLTZYJUUIWFFZU + WDUWHYHUWKYIYHUUIUVPVTTZYKUUIWAWBYLYKBCWCWDUWEYHUWFYPUWGFZYQUUFRUWIUWJYJU + WKUWDUWMUWLYKUUIWEWBYLYPBCWCWDWGWHWIUVDUUJYNJZUUSYNJZNZYSFZYJUUOYTUWQEUUJ + UUILOWJYKUUJRZYRUWPYSUWRYOUWNYQUWOYKUUJYNWKYKUUJLYNMWLWGWHWMYJUWPUUNYSYJU + WNUUMUWOYLYJUWNUUJCJZUUMYJYHUWFUUJUWGFZUWNUWSRUVKUVLYJUWBUWTUWCUUIWNSYLUU + JBCWCWDYHUUMUWSRYICYGXGTWOYJUWOUUIYNJZYLYJUUSUUIYNYHUUSUUIRZYIYHUVOUXBUVQ + UUIWPSTWQYJYHUWFUUIUUIRUXAYLRUVKUVLYJUUIWRYLUUIBCWSWDVEWGWHWTXAXBYJUUQYNG + UCZYNYGFZHZUUDYJUXCUXDUVJYJUXCUVNUVJYIUXCYHUVJYNGCGUVJYNGRCGRZYMGRZHUXFBC + YMXCUXFUXGUKXHXDXEXFYJUXDYHUVIUVKUVMYHUXDUVJXIZYIYHCXJUBZFYMUXIFUXHBCXKYL + XLCYMBXMXNTXOVFUUQUXEUUDHZXIYJUUQUXCUXDUUDXPUXJEYSABYNDYSXQZXRUXCUXDUUDXS + XTYAYBUURYJUUPUURYJUULUUOXPYJUUPHEYSABCDUXKYCYJUULUUOYDXTYEYF $. $( A word over vertices represents a closed walk of a fixed length ` N ` greater than zero iff the word concatenated with its first symbol @@ -415824,8 +415825,8 @@ closed walk as word of the same length (in an undirected graph). id ccatlen syl2an oveqan12d eqtrd 3adant3 sylanbrc ) ADCFGHZBECFGHZIAJIBJKZ RABLGZCUAJZHZUSMJZDENGZKZUSVCCFGHUPAUTHZAMJZDKZOZUQBUTHZBMJZEKZOZURURVACDAP CEBPURUBVHVEVLVIURURVAVEVGSVIVKSURUIABCUCUDUEUPUQVDURUPUQOVBVFVJNGZVCUPACUF - JZUGZHBVOHVBVMKUQCDVNAVNUHZQCEVNBVPQVNABUJUKUPUQVFDVJENCDATCEBTULUMUNCVCUSP - UO $. + JZUGZHBVOHVBVMKUQCDVNAVNUHZQCEVNBVPQVNVNABUJUKUPUQVFDVJENCDATCEBTULUMUNCVCU + SPUO $. ${ $d G i $. $d N i $. $d W i $. @@ -425532,7 +425533,7 @@ Below is the final Metamath proof (which reorders some steps). $( Example for ~ df-ss . Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) $) ex-ss $p |- { 1 , 2 } C_ { 1 , 2 , 3 } $= - ( c1 c2 cpr c3 csn cun ctp ssun1 df-tp sseqtr4i ) ABCZKDEZFABDGKLHABDIJ $. + ( c1 c2 cpr c3 csn cun ctp ssun1 df-tp sseqtrri ) ABCZKDEZFABDGKLHABDIJ $. $( Example for ~ df-pss . Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) $) @@ -430935,7 +430936,7 @@ a normed complex vector space (normally a Hilbert space). @) ( ( t ` x ) Q y ) = ( x P ( s ` y ) ) ) } ) $= ( co cv cfv cba vu vw cnv wcel wa caj wf wceq wral w3a copab cdip syl6eqr fveq2 feq2d feq3d oveqd eqeq2d ralbidv raleqbidv 3anbi123d opabbidv df-aj - eqeq1d cmap cxp ovex xpex fvexi anbi12i biimpri 3adant3 ssopab2i sseqtr4i + eqeq1d cmap cxp ovex xpex fvexi anbi12i biimpri 3adant3 ssopab2i sseqtrri elmap df-xp ssexi ovmpo syl5eq ) GUCUDHUCUDUEDGHUFQIJCRZUGZJIKRZUGZARZVTS ZBRZFQZWDWFWBSZEQZUHZBJUIZAIUIZUJZCKUKZPUAUBGHUCUCUARZTSZUBRZTSZVTUGZWRWP WBUGZWEWFWQULSZQZWDWHWOULSZQZUHZBWRUIZAWPUIZUJZCKUKWNUFIWRVTUGZWRIWBUGZXB @@ -431498,7 +431499,7 @@ Inner product (pre-Hilbert) spaces ( wcel cq co vx cc0 cc ccnv csn cima wss cioo crn ctg cfv ccl cr wf 0cn wceq cv wral cmul cmin qre oveq1 oveq1d oveq12d ovex fvmpt syl wa phnvi qcn cnv nvscl mp3an1 sylan dipcl mp3an13 ipasslem5 subeq0bd mpan2 eqtrd - rgen wfun cdm wb funmpt2 qssre dmmpti sseqtr4i funconstss mpbi qdensere + rgen wfun cdm wb funmpt2 qssre dmmpti sseqtrri funconstss mpbi qdensere mp2an ccnfld ctopn ct1 ccn w3a cha eqid cnfldhaus haust1 ax-mp uniretop ipasslem7 cnfldtopon toponunii dnsconst mpanl12 mp3an ) UBUCRZSGUDUBUEZ UFUGZSUHUIUJUKZULUKUKUMUPZUMXKGUNZUOUAUQZGUKZUBUPZUASURZXLXRUASXPSRZXQX @@ -443074,7 +443075,7 @@ negative of a vector from this axiom (see ~ hvsubid and ~ hvsubval ). hlimcaui $p |- ( F ~~>v A -> F e. Cauchy ) $= ( chli wbr cdm ccauold cva csm cop cno cims cfv ccau chba cin eqsstri ax-mp wss eqid wrel cn cmap co cmopn cres hhlm resss dmss cxmet wcel hhxmet lmcau - clm sstri dmeqi dmres eqtri inss1 ssini hhcau sseqtr4i relres releqi sseldi + clm sstri dmeqi dmres eqtri inss1 ssini hhcau sseqtrri relres releqi sseldi mpbir releldmi ) BACDCEZFBVGGHIJIZKLZMLZNUAUBUCZOFVGVJVKVGVIUDLZUMLZEZVJCVM RVGVNRCVMVKUEZVMVIVHVLVHSZVISZVLSZUFZVMVKUGPCVMUHQVINUILUJVNVJRVIVHVPVQUKVI VLNVRULQUNVGVKVNOZVKVGVOEVTCVOVSUOVMVKUPUQVKVNURPUSVIVHVPVQUTVABACCTVOTVMVK @@ -443177,7 +443178,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] (New usage is discouraged.) $) shsspwh $p |- SH C_ ~P ~H $= ( vx csh cuni cpw chba pwuni wcel cv wss wral wceq helsh shss ssunieq mp2an - rgen pweqi sseqtr4i ) BBCZDEDBFESEBGAHZEIZABJESKLUAABTMPAEBNOQR $. + rgen pweqi sseqtrri ) BBCZDEDBFESEBGAHZEIZABJESKLUAABTMPAEBNOQR $. $( Closed subspaces are subsets of Hilbert space. (Contributed by NM, 24-Nov-2004.) (New usage is discouraged.) $) @@ -443362,7 +443363,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] shortened by AV, 27-Aug-2021.) (New usage is discouraged.) $) hhssabloi $p |- ( +h |` ( H X. H ) ) e. AbelOp $= ( vx vy cva cxp cres cgr wcel wss hhssabloilem simp2i wceq chba shssii cv - cdm co sheli ovres mp2an ax-hfvadd fdmi sseqtr4i ssdmres mpbi wa ax-hvcom + cdm co sheli ovres mp2an ax-hfvadd fdmi sseqtrri ssdmres mpbi wa ax-hvcom xpss12 syl2an ancoms 3eqtr4d isabloi ) CDEAAFZGZAEHIUOHIUOEJABKLUNEQZJUOQ UNMUNNNFZUPANJZURUNUQJABOZUSANANUIUAUQNEUBUCUDUNEUEUFCPZAIZDPZAIZUGUTVBER ZVBUTERZUTVBUORVBUTUORZVAUTNIVBNIVDVEMVCUTABSVBABSUTVBUHUJUTVBAAETVCVAVFV @@ -443387,7 +443388,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] hhssnv $p |- W e. NrmCVec $= ( vx vy csm cc cva cno c0v wcel wss wceq chba mp2an cfv co ovres eqtrd cxp cres cablo cgr hhssabloi ablogrpo ax-mp cdm shssii xpss12 ax-hfvadd - vz fdmi sseqtr4i ssdmres grporn cgi csh sh0 ax-hv0cl hvaddid2i eqtri wb + vz fdmi sseqtrri ssdmres grporn cgi csh sh0 ax-hv0cl hvaddid2i eqtri wb mpbi eqid grpoid mpbir cop wf wfn crn ax-hfvmul ffn ssid fnssres ovelrn cv wrex wa shmulcl mp3an1 eqeltrd eleq1 syl5ibrcom rexlimivv sylbi df-f ssriv mpbir2an c1 ax-1cn sheli ax-hvmulid syl w3a id ax-hvdistr1 syl3an @@ -443461,7 +443462,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] ( cfv cpv eqid cdm wcel mp2an cxp cva cno c1st chba cvv eqtri wceq hhnv cba crn bafval cnv cgr css sspnv nvgrp grporndm mp2b cres csm cc fveq2i cop vafval opex cr wf normf ax-hilex fex resex op1st cablo resexg ax-mp - hilablo hvmulex dmeqi wss xpss12 ax-hfvadd fdmi sseqtr4i ssdmres dmxpid + hilablo hvmulex dmeqi wss xpss12 ax-hfvadd fdmi sseqtrri ssdmres dmxpid mpbi eqcomi ) CUCHZBWBCIHZUDZBCWCWBWBJWCJZUEWDWCKZKZBCUFLZWCUGLWDWGUAAU FLCAUHHZLWHADUBFAWICWIJUIMCWCWEUJWCUKULWGBBNZKBWFWJWFOWJUMZKZWJWCWKWCWK UNUOBNZUMZUQZPBUMZUQZIHZWKCWQIEUPWRWQQHZQHZWKWQWRWRJURWTWOQHWKWSWOQWOWP @@ -444463,7 +444464,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] (New usage is discouraged.) $) shunssji $p |- ( A u. B ) C_ ( A vH B ) $= ( cun cort cfv chj co chba wss shssii unssi ococss ax-mp wcel wceq shjval - csh mp2an sseqtr4i ) ABEZUBFGFGZABHIZUBJKUBUCKABJACLBDLMUBNOASPBSPUDUCQCD + csh mp2an sseqtrri ) ABEZUBFGFGZABHIZUBJKUBUCKABJACLBDLMUBNOASPBSPUDUCQCD ABRTUA $. $( Subspace sum is smaller than Hilbert lattice join. Remark in [Kalmbach] @@ -446120,7 +446121,7 @@ equals the join of their closures (double orthocomplements). ( ( _|_ ` ( _|_ ` A ) ) vH ( _|_ ` ( _|_ ` B ) ) ) $= ( chj co cort cfv cun wss chba ococss ax-mp mp2an cch wcel choccli chssii unssi occon2i unss12 occl wceq sshjval chjvali 3sstr4i ssun1 sstri sshjcl - sseqtr4i ssun2 chlubii ococi sseqtri eqssi ) ABEFZAGHZGHZBGHZGHZEFZABIZGH + sseqtrri ssun2 chlubii ococi sseqtri eqssi ) ABEFZAGHZGHZBGHZGHZEFZABIZGH GHZURUTIZGHGHZUPVAVBVDJZVCVEJAURJZBUTJZVFAKJZVGCALMBKJZVHDBLMAURBUTUANVBV DABKCDSZURUTKURUQVIUQOPCAUBMQZRUTUSVJUSOPDBUBMQZRSTMVIVJUPVCUCCDABUDNZURU TVLVMUEUFVAUPGHZGHZUPURVPJZUTVPJZVAVPJAUPJVQAVCUPAVBVCABUGVBKJVBVCJVKVBLM @@ -446675,7 +446676,7 @@ equals the join of their closures (double orthocomplements). pjoml6i $p |- ( A C_ B -> E. x e. CH ( A C_ ( _|_ ` x ) /\ ( A vH x ) = B ) ) $= ( wss cort cfv cin cch wcel chj co wceq wa wrex choccli chincli pjoml2i - cv chub1i chdmm2i sseqtr4i jctil fveq2 sseq2d oveq2 eqeq1d anbi12d rspcev + cv chub1i chdmm2i sseqtrri jctil fveq2 sseq2d oveq2 eqeq1d anbi12d rspcev sylancr ) BCFZBGHZCIZJKBUNGHZFZBUNLMZCNZOZBATZGHZFZBUTLMZCNZOZAJPUMCBDQER ULURUPBCDESBBCGHZLMUOBVFDCEQUABCDEUBUCUDVEUSAUNJUTUNNZVBUPVDURVGVAUOBUTUN GUEUFVGVCUQCUTUNBLUGUHUIUJUK $. @@ -447299,7 +447300,7 @@ Note that the (countable) Axiom of Choice is used for this proof via (New usage is discouraged.) $) osumcori $p |- ( ( A i^i B ) +H ( A i^i ( _|_ ` B ) ) ) = ( ( A i^i B ) vH ( A i^i ( _|_ ` B ) ) ) $= - ( cin cort cfv wss cph co chj inss2 choccli chub2i sstri chdmm3i sseqtr4i + ( cin cort cfv wss cph co chj inss2 choccli chub2i sstri chdmm3i sseqtrri wceq chincli osumi ax-mp ) ABEZABFGZEZFGZHUBUDIJUBUDKJRUBAFGZBKJZUEUBBUGA BLBUFDACMNOABCDPQUBUDABCDSAUCCBDMSTUA $. @@ -448504,7 +448505,7 @@ Note that the (countable) Axiom of Choice is used for this proof via oveq1d inss2 sseli elin2 cort pjdsi mpan2 oveqan12d inss1 pjhcli syl22anc hvadd4 3syl eqtrd hvaddcl syl2anc 3eqtrd syl6eleq mpanr12 sylancl oveq12d pjds3i hvmulcl mpan hvpncan mpancom hvpncan2 chshii shsvai shscli eqeltrd - wss pjcli csh shmulcl ssriv chsleji shlessi ax-mp sstri sseqtr4i ) GHUFZB + wss pjcli csh shmulcl ssriv chsleji shlessi ax-mp sstri sseqtrri ) GHUFZB DUGUHZFUGUHZIYGYHFUIUHZYIYGBDUIUHZFUIUHZYJUEYGYLUEUJZYGUKZYMULUMUNUHZUMYM UOUHZUOUHZYLYNYMUPUKZYMYQUQYNYMGUKZYMHUKZURYRYMGHUSYSYRYTYRYMACUGUHZEUGUH ZGYMUUBUUAEACJLUTNUTVAUBVBVCVDZYRULYMUOUHZYMYQYMVEYRUUDYOUMVFUHZYMUOUHZYQ @@ -448553,7 +448554,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. mayetes3i $p |- ( ( X vH R ) i^i Y ) C_ ( Z vH R ) $= ( chj co cin wss chjcli chjcomi eqimssi chub1i chjassi chub2i sstri ss2in sseqtri mp2an chincli ccm wbr cfv cch eqeltri choccli lecmii cmcm2i mpbir - cort breqtri cm2mi fh3i ineq1i ineq2i eqtr2i sseqtr4i chsscon3i chsscon2i + cort breqtri cm2mi fh3i ineq1i ineq2i eqtr2i sseqtrri chsscon3i chsscon2i eqtri mpbi ssini chdmj1i chjjdiri oveq1i mayete3i chlubii ineq12i 3sstr4i eqid ) ACUHUIZFUHUIZEUHUIZABUHUIZCDUHUIZUJZFGUHUIZUJZUJZBDUHUIZGUHUIZEUHU IZHEUHUIZIUJJEUHUIXAEWNABEUHUIZUHUIZCDEUHUIZUHUIZUJZFGEUHUIZUHUIZUJZUJZUH @@ -452364,7 +452365,7 @@ problem in Remark (b) after Theorem 7.1 of [Mayet3] p. 1240. imaelshi $p |- ( T " A ) e. SH $= ( vu vv vx vy wcel chba wss c0v wa cv cva co wral ax-mp cfv mp2an cc wf cima csh csm crn imassrn lnopfi frn sstri lnop0i sh0 wfun cdm wi shssii - ffun fdmi sseqtr4i funfvima2 eqeltrri pm3.2i wfn ffn wceq oveq1 ralbidv + ffun fdmi sseqtrri funfvima2 eqeltrri pm3.2i wfn ffn wceq oveq1 ralbidv wb eleq1d ralima sheli lnopaddi syl2an shaddcl eqeltrrd ralrimiva oveq2 mp3an1 syl sylibr mprgbir lnopmuli sylan2 shmulcl rgen issh2 mpbir2an ) BAUCZUDIWHJKZLWHIZMENZFNZOPZWHIZFWHQZEWHQZWKWLUEPZWHIZFWHQZEUAQZMWIWJWH @@ -454454,7 +454455,7 @@ Positive operators (cont.) pjclem1 $p |- ( G C_H H -> ( ( projh ` G ) o. ( projh ` H ) ) = ( projh ` ( G i^i H ) ) ) $= ( cpjh cfv ccom cort chos co cin ch0o chj wceq sylbi chincli ax-mp eqeq2i - wss pjfi ccm wbr cmbri fveq2 inss2 choccli chub2i chdmm3i sseqtr4i pjscji + wss pjfi ccm wbr cmbri fveq2 inss2 choccli chub2i chdmm3i sseqtrri pjscji sstri coeq2 pjsdii pjss1coi mpbi pjorthcoi oveq12i hoaddid1i 3eqtri inss1 syl6eq syl cmcm3i bitri chdmm4i chub1i chdmm2i oveq12d chba df-iop coeq2i chio hoid1i eqtr3i pjtoi hocofi eqtr2i 3eqtr3g ) ABUAUBZAEFZBEFZVTGZGZVTW @@ -454535,7 +454536,7 @@ Positive operators (cont.) ) $= ( cpjh cfv ccom wceq cin cort chj chos pjclem4 choccli chio coeq2i eqtr3i co pjfi chincli ccm wbr pjclem2 pjclem3 oveq12d chba df-iop hoid1i pjsdii - syl pjtoi inss2 chub2i sstri chdmm3i sseqtr4i pjscji ax-mp 3eqtr4g chjcli + syl pjtoi inss2 chub2i sstri chdmm3i sseqtrri pjscji ax-mp 3eqtr4g chjcli wss pj11i sylib cmbri sylibr impbii ) ABUAUBZAEFZBEFZGZVIVHGHZABCDUCVKAAB IZABJFZIZKRZHZVGVKVHVOEFZHVPVKVJVHVMEFZGZLRZVLEFZVNEFZLRZVHVQVKVJWAVSWBLA BCDMVKVSVRVHGHVSWBHABCDUDAVMCBDNZMUJUEVHUFEFZGZVHVTVHOGWFVHOWEVHUGPVHACSU @@ -457798,7 +457799,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, mdcompli $p |- ( A MH B <-> ( A i^i ( _|_ ` ( A i^i B ) ) ) MH ( B i^i ( _|_ ` ( A i^i B ) ) ) ) $= ( cin cort cfv cmd wbr cdmd wss chj co chincli mdoc1i dmdoc2i ssid chjcli - wb chba chssii chjoi sseqtr4i choccli mdslmd1i mp4an ) ABEZUGFGZHIUHUGJIU + wb chba chssii chjoi sseqtrri choccli mdslmd1i mp4an ) ABEZUGFGZHIUHUGJIU GUGKABLMZUGUHLMZKABHIAUHEBUHEHISUGABCDNZOUGUKPUGQUITUJUIABCDRUAUGUKUBUCUG UHABUKUGUKUDCDUEUF $. @@ -457807,7 +457808,7 @@ the later (1970) definition given in Remark 29.6 of [MaedaMaeda] p. 130, dmdcompli $p |- ( A MH* B <-> ( A i^i ( _|_ ` ( A i^i B ) ) ) MH* ( B i^i ( _|_ ` ( A i^i B ) ) ) ) $= ( cin cort cfv cmd wbr cdmd wss chj co chincli mdoc1i dmdoc2i ssid chjcli - wb chba chssii chjoi sseqtr4i choccli mdsldmd1i mp4an ) ABEZUGFGZHIUHUGJI + wb chba chssii chjoi sseqtrri choccli mdsldmd1i mp4an ) ABEZUGFGZHIUHUGJI UGUGKABLMZUGUHLMZKABJIAUHEBUHEJISUGABCDNZOUGUKPUGQUITUJUIABCDRUAUGUKUBUCU GUHABUKUGUKUDCDUEUF $. $} @@ -460378,7 +460379,7 @@ Class abstractions (a.k.a. class builders) GVHWLVI $. $} - $( A function of non-empty domain is not empty. (Contributed by Thierry + $( A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.) $) eldmne0 $p |- ( X e. dom F -> F =/= (/) ) $= ( cdm wcel c0 wne ne0i wceq dmeq dm0 syl6eq necon3i syl ) BACZDNEFAEFNBGAEN @@ -460708,7 +460709,7 @@ Class abstractions (a.k.a. class builders) ( vx wfun cvv wss wa ccnv cima c1st wcel c2nd cdm crab cin wceq adantr wb cfv crn cxp ccom wfn funfn fncnvima2 sylbi cop fvco opeq12d eqeq2d eleq1d anbi12d elxp6 syl6rbbr adantlr opfv biantrurd wfo fo1st fofun ax-mp funco - cv mpan ssv wf fof fdm mp2b sseqtr4i ssid funimass3 mpan2 mpbii syl6eleqr + cv mpan ssv wf fof fdm mp2b sseqtrri ssid funimass3 mpan2 mpbii syl6eleqr dmco fvimacnv syl2anc fo2nd 3bitr2d rabbidva eqtrd dfin5 ineq12i cnvimass sselda dmcoss sstri sseqin2 mpbi inrab 3eqtr3ri syl6eq ) AEZAUAFFUBGZHZAI ZBCUBZJZDVDZKAUCZIBJZLZXAMAUCZICJZLZHZDANZOZXCXFPZWQWTXAATZWSLZDXIOZXJWOW @@ -462814,7 +462815,7 @@ its graph has a given second element (that is, function value). -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- $) - $( Non-zero extended nonnegative integers are strictly greater than zero. + $( Nonzero extended nonnegative integers are strictly greater than zero. (Contributed by Thierry Arnoux, 30-Jul-2023.) $) xnn0gt0 $p |- ( ( N e. NN0* /\ N =/= 0 ) -> 0 < N ) $= ( cxnn0 wcel cn0 cpnf wceq wo cc0 wne clt elxnn0 wa cn elnnne0 nngt0 sylbir @@ -464867,46 +464868,46 @@ its graph has a given second element (that is, function value). $( Conditions for a concatenation to be injective. (Contributed by Thierry Arnoux, 11-Dec-2023.) $) ccatf1 $p |- ( ph -> ( A ++ B ) : dom ( A ++ B ) -1-1-> S ) $= - ( co cfv wceq wcel syl2anc syl wa ad5antr syl3anc vi vj cconcat cdm wf cv - wi wral wf1 cc0 chash cfzo cword ccatcl wrdf ffdmd simpllr ccatval1 simpr - simplr 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi ad3antrrr r19.21bi mpd c0 - crn wfun f1fun eleqtrrd fvelrn cmin caddc ccatlen oveq2d eleqtrd ccatval2 - cin cz cn0 lencl nn0zd fzosubel3 eqeltrd elind wn noel pm2.21dd wo eleq2d - a1i adantr fzospliti ad5ant13 mpjaodan 3eqtr3rd fzossz sseldi zcnd nn0cnd - cc f1veqaeq anassrs imp syl1111anc subcan2d ad2antrr ex anasss ralrimivva - sylanbrc ) ABCUCLZUDZDXPUEUAUFZXPMZUBUFZXPMZNZXRXTNZUGZUBXQUHUAXQUHXQDXPU - IAUJXPUKMZULLZDXPAXPDUMZOZYFDXPUEABYGOZCYGOZYHGHDBCUNPZDXPUOQUPAYDUAUBXQX - QAXRXQOZXTXQOZYDAYLRZYMRZYBYCYOYBRZXRUJBUKMZULLZOZYCXRYQYEULLZOZYPYSRZXTY - ROZYCXTYTOZUUBUUCRZXRBMZXTBMZNZYCUUEXSYAUUFUUGYOYBYSUUCUQUUEYIYJYSXSUUFNZ - AYIYLYMYBYSUUCGSZAYJYLYMYBYSUUCHSZYPYSUUCUTDBCXRURZTUUEYIYJUUCYAUUGNZUUJU - 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F ) \ _I ) ) $= $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) cycpmco2lem3 $p |- ( ph -> ( ( # ` U ) - 1 ) = ( # ` W ) ) $= - ( co wcel vw chash cfv c1 cc0 cfz cz cword cn0 cv cdm wf1 crab ssrab2 cbs - eqid tocycf syl fdmd eleqtrd sseldi lencl nn0fz0 biimpi 3syl elfzelz zcnd - 1cnd caddc cmin cpfx cs1 cconcat cop csubstr cotp csplice wceq ccnv ovexd - cvv eqeltrid crn eldifad s1cld splval syl13anc syl5eq fveq2d pfxcl ccatcl - wf syl2anc swrdcl ccatlen ccatws1len cfzo wf1o wa id eqidd f1eq123d elrab - dmeq sylib simprd f1cnv f1of ffvelrnd wrddm fzofzp1 pfxlen oveq1d swrdlen - eqtrd syl3anc oveq12d 3eqtrd fz0ssnn0 peano2zd addsubassd addassd 3eqtr2d - nn0zd nn0cnd addcld pncan2d addcomd mvrraddd ) ADUBUCZJUBUCZUDAYKAYKUEYKU - FSZTZYKUGTAJBUHZTZYKUITZYMAUAUJZUKZBYQULZUAYNUMZYNJYSUAYNUNAJHUKZYTNAYTCU - OUCZHABITYTUUBHWLMUAUUBHBCIKLUUBUPUQURUSUTZVAZBJVBYPYMYKVCVDVEZYKUEYKVFUR - VGZAVHZAYJEUDYKVISZVISZEVJSZUUHYKUDVISAYJEUDVISZYKEVJSZVISZUUKYKVISZEVJSU - UJAYJJEVKSZFVLZVMSZJEYKVNVOSZVMSZUBUCZUUQUBUCZUURUBUCZVISZUUMADUUSUBADJEE - UUPVPVQSZUUSRAJUUATEWATZUVEUUPYNTZUVDUUSVRNAEGJVSZUCZUDVISZWAQAUVHUDVIVTW - BZUVJAFBAFBJWCZOWDWEZUUPJEEUUAWAWAYNWFWGWHWIAUUQYNTZUURYNTZUUTUVCVRAUUOYN - TZUVFUVMAYOUVOUUDBJEWJURZUVLBUUOUUPWKWMAYOUVNUUDBJEYKWNURBUUQUURWOWMAUVAU - UKUVBUULVIAUVAUUOUBUCZUDVISZUUKAUVOUVAUVRVRUVPBUUOFWPURAUVQEUDVIAYOEYLTZU - VQEVRUUDAEUVIYLQAUVHUEYKWQSZTUVIYLTAUVHJUKZUVTAUVKUWAGUVGAUWABJULZUVKUWAU - VGWRUVKUWAUVGWLAYOUWBAJYTTYOUWBWSUUCYSUWBUAJYNYQJVRZYRUWABBYQJUWCWTYQJXDU - WCBXAXBXCXEXFUWABJXGUVKUWAUVGXHVEPXIAYOUWAUVTVRUUDBJXJURUTUEYKUVHXKURWBZB - JEXLWMXMXOAYOUVSYMUVBUULVRUUDUWDUUEBJEYKXNXPXQXRAUUKYKEAUUKAEAEAYLUIEYKXS - UWDVAZYDXTVGUUFAEUWEYEZYAAUUNUUIEVJAEUDYKUWFUUGUUFYBXMYCAEUUHUWFAUDYKUUGU - UFYFYGAUDYKUUGUUFYHXRYI $. + ( wcel co vw chash cfv c1 cword cn0 cv cdm wf1 crab ssrab2 wf eqid tocycf + cbs syl fdmd eleqtrd sseldi lencl nn0cnd 1cnd caddc cmin cpfx cs1 cconcat + cop csubstr cotp csplice cvv wceq ovexd eqeltrid crn eldifad s1cld splval + syl13anc syl5eq fveq2d pfxcl ccatcl syl2anc swrdcl ccatlen ccatws1len cc0 + ccnv cfz cfzo wf1o wa dmeq eqidd f1eq123d elrab sylib f1cnv simpl2im f1of + ffvelrnd wrddm fzofzp1 pfxlen oveq1d eqtrd nn0fz0 swrdlen syl3anc oveq12d + id 3eqtrd fz0ssnn0 nn0zd peano2zd zcnd addsubassd addassd 3eqtr2d pncan2d + addcld addcomd mvrraddd ) ADUBUCZJUBUCZUDAYGAJBUEZSZYGUFSZAUAUGZUHZBYKUIZ + UAYHUJZYHJYMUAYHUKAJHUHZYNNAYNCUOUCZHABISYNYPHULMUAYPHBCIKLYPUMUNUPUQURZU + SZBJUTUPZVAZAVBZAYFEUDYGVCTZVCTZEVDTZUUBYGUDVCTAYFEUDVCTZYGEVDTZVCTZUUEYG + VCTZEVDTUUDAYFJEVETZFVFZVGTZJEYGVHVITZVGTZUBUCZUUKUBUCZUULUBUCZVCTZUUGADU + UMUBADJEEUUJVJVKTZUUMRAJYOSEVLSZUUSUUJYHSZUURUUMVMNAEGJWJZUCZUDVCTZVLQAUV + BUDVCVNVOZUVDAFBAFBJVPZOVQVRZUUJJEEYOVLVLYHVSVTWAWBAUUKYHSZUULYHSZUUNUUQV + MAUUIYHSZUUTUVGAYIUVIYRBJEWCUPZUVFBUUIUUJWDWEAYIUVHYRBJEYGWFUPBBUUKUULWGW + EAUUOUUEUUPUUFVCAUUOUUIUBUCZUDVCTZUUEAUVIUUOUVLVMUVJBUUIFWHUPAUVKEUDVCAYI + EWIYGWKTZSZUVKEVMYRAEUVCUVMQAUVBWIYGWLTZSUVCUVMSAUVBJUHZUVOAUVEUVPGUVAAUV + EUVPUVAWMZUVEUVPUVAULAYIUVPBJUIZUVQAJYNSYIUVRWNYQYMUVRUAJYHYKJVMZYLUVPBBY + KJUVSXMYKJWOUVSBWPWQWRWSUVPBJWTXAUVEUVPUVAXBUPPXCAYIUVPUVOVMYRBJXDUPURWIY + GUVBXEUPVOZBJEXFWEXGXHAYIUVNYGUVMSZUUPUUFVMYRUVTAYJUWAYSYGXIWSBJEYGXJXKXL + XNAUUEYGEAUUEAEAEAUVMUFEYGXOUVTUSZXPXQXRYTAEUWBVAZXSAUUHUUCEVDAEUDYGUWCUU + AYTXTXGYAAEUUBUWCAUDYGUUAYTYCYBAUDYGUUAYTYDXNYE $. $( Lemma for ~ cycpmco2 . (Contributed by Thierry Arnoux, 4-Jan-2024.) $) cycpmco2lem4 $p |- ( ph -> ( ( M ` W ) ` ( ( M ` <" I J "> ) ` I ) ) = ( ( @@ -467821,58 +467821,57 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $= wceq cs1 cotp csplice cv cdm wf1 crab ssrab2 cbs eqid tocycf fdmd eleqtrd syl sseldi crn eldifad s1cld splcl syl2anc eqeltrid cycpmco2f1 cmin simpr cycpmco2lem3 oveq2d eleqtrrd cycpmfv1 cycpmco2lem2 fveq2d cop csubstr cfz - cn0 lencl nn0fz0 biimpi 3syl swrdfv0 syl3anc cpfx cconcat cvv ccnv splval - wf ovexd syl13anc syl5eq fveq1d pfxcl ccatcl swrdcl cz fzoaddel elfzolt2b - 1zzd ccatws1len wf1o id eqidd f1eq123d elrab3 biimpa f1cnv ffvelrnd wrddm - dmeq f1of fzofzp1 oveq1d eqtrd oveq12d 3eqtrd nn0cnd 3eqtr2d 3eqtr3rd clt - zcnd wbr nn0p1gt0 cycpmfv2 cn nn0p1nn lbfzo0 sylibr ccatval1 pfxlen nn0zd - ccatlen swrdlen fz0ssnn0 peano2zd elfzelz addsubassd 1cnd addassd pncan2d - addcld addcomd ccatval2 subidd a1i pncand eqtr2d wne breqtrrdi fzo1fzo0n0 - elfzonn0 gt0ne0d sylanbrc elfzo1elm1fzo0 eqeltrd f1f1orn 3eqtr2rd 3eqtr4d - f1ocnvfv2 breqtrd eqtr3d breqtrrd eqtr4d fzne1 addid2d pfxfv0 wo mpjaodan - elfzr ) AFFGUBHUCUCJHUCZUCGUWAUCZFDHUCZUCZABCDEFGHIJKLMNOPQRUDAESJUEUCZUF - 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F ) \ _I ) ) $= s1cld splval syl13anc syl5eq fveq2d cv wf1 crab ssrab2 wf eqid tocycf cbs syl eleqtrd sseldi pfxcl ccatcl syl2anc swrdcl ccatlen ccatws1len fdmd cc0 cfz cfzo wf1o id dmeq eqidd f1eq123d elrab sylib simprd f1of - f1cnv 3syl ffvelrnd wrddm pfxlen oveq1d eqtrd cn0 lencl nn0fz0 biimpi - fzofzp1 swrdlen syl3anc oveq12d 3eqtrd fz0ssnn0 addcomd simpr 3eqtr3d - zcnd 1cnd wne clt wbr nn0p1gt0 breqtrrd cycpmfv2 a1i oveq2d npcand cc - cn nn0p1nn lbfzo0 sylibr eleqtrrd ccatval1 peano2zd cz elfzelz nn0cnd - nn0zd addsubassd addassd 3eqtr2d addcld pncan2d pncand cycpmco2f1 csn - cun ssun1 cycpmco2rn sseqtrrid sselda f1ocnvfv2 syl2an2r mpdan eqtr3d - f1f1orn cycpmco2lem2 wn eldifbd eqneltrd pm2.21dd splcl c0 frnd ssexd - f1f ne0d hashgt0 hashf1rn cycpmco2lem3 subcld nppcan3d eqtr4d fveq12d - dmexd sub32d fznn0sub subne0nn fzo0end eqeltrd eqcomd splfv3 elfzonn0 - s1len fveq1d breqtrrdi gt0ne0d addid2d pfxfv0 3eqtr4rd pm2.61dane - fzne1 ) ADUCUDZUEUFUGZDUDZDIUDZUDZUXBKIUDZUDZHUXCUDHUXEUDAUXDUXFUHZKU - 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NYAYBYSYTAVVBVVHVVKYOZWXDVVAUYSVVJYPZTAVVBVVAUYSVVJUXQAWXEBVVAAUYSBUGWXEB + VEVUHUYSBUXRXNYJUXSVLUXTXMUYA $. $} ${ @@ -468332,7 +468330,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry biimpa wfn wf1o csymg cbs eqid cycpmcl elsymgbas f1ofn wf wss cres 3eqtri wrdf frn sselda fnelnfp mpbird ssrdv ccsh ccnv ccom tocycfv difeq1d dmeqd difundir resdifcom difid reseq1i 0res uneq1i 0un dmeqi difss ax-mp dmcoss - ex dmss df-rn sseqtr4i sstri eqsstri syl6eqss eqssd ) AEUCZECKZLMZUHZAUAU + ex dmss df-rn sseqtrri sstri eqsstri syl6eqss eqssd ) AEUCZECKZLMZUHZAUAU VLUVOAUAUDZUVLNZUVPUVONZAUVQOZUVRUVPUVMKZUVPPZUVSUVPUBUDZEKZUEZUWAUBEUHZU VSUWBUWENZOZUWDOZUWBQEUFKZRUGSZUISZNZUWAUWBUWJUEZUWHUWLOZUWCUVMKZUWCUVTUV PUWNUWCUWOUWNUWCUWBRUJSZEKZUWOUWNUWEBEUKZUWFUWPUWENZUWBUWPPZUWCUWQPZAUWRU @@ -471732,7 +471730,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a $d .0. y $. $d B y $. $d W x y $. $d X x y $. lindssn.1 $e |- B = ( Base ` W ) $. lindssn.2 $e |- .0. = ( 0g ` W ) $. - $( Any singleton of a non-zero element is an independent set. (Contributed + $( Any singleton of a nonzero element is an independent set. (Contributed by Thierry Arnoux, 5-Aug-2023.) $) lindssn $p |- ( ( W e. LVec /\ X e. B /\ X =/= .0. ) -> { X } e. ( LIndS ` W ) ) $= @@ -474770,7 +474768,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a circtopn $p |- ( J qTop F ) = ( TopOpen ` ( F "s RRfld ) ) $= ( co cpw crefld ctopn cfv wceq cuni cr wtru ccms a1i cqtop wss cimas ctop pwuni wcel wfo crn ctg retop eqeltri efifo uniretop unieqi eqtr4i qtopuni - cioo mp2an pweqi sseqtr4i cbs eqidd rebase recms imasbas mptru cts retopn + cioo mp2an pweqi sseqtrri cbs eqidd rebase recms imasbas mptru cts retopn eqtri eqid imastset eqcomi topnid ax-mp ) ECUAJZBKZUBVOCLUCJZMNOVOVOPZKVP VOUEBVREUDUFQBCUGZBVROEUQUHUINZUDGUJUKABCHIULZCEQBQVTPEPUMEVTGUNUOUPURUSU TBVOVQBVQVANORBLVQCQSRVQVBZQLVANORVCTZVSRWATZLSUFRVDTZVEVFVQVGNZVOWFVOORB @@ -479430,7 +479428,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be wss sylan2 ex imp breq1 rexlimdva sylan2b simplr fsumrecl ad2antrr simprr wne wfn mpbir rspceeqv simpr1r 3anassrs w3a 0xr pnfxr mp2an elin mpbir2an wb sumeq1 wo xrlelttric mpan mpjaodan cvv cgsu cc eqcomd mpteq2dva eqtr2d - eqeltrrd isumrecl sseqtr4i isumless eqbrtrrd brralrspcev climsup climrecl + eqeltrrd isumrecl sseqtrri isumless eqbrtrrd brralrspcev climsup climrecl rexrd eqid sumex ssnnssfz fsumless reximdv rexbidv syl5ibrcom inss2 inss1 fzssuz elpwid sseldd eqeltrd r19.29an frnd 1nn dm0rn0 fdmd eqeq1d syl5bbr ne0ii necon3bid mpbiri cz 1z seqfn ax-mp fneq2i mpbi fvex r19.29 biimparc @@ -479682,7 +479680,7 @@ embedding at each step ( ` ZZ ` , ` QQ ` and ` RR ` ). It would be eqtr4d eqtr4i w3a simpr3 3anassrs peano2nnd 3brtr4d lmdvglim cpw cin csup crn ccnfld cress inss1 elpwid sseldd esumpfinvallem inss2 gsumfsum eqtr3d esumval lmdvg r19.21bi nnz fveq2d rspcdv reximia ad2antrr ffvelrnd esumex - cxrs ltle sylibd mpd fzssuz sseqtr4i elpw fzfi elin mpbir2an sumex sumeq1 + cxrs ltle sylibd mpd fzssuz sseqtrri elpw fzfi elin mpbir2an sumex sumeq1 fvmptd3 elrnmpt1s breq2 mpan rexlimivw adantllr rexrd supxrunb1 pm2.61dan fsumrecl frn csn reseq1i wsb sbequ12r anbi12d fveq2 reseq2d eqeq12d nfs1v xpeq1d simpllr elnnuz eluzfz sylanb sbequ12 mpteq12 sylan uznnssnn resmpt @@ -481123,7 +481121,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $( Lemma for ~ ldgenpisys . (Contributed by Thierry Arnoux, 18-Jul-2020.) $) ldgenpisyslem3 $p |- ( ph -> E C_ { b e. ~P O | ( A i^i b ) e. E } ) $= - ( wcel cv wss crab cint wi wral id rgenw ssintrab sseqtr4i sseldi cpw + ( wcel cv wss crab cint wi wral id rgenw ssintrab sseqtrri sseldi cpw mpbir cin cfi cfv ispisys sylib simpld elpwi syl adantr simpr syl3anc wa inelpisys ralrimiva jca ssrab sylibr ldgenpisyslem2 ) ABCDEFGHIJKL MNOPQRAGHEGGDUAUBZDIUCUDZHGVMUBVLVLUEZDIUFVNDIVLUGUHVLDGIUIUMQUJZSUKA @@ -482833,7 +482831,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry elmbfmvol2 $p |- ( F e. ( dom vol MblFnM BrSiga ) -> F e. MblFn ) $= ( vx cvol cbrsiga co wcel crn wral wss cfv ctb ax-mp ctop cuni cmap wb cr csiga elrnsiga mp1i cdm cmbfm cmbf ccnv cima cioo csigagen retopbas bastg - cv ctg retop sssigagen sstri df-brsiga sseqtr4i wceq wa dmvlsiga brsigarn + cv ctg retop sssigagen sstri df-brsiga sseqtrri wceq wa dmvlsiga brsigarn eqid ismbfm simprbi ssralv mpsyl simplbi elmapi unibrsiga unidmvol eleq2s wf oveq12i ismbf 3syl mpbird ) ACUAZDUBEFZAUCFZAUDBUJUEVPFZBUFGZHZVTDIVQV SBDHZWAVTVTUKJZUGJZDVTWCWDVTKFVTWCIUHVTKUILWCMFWCWDIULWCMUMLUNUOUPVQADNZV @@ -482911,7 +482909,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry sylibr mprg sseli elpwid xpss2 jaoi mprgbir wfun c1st funmpt cvv c2nd clt sylbi wbr rexr a1i ltpnf lbico1 syl3anc anim1i anim2i elxp7 3imtr4i xp1st wceq oveq1 xpeq1d fvmpt eleqtrrd elunirn2 sylancr ssriv ssun3 ax-mp uniun - sseqtr4i eqssi ) ADAEZFGHZDIZJZUAZBDDBEZFGHZIZJZUAZUBZKZDDIZXCXDLCEZXDLZC + sseqtrri eqssi ) ADAEZFGHZDIZJZUAZBDDBEZFGHZIZJZUAZUBZKZDDIZXCXDLCEZXDLZC XBCXBXDUCXEXBMXEWPMZXEXAMZUDXFXEWPXAUEXGXFXHXGXEXDWPXDUFZXEWNXIMZWPXILADA DWNXIWOWOUGZUHWLDMZWNXDLZXJXLWMDLZXMXLFUIMZXNNWLFUJUKWMDDULOWNXDWMDWLFGPQ TUMUNUOUPUQXHXEXDXAXIXEWSXIMZXAXILBDBDWSXIWTWTUGUHWQDMZWSXDLZXPXQWRDLZXRX @@ -483770,7 +483768,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry syl6eqss ) ABIJGKZHKZLBJUCUDMBJNOUCBJPGCQZRZHUESUEAGBCDHEFTUFHUEUAUB $. $d A a e $. - $( Property of being a Catatheodory measurable set. (Contributed by + $( Property of being a Caratheodory measurable set. (Contributed by Thierry Arnoux, 17-May-2020.) $) elcarsg $p |- ( ph -> ( A e. ( toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O @@ -484265,7 +484263,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry omsmeas.r $e |- ( ph -> R : Q --> ( 0 [,] +oo ) ) $. omsmeas.d $e |- ( ph -> (/) e. dom R ) $. omsmeas.0 $e |- ( ph -> ( R ` (/) ) = 0 ) $. - $( The restriction of a constructed outer measure to Catatheodory + $( The restriction of a constructed outer measure to Caratheodory measurable sets is a measure. This theorem allows to construct measures from pre-measures with the required characteristics, as for the Lebesgue measure. (Contributed by Thierry Arnoux, 17-May-2020.) $) @@ -485323,7 +485321,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $( @{ @d w m f e n i @. - @( TODO with the previous definiton, fundamental sequences shall be the + @( TODO with the previous definition, fundamental sequences shall be the Cauchy sequences for ` ( W sitm M ) ` : ` U. ( Cau `` ( W sitm M ) ) ` Define the 'fundamental in the mean' sequences, in the sense of the @@ -485560,7 +485558,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry eulerpartlemsv3 $p |- ( A e. ( ( NN0 ^m NN ) i^i R ) -> ( S ` A ) = sum_ k e. ( 1 ... ( S ` A ) ) ( ( A ` k ) x. k ) ) $= ( vt cn0 cn co wcel cfv cv cmul csu c1 cc0 wceq wral cmap eulerpartlemsv1 - cin cfz wss cuz fzssuz nnuz sseqtr4i a1i wa ccnv cima cfn eulerpartlemelr + cin cfz wss cuz fzssuz nnuz sseqtrri a1i wa ccnv cima cfn eulerpartlemelr wf simpld adantr sselda ffvelrnd nn0cnd nncnd mulcld cdif caddc ralrimiva eulerpartlems fveqeq2 cbvralv sylibr eulerpartlemsf ffvelrni nndiffz1 syl raleqdv mpbird r19.21bi oveq1d simpr eldifad mul02d eqtrd eqimssi eqtr4d @@ -489380,7 +489378,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ( vb c1 cv wcel cdif crab wn cfv cima funmpt2 ballotlemrinv wss wral wa rabid ballotlemrc adantr ballotlem1c ex ballotlem1ri notbid sylibrd imp jca sylbi rgen wceq eleq2 cbvrabv eleq2i bitr3i ralbii mpbi wfun cdm wb - elrab ssrab2 cvv fvex imaexg ax-mp dmmpti sseqtr4i nfrab1 nfmpt1 nfcxfr + elrab ssrab2 cvv fvex imaexg ax-mp dmmpti sseqtrri nfrab1 nfmpt1 nfcxfr cmpt funimass4f mp2an mpbir ballotlemic rinvf1o ) UEMUFZUGZMLGUHZUIZWRU JZMWSUIZCMWSWQDUKZWQULZCUCUMZABCDEFGHIJKLMNOPQRSTUAUBUCUNCWTULXBUOZWQCU KZXBUGZMWTUPZXGWSUGZUEXGUGZUJZUQZMWTUPXIXMMWTWQWTUGWQWSUGZWRUQZXMWRMWSU @@ -489726,21 +489724,20 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry Arnoux, 5-Oct-2018.) $) ofcccat $p |- ( ph -> ( ( F ++ G ) oFC R K ) = ( ( F oFC R K ) ++ ( G oFC R K ) ) ) $= - ( co cc0 chash cfv cfzo wcel cmul cfn wceq syl cconcat csn cxp cofc cword - wf fconst6g iswrdi 3syl fzofi snfi hashxp mp2an c1 wrdfin hashcl hashfzo0 - cof cn0 hashsng oveq12d nn0cnd mulid1d syl5req ofccat ccatcl syl2anc wrdf - eqtrd cvv ovexd ofcof caddc ccatlen oveq2d xpeq1d eqid ccatmulgnn0dir a1i - eqtr4d 3eqtr4d ) AEFUAKZLEMNZOKZGUBZUCZLFMNZOKZWEUCZUAKZBURZKZEWFWKKZFWIW - KKZUAKWBGBUDZKZEGWOKZFGWOKZUAKABCDEFWFWIHIAGDPZWDDWFUFWFDUEZPJWDGDUGDWCWF - UHUIAWSWHDWIUFWIWTPJWHGDUGDWGWIUHUIAWFMNZWDMNZWEMNZQKZWCWDRPZWERPZXAXDSLW - CUJZGUKZWDWEULUMAXDWCUNQKWCAXBWCXCUNQAWCUSPZXBWCSAECUEZPZERPXIHCEUOEUPUIZ - WCUQTAWSXCUNSJGDUTTZVAAWCAWCXLVBVCVIVDAWIMNZWHMNZXCQKZWGWHRPZXFXNXPSLWGUJ - ZXHWHWEULUMAXPWGUNQKWGAXOWGXCUNQAWGUSPZXOWGSAFXJPZFRPXSICFUOFUPUIZWGUQTXM - VAAWGAWGYAVBVCVIVDVEAWPWBLWBMNZOKZWEUCZWKKWLAYCCGBWBVJDAWBXJPZYCCWBUFAXKX - TYEHICEFVFVGCWBVHTALYBOVKJVLAYDWJWBWKAYDLWCWGVMKZOKZWEUCZWJAYCYGWEAYBYFLO - AXKXTYBYFSHICEFVNVGVOVPAWFWIYHDGWCWGWFVQWIVQYHVQJXLYAVRVTVOVIAWQWMWRWNUAA - WDCGBERDAXKWDCEUFHCEVHTXEAXGVSJVLAWHCGBFRDAXTWHCFUFICFVHTXQAXRVSJVLVAWA - $. + ( cconcat co cc0 chash cfv cfzo wcel cmul wceq syl csn cxp cof cofc cword + wf fconst6g iswrdi 3syl cfn fzofi snfi hashxp mp2an c1 cn0 lencl hashfzo0 + hashsng oveq12d nn0cnd mulid1d eqtrd syl5req ofccat cvv ccatcl wrdf ovexd + syl2anc ofcof caddc ccatlen oveq2d xpeq1d ccatmulgnn0dir 3eqtr4a 3eqtr4d + eqid ) AEFKLZMENOZPLZGUAZUBZMFNOZPLZWCUBZKLZBUCZLZEWDWILZFWGWILZKLVTGBUDZ + LZEGWMLZFGWMLZKLABCDEFWDWGHIAGDQZWBDWDUFWDDUEZQJWBGDUGDWAWDUHUIAWQWFDWGUF + WGWRQJWFGDUGDWEWGUHUIAWDNOZWBNOZWCNOZRLZWAWBUJQWCUJQZWSXBSMWAUKGULZWBWCUM + UNAXBWAUORLWAAWTWAXAUORAECUEZQZWAUPQZWTWASHCEUQZWAURUIAWQXAUOSJGDUSTZUTAW + AAWAAXFXGHXHTZVAVBVCVDAWGNOZWFNOZXARLZWEWFUJQXCXKXMSMWEUKXDWFWCUMUNAXMWEU + ORLWEAXLWEXAUORAFXEQZWEUPQZXLWESICFUQZWEURUIXIUTAWEAWEAXNXOIXPTZVAVBVCVDV + EAWNVTMVTNOZPLZWCUBZWILWJAXSCGBVTVFDAVTXEQZXSCVTUFAXFXNYAHICEFVGVJCVTVHTA + MXRPVIJVKAXTWHVTWIAMWAWEVLLZPLZWCUBZYDXTWHYDVSZAXSYCWCAXRYBMPAXFXNXRYBSHI + CCEFVMVJVNVOAWDWGYDDGWAWEWDVSWGVSYEJXJXQVPVQVNVCAWOWKWPWLKAWBCGBEVFDAXFWB + CEUFHCEVHTAMWAPVIJVKAWFCGBFVFDAXNWFCFUFICFVHTAMWEPVIJVKUTVR $. $} ${ @@ -490286,36 +490283,35 @@ strict in the case where the sets B(x) overlap. 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(Contributed by Thierry -> ( ( T ` ( F ++ <" K "> ) ) ` N ) = ( ( T ` F ) ` N ) ) $= ( wcel cfv co cr cword cc0 chash cfzo w3a cfz cv cconcat csgn cmpt cgsu cs1 wa wceq simp1 adantr 3ad2ant2 wss simp3 fzssfzo syl sselda ccatval1 - syl3anc fveq2d mpteq2dva oveq2d ccatcl syl2anc c1 caddc cuz lencl nn0zd - s1cl cz uzid peano2uz fzoss2 4syl 3ad2ant1 sseldd ccatlen oveq2i syl6eq - s1len eleqtrrd signstfval 3eqtr4d ) GUAUBZRZHUARZIUCGUDSZUETZRZUFZKDUCI - UGTZDUHZGHUMZUITZSZUJSZUKZULTZKDWRWSGSZUJSZUKZULTZIXABSSZIGBSSZWQXDXHKU - LWQDWRXCXGWQWSWRRZUNZXBXFUJXMWLWTWKRZWSWORXBXFUOWQWLXLWLWMWPUPZUQWQXNXL - WMWLXNWPHUAVPURZUQWQWRWOWSWQWPWRWOUSWLWMWPUTZIUCWNVAVBVCUAGWTWSVDVEVFVG - VHWQXAWKRZIUCXAUDSZUETZRXJXEUOWQWLXNXRXOXPUAGWTVIVJWQIUCWNVKVLTZUETZXTW - QWOYBIWLWMWOYBUSZWPWLWNVQRWNWNVMSZRYAYDRYCWLWNUAGVNVOWNVRWNWNVSWNUCYAVT - WAWBXQWCWQXSYAUCUEWQXSWNWTUDSZVLTZYAWQWLXNXSYFUOXOXPUAGWTWDVJYEVKWNVLHW - GWEWFVHWHABCDEFXAIJKLMNOPQWIVJWQWLWPXKXIUOXOXQABCDEFGIJKLMNOPQWIVJWJ $. + syl3anc fveq2d mpteq2dva oveq2d ccatws1cl 3adant3 caddc cuz lencl nn0zd + s1cl c1 uzidd peano2uz fzoss2 3syl 3ad2ant1 sseldd ccatlen s1len oveq2i + syl2anc syl6eq eleqtrrd signstfval 3eqtr4d ) GUAUBZRZHUARZIUCGUDSZUETZR + ZUFZKDUCIUGTZDUHZGHUMZUITZSZUJSZUKZULTZKDWRWSGSZUJSZUKZULTZIXABSSZIGBSS + ZWQXDXHKULWQDWRXCXGWQWSWRRZUNZXBXFUJXMWLWTWKRZWSWORXBXFUOWQWLXLWLWMWPUP + ZUQWQXNXLWMWLXNWPHUAVOURZUQWQWRWOWSWQWPWRWOUSWLWMWPUTZIUCWNVAVBVCUAGWTW + SVDVEVFVGVHWQXAWKRZIUCXAUDSZUETZRXJXEUOWLWMXRWPUAGHVIVJWQIUCWNVPVKTZUET + ZXTWQWOYBIWLWMWOYBUSZWPWLWNWNVLSZRYAYDRYCWLWNWLWNUAGVMVNVQWNWNVRWNUCYAV + SVTWAXQWBWQXSYAUCUEWQXSWNWTUDSZVKTZYAWQWLXNXSYFUOXOXPUAUAGWTWCWFYEVPWNV + KHWDWEWGVHWHABCDEFXAIJKLMNOPQWIWFWQWLWPXKXIUOXOXQABCDEFGIJKLMNOPQWIWFWJ + $. $d e f i k m n $. $d g m F $. $d m N $. $d a b e g k m n T $. $( In case the first letter is not zero, the zero skipping sign is never @@ -490467,25 +490464,25 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry signstfvc $p |- ( ( F e. Word RR /\ G e. Word RR /\ N e. ( 0 ..^ ( # ` F ) ) ) -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) ) $= - ( wcel cfv wceq vg ve vk cr cword cc0 chash cfzo co cconcat wa cv wi c0 - cs1 oveq2 fveq2d fveq1d eqeq1d imbi2d adantr simprl simpll simplr s1cld - ccatrid ccatass syl3anc ccatcl syl2anc cuz wss cz cle wbr cn0 lencl syl - nn0zd caddc nn0red nn0addge1 ccatlen breqtrrd eluz2 syl3anbrc signstfvp - fzoss2 simprr sseldd eqtr3d simpr eqtrd exp31 a2d wrdind 3impib 3com12 - ) HUDUEZRZGWSRZIUFGUGSZUHUIZRZIGHUJUIZBSZSZIGBSZSZTZWTXAXDXJXAXDUKZIGUA - ULZUJUIZBSZSZXITZUMXKIGUNUJUIZBSZSZXITZUMXKIGUBULZUJUIZBSZSZXITZUMXKIGY - AUCULZUOZUJUIZUJUIZBSZSZXITZUMXKXJUMUAUBUCHUDXLUNTZXPXTXKYMXOXSXIYMIXNX - RYMXMXQBXLUNGUJUPUQURUSUTXLYATZXPYEXKYNXOYDXIYNIXNYCYNXMYBBXLYAGUJUPUQU - RUSUTXLYHTZXPYLXKYOXOYKXIYOIXNYJYOXMYIBXLYHGUJUPUQURUSUTXLHTZXPXJXKYPXO - XGXIYPIXNXFYPXMXEBXLHGUJUPUQURUSUTXAXTXDXAIXRXHXAXQGBUDGVFUQURVAYAWSRZY - FUDRZUKZXKYEYLYSXKYEYLYSXKUKZYEUKYKYDXIYTYKYDTYEYTIYBYGUJUIZBSZSZYKYDYT - IUUBYJYTUUAYIBYTXAYQYGWSRUUAYITYSXAXDVBZYQYRXKVCZYTYFUDYQYRXKVDZVEUDGYA - YGVGVHUQURYTYBWSRZYRIUFYBUGSZUHUIZRUUCYDTYTXAYQUUGUUDUUEUDGYAVIVJZUUFYT - XCUUIIYTUUHXBVKSRZXCUUIVLYTXBVMRUUHVMRXBUUHVNVOUUKYTXBYTXAXBVPRUUDUDGVQ - VRZVSYTUUHYTUUGUUHVPRUUJUDYBVQVRVSYTXBXBYAUGSZVTUIZUUHVNYTXBUDRUUMVPRZX - BUUNVNVOYTXBUULWAYTYQUUOUUEUDYAVQVRXBUUMWBVJYTXAYQUUHUUNTUUDUUEUDGYAWCV - JWDXBUUHWEWFXBUFUUHWHVRYSXAXDWIWJABCDEFYBYFIJKLMNOPQWGVHWKVAYTYEWLWMWNW - OWPWQWR $. + ( cr wcel cfv vg ve vk cword cc0 chash cfzo co cconcat wceq wa cv wi c0 + cs1 fveq2d fveq1d eqeq1d imbi2d weq ccatrid adantr s1cl ccatass syl3an3 + oveq2 3expb adantlr ccatcl ad2ant2r simprr wss cuz cz cle wbr lencl cn0 + nn0zd syl caddc nn0red nn0addge1 syl2an breqtrrd eluz2 syl3anbrc fzoss2 + ccatlen simplr sseldd signstfvp eqtr3d id sylan9eq ex expcom a2d wrdind + syl3anc 3impib 3com12 ) HRUDZSZGXCSZIUEGUFTZUGUHZSZIGHUIUHZBTZTZIGBTZTZ + UJZXDXEXHXNXEXHUKZIGUAULZUIUHZBTZTZXMUJZUMXOIGUNUIUHZBTZTZXMUJZUMXOIGUB + ULZUIUHZBTZTZXMUJZUMXOIGYEUCULZUOZUIUHZUIUHZBTZTZXMUJZUMXOXNUMUAUBUCHRX + PUNUJZXTYDXOYQXSYCXMYQIXRYBYQXQYABXPUNGUIVFUPUQURUSUAUBUTZXTYIXOYRXSYHX + MYRIXRYGYRXQYFBXPYEGUIVFUPUQURUSXPYLUJZXTYPXOYSXSYOXMYSIXRYNYSXQYMBXPYL + GUIVFUPUQURUSXPHUJZXTXNXOYTXSXKXMYTIXRXJYTXQXIBXPHGUIVFUPUQURUSXEYDXHXE + IYBXLXEYAGBRGVAUPUQVBYEXCSZYJRSZUKZXOYIYPXOUUCYIYPUMXOUUCUKZYIYPUUDYIYO + YHXMUUDIYFYKUIUHZBTZTZYOYHUUDIUUFYNUUDUUEYMBXEUUCUUEYMUJZXHXEUUAUUBUUHU + UBXEUUAYKXCSUUHYJRVCRGYEYKVDVEVGVHUPUQUUDYFXCSZUUBIUEYFUFTZUGUHZSUUGYHU + JXEUUAUUIXHUUBRGYEVIZVJXOUUAUUBVKUUDXGUUKIXEUUAXGUUKVLZXHUUBXEUUAUKZUUJ + XFVMTSZUUMUUNXFVNSZUUJVNSXFUUJVOVPUUOXEUUPUUAXEXFRGVQZVSVBUUNUUJUUNUUIU + UJVRSUULRYFVQVTVSUUNXFXFYEUFTZWAUHZUUJVOXEXFRSUURVRSXFUUSVOVPUUAXEXFUUQ + WBRYEVQXFUURWCWDRRGYEWIWEXFUUJWFWGXFUEUUJWHVTVJXEXHUUCWJWKABCDEFYFYJIJK + LMNOPQWLWTWMYIWNWOWPWQWRWSXAXB $. $} ${ @@ -490619,39 +490616,39 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry /\ K e. RR ) -> ( V ` ( F ++ <" K "> ) ) = ( ( V ` F ) + if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) ) $= ( wcel cc0 cfv c1 cr cword c0 csn cdif wne wa cs1 cconcat co chash cfzo - cmin cif csu caddc cmul clt wbr wceq simpl eldifad simpr ccatcl syl2anc - cv s1cld signsvvfval syl ccatlen s1len oveq2i syl6eq oveq2d sumeq1d cuz - cn eldifsn lennncl sylbi nnuz syl6eleq adantr cfz cc 1cnd wn 0cnd fveq2 - ifclda fvoveq1 neeq12d ifbid fzosump1 3eqtrd adantlr wss fzo0ss1 sselda + cv cmin cif csu cmul clt wbr wceq eldifi s1cl ccatcl syl2an signsvvfval + caddc syl ccatlen s1len oveq2i syl6eq oveq2d sumeq1d cn eldifsn lennncl + cuz sylbi nnuz syl6eleq adantr cfz cc 1cnd wn 0cnd ifclda fveq2 fvoveq1 + neeq12d ifbid fzosump1 3eqtrd adantlr eldifad simplr wss fzo0ss1 sselda a1i signstfvp syl3anc cz cn0 elfzoel2 adantl 1nn0 eluzmn sylancl fzoss2 - wb elfzoelz elfzom1b mpbid sseldd sumeq2dv eqtr4d csgn ad2antrr fzo0end - signstfvn cneg cpr signstfvcl syldan rexrd sgncl signswch sgnsgn breq1d - ctp cxr neg1rr 1re prssi mp2an sseldi sgnclre sgnmulsgn 3bitr4d oveq12d - 3bitrd eqtrd ) GUAUBZUCUDZUEQZRGSRUFZUGZHUAQZUGZGHUHZUIUJZISZTGUKSZULUJ - ZEVFZUULBSZSZUUPTUMUJZUUQSZUFZTRUNZEUOZUUNUUQSZUUNTUMUJZUUQSZUFZTRUNZUP - UJZGISZUVEGBSZSZHUQUJRURUSZTRUNZUPUJUUFUUIUUMUVIUTUUGUUFUUIUGZUUMTUULUK - SZULUJZUVBEUOZTUUNTUPUJZULUJZUVBEUOUVIUVOUULUUDQZUUMUVRUTUVOGUUDQZUUKUU - DQZUWAUVOGUUDUUEUUFUUIVAVBZUVOHUAUUFUUIVCZVGZUAGUUKVDVEABCDEFUULIJKLMNO - PVHVIUVOUVQUVTUVBEUVOUVPUVSTULUVOUVPUUNUUKUKSZUPUJZUVSUVOUWBUWCUVPUWHUT - UWDUWFUAGUUKVJVEUWGTUUNUPHVKVLVMVNVOUVOUVBUVHETUUNUUFUUNTVPSZQUUIUUFUUN - VQUWIUUFUWBGUCUFUGUUNVQQZGUUDUCVRUAGVSVTZWAWBWCUVOUUPTUUNWDUJQUGZUVATRW - EUWLUVAUGWFUWLUVAWGUGWHWJUUPUUNUTZUVAUVGTRUWMUURUVDUUTUVFUUPUUNUUQWIUUP - UUNTUUQUMWKWLWMWNWOWPUUJUVCUVJUVHUVNUPUUFUUIUVCUVJUTUUGUVOUVCUUOUUPUVKS - ZUUSUVKSZUFZTRUNZEUOZUVJUVOUUOUVBUWQEUVOUUPUUOQZUGZUVAUWPTRUWTUURUWNUUT - UWOUWTUWBUUIUUPRUUNULUJZQUURUWNUTUVOUWBUWSUWDWCZUVOUUIUWSUWEWCZUVOUUOUX - AUUPUUOUXAWQUVOUUNWRWTWSABCDEFGHUUPIJKLMNOPXAXBUWTUWBUUIUUSUXAQUUTUWOUT - UXBUXCUWTRUVEULUJZUXAUUSUWTUUNUVEVPSQZUXDUXAWQUWTUUNXCQZTXDQUXEUWSUXFUV - OUUPTUUNXEXFZXGUUNTXHXIUVERUUNXJVIUWTUWSUUSUXDQZUVOUWSVCUWTUUPXCQZUXFUW - SUXHXKUWSUXIUVOUUPTUUNXLXFUXGUUPUUNXMVEXNXOABCDEFGHUUSIJKLMNOPXAXBWLWMX - PUVOUWBUVJUWRUTUWDABCDEFGIJKLMNOPVHVIXQWPUUJUVGUVMTRUUJUVGUVLHXRSZAUJZU - VLUFZUVLUXJUQUJRURUSZUVMUUJUVDUXKUVFUVLUUFUUIUVDUXKUTUUGABCDEFGHIJKLMNO - PYAWPUUJUWBUUIUVEUXAQZUVFUVLUTUUFUUIUWBUUGUWDWPUUHUUIVCZUUJUWJUXNUUFUWJ - UUGUUIUWKXSUUNXTVIZABCDEFGHUVEIJKLMNOPXAXBWLUUJUVLTYBZTYCZQZUXJUXQRTYKQ - ZUXLUXMXKUUHUUIUXNUXSUXPABCDEFGUVEIJKLMNOPYDYEZUUJHYLQZUXTUUJHUXOYFZHYG - VIAJUVLUXJKLMNYHVEUUJUVLXRSZUXJXRSZUQUJZRURUSZUYDUXJUQUJZRURUSZUXMUVMUU - JUYFUYHRURUUJUYEUXJUYDUQUUJUYBUYEUXJUTUYCHYIVIVNYJUUJUVLUAQZUXJUAQZUXMU - YGXKUUJUXRUAUVLUXQUAQTUAQUXRUAWQYMYNUXQTUAYOYPUYAYQZUUIUYKUUHHYRXFUVLUX - JYSVEUUJUYJUUIUVMUYIXKUYLUXOUVLHYSVEYTUUBWMUUAUUC $. + simpl elfzo1elm1fzo0 sseldd sumeq2dv eqtr4d csgn simpr fzo0end ad2antrr + signstfvn cneg cpr ctp wb signstfvcl syldan rexr sgncl signswch syl2anc + cxr rexrd sgnsgn breq1d neg1rr 1re prssi mp2an sseldi sgnmulsgn sgnclre + sylancom 3bitr4d 3bitrd oveq12d eqtrd ) GUAUBZUCUDZUEQZRGSRUFZUGZHUAQZU + GZGHUHZUIUJZISZTGUKSZULUJZEUMZUUOBSZSZUUSTUNUJZUUTSZUFZTRUOZEUPZUUQUUTS + ZUUQTUNUJZUUTSZUFZTRUOZVFUJZGISZUVHGBSZSZHUQUJRURUSZTRUOZVFUJUUIUULUUPU + VLUTUUJUUIUULUGZUUPTUUOUKSZULUJZUVEEUPZTUUQTVFUJZULUJZUVEEUPUVLUVRUUOUU + GQZUUPUWAUTUUIGUUGQZUUNUUGQZUWDUULGUUGUUHVAZHUAVBZUAGUUNVCVDABCDEFUUOIJ + KLMNOPVEVGUVRUVTUWCUVEEUVRUVSUWBTULUVRUVSUUQUUNUKSZVFUJZUWBUUIUWEUWFUVS + UWJUTUULUWGUWHUAUAGUUNVHVDUWITUUQVFHVIVJVKVLVMUVRUVEUVKETUUQUUIUUQTVQSZ + QUULUUIUUQVNUWKUUIUWEGUCUFUGUUQVNQZGUUGUCVOUAGVPVRZVSVTWAUVRUUSTUUQWBUJ + QUGZUVDTRWCUWNUVDUGWDUWNUVDWEUGWFWGUUSUUQUTZUVDUVJTRUWOUVAUVGUVCUVIUUSU + UQUUTWHUUSUUQTUUTUNWIWJWKWLWMWNUUMUVFUVMUVKUVQVFUUIUULUVFUVMUTUUJUVRUVF + UURUUSUVNSZUVBUVNSZUFZTRUOZEUPZUVMUVRUURUVEUWSEUVRUUSUURQZUGZUVDUWRTRUX + BUVAUWPUVCUWQUXBUWEUULUUSRUUQULUJZQUVAUWPUTUVRUWEUXAUVRGUUGUUHUUIUULXKW + OZWAZUUIUULUXAWPZUVRUURUXCUUSUURUXCWQUVRUUQWRWTWSABCDEFGHUUSIJKLMNOPXAX + BUXBUWEUULUVBUXCQUVCUWQUTUXEUXFUXBRUVHULUJZUXCUVBUXBUUQUVHVQSQZUXGUXCWQ + UXBUUQXCQZTXDQUXHUXAUXIUVRUUSTUUQXEXFXGUUQTXHXIUVHRUUQXJVGUXAUVBUXGQUVR + UUSUUQXLXFXMABCDEFGHUVBIJKLMNOPXAXBWJWKXNUVRUWEUVMUWTUTUXDABCDEFGIJKLMN + OPVEVGXOWNUUMUVJUVPTRUUMUVJUVOHXPSZAUJZUVOUFZUVOUXJUQUJRURUSZUVPUUMUVGU + XKUVIUVOUUIUULUVGUXKUTUUJABCDEFGHIJKLMNOPXTWNUUMUWEUULUVHUXCQZUVIUVOUTU + UIUULUWEUUJUXDWNUUKUULXQZUUIUXNUUJUULUUIUWLUXNUWMUUQXRVGXSZABCDEFGHUVHI + JKLMNOPXAXBWJUUMUVOTYAZTYBZQZUXJUXQRTYCQZUXLUXMYDUUKUULUXNUXSUXPABCDEFG + UVHIJKLMNOPYEYFZUULUXTUUKUULHYKQZUXTHYGHYHVGXFAJUVOUXJKLMNYIYJUUMUVOXPS + ZUXJXPSZUQUJZRURUSZUYCUXJUQUJZRURUSZUXMUVPUUMUYEUYGRURUUMUYDUXJUYCUQUUM + UYBUYDUXJUTUUMHUXOYLHYMVGVLYNUUMUVOUAQZUXJUAQZUXMUYFYDUUMUXRUAUVOUXQUAQ + TUAQUXRUAWQYOYPUXQTUAYQYRUYAYSZUULUYJUUKHUUAXFUVOUXJYTYJUUKUULUYIUVPUYH + YDUYKUVOHYTUUBUUCUUDWKUUEUUF $. $} ${ @@ -490843,19 +490840,17 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry signshf $p |- ( ( F e. Word RR /\ C e. RR+ ) -> H : ( 0 ..^ ( ( # ` F ) + 1 ) ) --> RR ) $= ( cr co vx vy cword wcel crp wa cc0 chash cfv c1 caddc cfzo cs1 cconcat - cmul cofc cmin cof wf cv resubcl adantl 0red s1cld simpl ccatcl syl2anc - cvv wrdf syl wceq ccatlen oveq1i syl6eq 1cnd cfn cn0 wrdfin hashcl 3syl - s1len nn0cnd addcomd eqtrd oveq2d feq2d mpbid syldan oveq2i ovexd simpr - remulcl rpred ofcf inidm off feq1i sylibr ) HSUCZUDZAUEUDZUFZUGHUHUIZUJ - UKTZULTZSUGUMZHUNTZHXFUNTZAUOUPTZUQURTZUSXESIUSXBUAUBXEXEXEUQSSSXGXIVHV - HUAUTZSUDUBUTZSUDUFZXKXLUQTSUDXBXKXLVAVBXBUGXGUHUIZULTZSXGUSZXESXGUSXBX - GWSUDZXPXBXFWSUDZWTXQXBUGSXBVCVDZWTXAVEZSXFHVFVGSXGVIVJXBXOXESXGXBXNXDU - GULXBXNUJXCUKTZXDXBXNXFUHUIZXCUKTZYAXBXRWTXNYCVKXSXTSXFHVLVGYBUJXCUKUGW - AZVMVNXBUJXCXBVOXBXCXBWTHVPUDXCVQUDXTSHVRHVSVTWBWCWDWEWFWGXBUAUBXEAUOSS - SXHVHXMXKXLUOTSUDXBXKXLWLVBXBUGXHUHUIZULTZSXHUSZXESXHUSXBXHWSUDZYGWTXAX - RYHXSSHXFVFWHSXHVIVJXBYFXESXHXBYEXDUGULXBYEXCYBUKTZXDWTXAXRYEYIVKXSSHXF - VLWHYBUJXCUKYDWIVNWEWFWGXBUGXDULWJZXBAWTXAWKWMWNYJYJXEWOWPXESIXJRWQWR - $. + cmul cofc cmin cof wf cvv cv resubcl adantl s1cl ax-mp ccatcl mpan wrdf + 0re syl 1cnd lencl nn0cnd wceq ccatlen s1len oveq1i syl6eq oveq2d feq2d + comraddd mpbid remulcl mpan2 ccatws1len ovexd rpre ofcf inidm off feq1i + adantr sylibr ) HSUCZUDZAUEUDZUFZUGHUHUIZUJUKTZULTZSUGUMZHUNTZHXAUNTZAU + OUPTZUQURTZUSWTSIUSWQUAUBWTWTWTUQSSSXBXDUTUTUAVAZSUDUBVAZSUDUFZXFXGUQTS + UDWQXFXGVBVCWOWTSXBUSZWPWOUGXBUHUIZULTZSXBUSZXIWOXBWNUDZXLXAWNUDZWOXMUG + SUDXNVIUGSVDVEZSXAHVFVGSXBVHVJWOXKWTSXBWOXJWSUGULWOXJUJWRWOVKWOWRSHVLVM + WOXJXAUHUIZWRUKTZUJWRUKTXNWOXJXQVNXOSSXAHVOVGXPUJWRUKUGVPVQVRWAVSVTWBWL + WQUAUBWTAUOSSSXCUTXHXFXGUOTSUDWQXFXGWCVCWOWTSXCUSZWPWOUGXCUHUIZULTZSXCU + SZXRWOXCWNUDZYAWOXNYBXOSHXAVFWDSXCVHVJWOXTWTSXCWOXSWSUGULSHUGWEVSVTWBWL + WQUGWSULWFZWPASUDWOAWGVCWHYCYCWTWIWJWTSIXERWKWM $. $} $( ` H ` , corresponding to the word ` F ` multiplied by ` ( x - C ) ` , is @@ -491302,7 +491297,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry Thierry Arnoux, 20-Dec-2021.) $) rpsqrtcn $p |- ( sqrt |` RR+ ) e. ( RR+ -cn-> RR+ ) $= ( vx csqrt crp cres ccncf co wcel wf cv cdm cc cr wceq sqrtf ax-mp wb mpbir - wss cc0 cpnf cfv wa wral rpssre ax-resscn sstri fdm sseqtr4i sseli rpsqrtcl + wss cc0 cpnf cfv wa wral rpssre ax-resscn sstri fdm sseqtrri sseli rpsqrtcl rgen wfun ffun ffvresb cico cioo ioossico eqsstrri resabs1 resqrtcn rescncf jca ioorp mp2 eqeltrri cncffvrn mp2an ) BCDZCCEFGZCCVHHZVJAIZBJZGZVKBUACGZU BZACUCZVOACVKCGVMVNCVLVKCKVLCLKUDUEUFZKKBHZVLKMNKKBUGOUHUIVKUJVBUKBULZVJVPP @@ -494310,8 +494305,8 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ). See ~ df-ofs . ( clpad co cfv chash cc0 cmin cfzo csn caddc wcel nn0cnd cconcat fveq2d cxp lpadval cword wceq lpadlem1 ccatlen syl2anc lpadlem2 oveq1d cn0 syl lencl npcand 3eqtrd eqtrd ) ADBEJKLZMLNDEMLZOKZPKBQUCZEUAKZMLZDAURVBMAB - CDEFGHUDUBAVCVAMLZUSRKZUTUSRKDAVACUEZSEVFSZVCVEUFABCDEHUGGCVAEUHUIAVDUT - USRABCDEFGHIUJUKADUSADFTAUSAVGUSULSGCEUNUMTUOUPUQ $. + CDEFGHUDUBAVCVAMLZUSRKZUTUSRKDAVACUEZSEVFSZVCVEUFABCDEHUGGCCVAEUHUIAVDU + TUSRABCDEFGHIUJUKADUSADFTAUSAVGUSULSGCEUNUMTUOUPUQ $. $} $( Length of a left-padded word, in the general case, expressed with an @@ -495652,7 +495647,7 @@ without the distinct variable restriction ($d ` A ` ` x ` ). $( First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) $) bnj931 $p |- B C_ A $= - ( cun ssun1 sseqtr4i ) BBCEABCFDG $. + ( cun ssun1 sseqtrri ) BBCEABCFDG $. $} ${ @@ -497636,9 +497631,9 @@ indirect lemma of the theorem in question (i.e. a lemma of a lemma... of bnj581.6 $e |- ( ch' <-> [. g / f ]. ch ) $. $( Technical lemma for ~ bnj580 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a - lemma... of the theorem). (Unnecessary distinct variable restrictions - were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan - Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) $) + lemma... of the theorem). (Contributed by Jonathan Ben-Naim, + 3-Jun-2011.) Remove unnecessary distinct variable conditions. (Revised + by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) $) bnj581 $p |- ( ch' <-> ( g Fn n /\ ph' /\ ps' ) ) $= ( cv wsbc wfn w3a sbcbii sbc3an bnj62 bicomi 3anbi123i bitr4i 3bitri ) IC DENZODNFNZPZABQZDUEOZUEUFPZGHQZMCUHDUEJRUIUGDUEOZADUEOZBDUEOZQUKUGABDUESU @@ -499775,7 +499770,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a bnj1137 $p |- ( ( R _FrSe A /\ X e. A ) -> _TrFo ( B , A , R ) ) $= ( vv wcel wa cv c-bnj14 wss wral c-bnj18 sseli bnj906 sylan2 sselda sstrd syl w-bnj15 w-bnj19 ciun wo cun eleq2i elun bitri bnj213 adantlr bnj18eq1 - ssiun2s bnj1147 rgenw iunss mpbir bnj1125 3expia ralrimiv sylibr sseqtr4i + ssiun2s bnj1147 rgenw iunss mpbir bnj1125 3expia ralrimiv sylibr sseqtrri jaodan ssun2 syl6ss sylan2b ralrimiva df-bnj19 ) BDUAZEBHZIZBDGJZKZCLZGCM BCDUBVJVMGCVKCHZVJVKBDEKZHZVKABDENZBDAJZNZUCZHZUDZVMVNVKVOVTUEZHWBCWCVKFU FVKVOVTUGUHVJWBIVLVTCVJVPVLVTLWAVJVPIZVLBDVKNZVTVPVJVKBHZVLWELZVOBVKBDEUI @@ -503548,7 +503543,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ( c1 caddc co cfz wf1o cfv wceq cpr cun cop cin c0 cz cvv f1oprswap wcel mp2an a1i chash cmin subfacp1lem1 simp1d f1oun syl22anc simp2d 1z wb f1oeq1 ax-mp f1oeq2 syl5bbr f1oeq3 bitrd syl mpbid csn f1ofun - wfun wss cdm snsspr1 ssun2 sseqtr4i sstri 1ex snid eleqtrri funssfv + wfun wss cdm snsspr1 ssun2 sseqtrri sstri 1ex snid eleqtrri funssfv dmsnop mp3an23 fvsn syl6eq snsspr2 3jca ) AUCMUCUDUEUFUEZWQIUGZUCIU HZLUILIUHZUCUIAKUCLUJZUKZXBJUCLULZLUCULZUJZUKZUGZWRAKKJUGXAXAXEUGZK XAUMUNUIZXIXGUBXHAUCUOURLUPURXHVHSUCLUOUPUQUSUTAXIXBWQUIZKVAUHMUCVB @@ -503565,7 +503560,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by 23-Jan-2015.) $) subfacp1lem2b $p |- ( ( ph /\ X e. K ) -> ( F ` X ) = ( G ` X ) ) $= ( wcel wa wfun wss cdm cfv wceq caddc cfz wf1o subfacp1lem2a simp1d - c1 co f1ofun syl adantr cop cpr cun ssun1 sseqtr4i a1i f1odm eleq2d + c1 co f1ofun syl adantr cop cpr cun ssun1 sseqtrri a1i f1odm eleq2d biimpar funssfv syl3anc ) ANKUDZUEZIUFZJIUGZNJUHZUDZNIUINJUIUJAVNVL AUPMUPUKUQULUQZVRIUMZVNAVSUPIUILUJLIUIUPUJABCDEFGHIJKLMOPQRSTUAUBUC UNUOVRVRIURUSUTVOVMJJUPLVALUPVAVBZVCIJVTVDUBVEVFAVQVLAVPKNAKKJUMVPK @@ -504135,7 +504130,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by erdszelem2 $p |- ( ( # " S ) e. Fin /\ ( # " S ) C_ NN ) $= ( vx chash cima cfn wcel cn wss cres c1 cv clt wiso mp2an cvv wfo cfz cpw co fzfi pwfi mpbi wa crab ssrab2 eqsstri ssfi wfun cdm cn0 cpnf csn hashf - cun ffun ax-mp ssv fdmi sseqtr4i fores fofi cfv wral funimass4 erdszelem1 + cun ffun ax-mp ssv fdmi sseqtrri fores fofi cfv wral funimass4 erdszelem1 wf wb w3a c0 wne 3ad2ant3 simp1 sylancr hashnncl syl mpbird sylbi mprgbir ne0i pm3.2i ) HCIZJKZWFLMZCJKZCWFHCNZUAZWGOBUBUDZUCZJKZCWMMWIWLJKZWNOBUEZ WLUFUGCAPZDWQIQEDWQNRBWQKUHZAWMUIWMFWRAWMUJUKWMCULSHUMZCHUNZMZWKTUOUPUQUS @@ -504774,7 +504769,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by 11-Feb-2015.) $) kur14lem7 $p |- ( N e. T -> ( N C_ X /\ { ( X \ N ) , ( K ` N ) } C_ T ) ) $= - ( cdif wss cfv cpr wa ctp cun wcel wo elun w3o eltpi ssun1 sseqtr4i sstri + ( cdif wss cfv cpr wa ctp cun wcel wo elun w3o eltpi ssun1 sseqtrri sstri wceq ctop topopn ax-mp elexi difss ssexi tpid2 sselii kur14lem1 kur14lem4 fvex tpid3 tpid1 eqeltri ssun2 kur14lem3 eqsstri eqeltrri kur14lem5 3jaoi syl difeq2i eqtri kur14lem2 eqtr4i fveq2i 3eqtr4i eqtr3i jaoi sylbi prid1 @@ -504831,7 +504826,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by Carneiro, 11-Feb-2015.) $) kur14lem9 $p |- ( S e. Fin /\ ( # ` S ) <_ ; 1 4 ) $= ( vs c1 c4 cdc cv wcel cdif cfv cpr wss wral wa cpw crab cint wi elintrab - vex ctp cun ssun1 sseqtr4i sstri topopn ax-mp elexi ssexi tpid1 kur14lem7 + vex ctp cun ssun1 sseqtrri sstri topopn ax-mp elexi ssexi tpid1 kur14lem7 ctop sselii simprd simpld elpw2 sylibr ssriv mpbir eleq2 sseq2 raleqbi1dv rgen pwex wceq anbi12d imbi12d rspccv mp2ani sylbi eqsstri kur14lem8 1nn0 mpi 4nn0 deccl hashsslei ) HGUDUEUFGCAUGZUHZLBUGZUIWTKUJUKZWRULZBWRUMZUNZ @@ -511239,56 +511234,56 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership A. c e. ( C \ V ) ( F ` <" c "> ) = <" c "> /\ A. x e. R A. y e. R ( F ` ( x ++ y ) ) = ( ( F ` x ) ++ ( F ` y ) ) ) ) ) $= - ( wcel cfv wceq co cvv c0 vw vr vv crn wf cv cs1 cdif wral cconcat mrsubf - w3a mrsubcn ralrimiva mrsubccat 3expb ralrimivva 3jca cmpt cun cfrmd ccom + ( vv wcel cfv wceq co c0 vw vr crn wf cv cs1 cdif wral cconcat w3a mrsubf + mrsubcn ralrimiva mrsubccat 3expb ralrimivva 3jca cmpt cun cfrmd cif ccom wa cgsu cword mrexval adantr s1eq fveq2d eqid fvex fvmpt adantl wn difun2 eleq2i eldif bitr3i simpr2 eqeq12d rspccva sylan2br anassrs eqcomd ifeqda - sylan mpteq2dva coeq1d oveq2d mpteq12dv elun2 simpr1 simpr s1cld ad2antrr - cif wss eleqtrrd ffvelrnd sylan2 cbvmptv ssid mrsubfval sylancl cmnd cmhm - fmptd cvrmd cmcn fvexi cmvar unex frmdmnd ax-mp a1i eleqtrd fmpttd cplusg - feq23d simpr3 wb simprl simprr cbs frmdbas eqcomi frmdadd syl2anc ffvelrn - mpbid ad2ant2lr ad2ant2l 2ralbidva raleqbidv bitr3d chash cc0 caddc fveq2 - raleqdv 3ad2antr1 cn0 wrd0 lencl syl nn0cnd 0cnd addid1d eleqtrrid oveq1d - fvoveq1 oveq2 ccatlid syl6eq rspc2va ccatlen 3eqtrrd hasheq0 sylib pm3.2i - 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(Contributed by Scott Fenton, 28-Feb-2011.) $) dfon2lem2 $p |- U. { x | ( x C_ A /\ ph /\ ps ) } C_ A $= - ( cv wss w3a cab cpw cuni simp1 ss2abi df-pw sseqtr4i sspwuni mpbi ) CEDF + ( cv wss w3a cab cpw cuni simp1 ss2abi df-pw sseqtrri sspwuni mpbi ) CEDF ZABGZCHZDIZFSJDFSQCHTRQCQABKLCDMNSDOP $. $} @@ -517338,7 +517333,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= wbr frrlem10 sylan2 adantlr cop fveq1i c0 wfun frrlem9 funres dmres df-fn wfn syl sylanblrc adantr vex ovex fnsn a1i wn eldifn disjsn sylibr adantl nsyl simpr fvun1 syl112anc syl5eq elinel1 fvresd eqtrd wss frrlem11 fnfun - ssun1 sseqtr4i wral eldifi rspa frrlem8 ssind syl6sseqr fun2ssres syl3anc + ssun1 sseqtrri wral eldifi rspa frrlem8 ssind syl6sseqr fun2ssres syl3anc syl2an resabs1d oveq2d 3eqtr4d ex fvsn vsnid fvun2 reseq1i resundir eqtri wfr predfrirr ressnop0 uneq12d un0 syl6eq 3eqtr4a fveq2 id predeq3 syl5bi reseq2d oveq12d eqeq12d syl5ibrcom jaod 3impia ) ADUGZHPUHZUIUJZEUGZLUUIU @@ -517623,6 +517618,7 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= KUEABCDFEUFUG $. $} + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Surreal Numbers @@ -517631,11 +517627,14 @@ C Fn ( ( S i^i dom F ) u. { z } ) ) $= $( Set up the new syntax for surreal numbers $) $c No ( ph <-> ps ) ) $. - $( Version of ~ cbval2 with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) - (Proof modification is discouraged.) $) - bj-cbval2v $p |- ( A. x A. y ph <-> A. z A. w ps ) $= - ( wal nfal weq wi nfv nfim wb cbvalv1 19.21v pm5.74d 3bitr3i pm5.74ri - expcom ) ADLZBFLZCEAEDGMBCFIMCENZUEUFUGAOZDLUGBOZFLUGUEOUGUFOUHUIDFUGAFUG - FPHQUGBDUGDPJQDFNZUGABUGUJABRKUDUASUGADTUGBFTUBUCS $. - - $( Version of ~ cbvex2 with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) - (Proof modification is discouraged.) $) - bj-cbvex2v $p |- ( E. x E. y ph <-> E. z E. w ps ) $= - ( wn wal wex nfn weq wa notbid bj-cbval2v 2exnaln notbii 3bitr4i ) ALZDMC - MZLBLZFMEMZLADNCNBFNENUDUFUCUECDEFAEGOAFHOBCIOBDJOCEPDFPQABKRSUAACDTBEFTU - B $. - $} - ${ $d z w ph $. $d x y ps $. $d x y z w $. bj-cbval2vv.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. - $( Version of ~ cbval2v with a disjoint variable condition, which does not + $( Version of ~ cbval2vv with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) $) bj-cbval2vv $p |- ( A. x A. y ph <-> A. z A. w ps ) $= - ( nfv bj-cbval2v ) ABCDEFAEHAFHBCHBDHGI $. + ( nfv cbval2v ) ABCDEFAEHAFHBCHBDHGI $. - $( Version of ~ cbvex2v with a disjoint variable condition, which does not + $( Version of ~ cbvex2vv with a disjoint variable condition, which does not require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) $) bj-cbvex2vv $p |- ( E. x E. y ph <-> E. z E. w ps ) $= - ( nfv bj-cbvex2v ) ABCDEFAEHAFHBCHBDHGI $. + ( nfv cbvex2v ) ABCDEFAEHAFHBCHBDHGI $. $} ${ @@ -532598,8 +532573,8 @@ replacing a nonfree hypothesis with a disjoint variable condition (see require ~ ax-13 . (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.) $) bj-cbvex4vv $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $= - ( wex weq wa 2exbidv bj-cbvex2vv 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJN - ZINHNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. + ( wex weq wa 2exbidv cbvex2vv 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN + HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. $} ${ @@ -532654,41 +532629,6 @@ universe has at least two objects (see ~ dtru ). (Contributed by BJ, ( wal wi axc11r bj-aecomsv ) ABDACDECBABCFG $. $} - ${ - $d x y $. - bj-dral1v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $. - $( Version of ~ dral1 with a disjoint variable condition, which does not - require ~ ax-13 . Remark: the corresponding versions for ~ dral2 and - ~ drex2 are instances of ~ albidv and ~ exbidv respectively. - (Contributed by BJ, 17-Jun-2019.) - (Proof modification is discouraged.) $) - bj-dral1v $p |- ( A. x x = y -> ( A. x ph <-> A. y ps ) ) $= - ( weq wal nfa1 albid bj-axc11v axc11r impbid bitrd ) CDFZCGZACGBCGZBDGZOA - BCNCHEIOPQBCDJBDCKLM $. - $} - - ${ - $d x y $. - bj-drex1v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $. - $( Version of ~ drex1 with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by BJ, 17-Jun-2019.) - (Proof modification is discouraged.) $) - bj-drex1v $p |- ( A. x x = y -> ( E. x ph <-> E. y ps ) ) $= - ( weq wal wn wex notbid bj-dral1v df-ex 3bitr4g ) CDFCGZAHZCGZHBHZDGZHACI - BDINPROQCDNABEJKJACLBDLM $. - $} - - ${ - $d x y $. - bj-drnf1v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $. - $( Version of ~ drnf1 with a disjoint variable condition, which does not - require ~ ax-13 . (Contributed by BJ, 17-Jun-2019.) - (Proof modification is discouraged.) $) - bj-drnf1v $p |- ( A. x x = y -> ( F/ x ph <-> F/ y ps ) ) $= - ( weq wal wi wnf bj-dral1v imbi12d nf5 3bitr4g ) CDFCGZAACGZHZCGBBDGZHZDG - ACIBDIPRCDNABOQEABCDEJKJACLBDLM $. - $} - ${ $d x y z $. bj-drnf2v.1 $e |- ( A. x x = y -> ( ph <-> ps ) ) $. @@ -532794,24 +532734,6 @@ universe has at least two objects (see ~ dtru ). (Contributed by BJ, GHBCIQBQQGZJRQBKCKDLQMNOP $. $} - ${ - $d x y $. - $( TODO: delete after having added to ~ dvdemo1 and ~ dvdemo2 the following - comment: - - "Note that ~ dvdemo1 and ~ dvdemo2 are "partially bundled" in that - ` x , z ` (resp. ` x , y ` ) and ` y , z ` need not be disjoint - variables, and in spite of that, neither requires ~ ax-11 nor ~ ax-13 , - and furthermore, ~ dvdemo2 does not require any of the auxiliary axioms - ( ~ ax-10 , ~ ax-11 , ~ ax-12 , ~ ax-13 ). - - (Contributed by BJ, 16-Jul-2019.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - bj-dvdemo1 $p |- E. x ( x = y -> z e. x ) $= - ( weq wn wel wi wex wal dtru exnal mpbir pm2.21 eximii ) ABDZEZOCAFZGAPAH - OAIEABJOAKLOQMN $. - $} - $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- @@ -533426,7 +533348,7 @@ The proof goes by induction on the complexity of the formula (see op. cit. $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- - Classes without extensionality + Classes without the axiom of extensionality -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- A few results about classes can be proved without using ~ ax-ext . One could @@ -534695,7 +534617,7 @@ FOL part ( ~ bj-ru0 ) and then two versions ( ~ bj-ru1 and ~ bj-ru ). $( The singletonization is included in the tagging. (Contributed by BJ, 6-Oct-2018.) $) bj-snglsstag $p |- sngl A C_ tag A $= - ( bj-csngl c0 csn cun bj-ctag ssun1 df-bj-tag sseqtr4i ) ABZJCDZEAFJKGAHI + ( bj-csngl c0 csn cun bj-ctag ssun1 df-bj-tag sseqtrri ) ABZJCDZEAFJKGAHI $. $( The singletonization is included in the tagging. (Contributed by BJ, @@ -535059,7 +534981,7 @@ sethood hypotheses (compare ~ opth ). (Contributed by BJ, 6-Oct-2018.) $) ( bj-c1upl cdif wne wpss c1o csn bj-ctag cxp cun wss difeq2i cin wceq ax-mp c0 mpbi 0pss bj-c2uple xpundi bj-disjsn01 xpdisj1 eqtr3i disjdif2 1oex snnz incom wa bj-tagn0 pm3.2i xpnz eqnetri eqnetrri mpbir ssun2 sscon df-bj-2upl - ssdif sstri df-bj-1upl uneq1i eqtri difeq1i sseqtr4i psssstr mp2an difn0 ) + ssdif sstri df-bj-1upl uneq1i eqtri difeq1i sseqtrri psssstr mp2an difn0 ) ABUAZCDZEZRFZVJVKFRVLGZVMRHIZBJZKZRIZAJZKZVRCJZKZLZEZGZWDVLMVNWEWDRFVQVRVSW ALZKZEZWDRWGWCVQVRVSWAUBNWHVQRVQWGOZRPWHVQPWGVQOZWIRWGVQUIVRVOORPWJRPUCVRVO WFVPUDQUEVQWGUFQVORFZVPRFZUJVQRFWKWLHUGUHBUKULVOVPUMSUNUOWDTUPWDVJWBEZVLWDV @@ -535687,14 +535609,14 @@ Moore collections (complements) of elements of ` A ` relative to ` X ` belong to some class ` B ` : the LHS singles out the empty intersection (the empty intersection relative to ` X ` is ` X ` and the intersection of a nonempty family of subsets - of ` X ` in included in ` X ` , so there is no need to intersect it with + of ` X ` is included in ` X ` , so there is no need to intersect it with ` X ` ). In typical applications, ` B ` is ` A ` itself. (Contributed by BJ, 7-Dec-2021.) $) bj-0int $p |- ( A C_ ~P X -> ( ( X e. B /\ A. x e. ( ~P A \ { (/) } ) |^| x e. B ) <-> A. x e. ~P A ( X i^i |^| x ) e. B ) ) $= ( cpw wss wcel cv cint c0 csn wral wa cin wb wceq df-ss a1i wi eleq1 cdif - cvv ssv int0 sseqtr4i mpbi eqcomi eleq1i wne eldifsn sstr2 bj-intss elpwi + cvv ssv int0 sseqtrri mpbi eqcomi eleq1i wne eldifsn sstr2 bj-intss elpwi syl6 syl11 impd syl5bi incom eqeq1i eqcom sylbb sylbi syl5 ralrimiv ralbi syld anbi12d biancomd 0elpw inteq ineq2 3syl bj-raldifsn ax-mp syl6bbr syl ) BDEZFZDCGZAHZIZCGZABEZJKUAZLZMZDWANZCGZAWDLZDJIZNZCGZMZWHAWCLZVRWFW @@ -536397,7 +536319,7 @@ Identity relation (complements) $( A variant of ~ relopabiv (which could be proved from it, similarly to ~ relxp from ~ xpss ). (Contributed by BJ, 28-Dec-2023.) $) bj-opabssvv $p |- { <. x , y >. | ph } C_ ( _V X. _V ) $= - ( copab cv cvv wcel wa cxp vex pm3.2i a1i ssopab2i df-xp sseqtr4i ) ABCDB + ( copab cv cvv wcel wa cxp vex pm3.2i a1i ssopab2i df-xp sseqtrri ) ABCDB EFGZCEFGZHZBCDFFIARBCRAPQBJCJKLMBCFFNO $. $} @@ -537104,12 +537026,12 @@ extended reals (which bypasses the intermediate definition of a temporary $( Complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) $) bj-ccssccbar $p |- CC C_ CCbar $= - ( cc cccinfty cun cccbar ssun1 df-bj-ccbar sseqtr4i ) AABCDABEFG $. + ( cc cccinfty cun cccbar ssun1 df-bj-ccbar sseqtrri ) AABCDABEFG $. $( Infinite extended complex numbers are extended complex numbers. (Contributed by BJ, 27-Jun-2019.) $) bj-ccinftyssccbar $p |- CCinfty C_ CCbar $= - ( cccinfty cc cun cccbar ssun2 df-bj-ccbar sseqtr4i ) ABACDABEFG $. + ( cccinfty cc cun cccbar ssun2 df-bj-ccbar sseqtrri ) ABACDABEFG $. $( Token for "plus infinity". $) $c pinfty $. @@ -537839,14 +537761,36 @@ singleton on a couple (with disjoint domain) at a point in the domain ZUTUMUOVAGUOVAULVAUOULUNEBNOPQACRUAUMUOUBUKUTUOUPUKUTUQUKUTHUOUPABCUCUDTUEU FTUGQUKUPUOSUMUKUPUOABCUHTUIUJ $. + ${ + bj-isvec.scal $e |- ( ph -> K = ( Scalar ` V ) ) $. + $( The predicate "is a vector space". (Contributed by BJ, 6-Jan-2024.) $) + bj-isvec $p |- ( ph -> ( V e. LVec <-> ( V e. LMod /\ K e. DivRing ) ) ) $= + ( clvec wcel clmod csca cfv cdr eqid islvec eqcomd eleq1d anbi2d syl5bb + wa ) CEFCGFZCHIZJFZQARBJFZQSCSKLATUARASBJABSDMNOP $. + $} + $( Fields are division rings. (Contributed by BJ, 6-Jan-2024.) $) - bj-fielddivring $p |- Field C_ DivRing $= + bj-flddrng $p |- Field C_ DivRing $= ( cfield cdr ccrg cin df-field inss1 eqsstri ) ABCDBEBCFG $. $( The field of real numbers is a division ring. (Contributed by BJ, 6-Jan-2024.) $) bj-rrdrg $p |- RRfld e. DivRing $= - ( cfield cdr crefld bj-fielddivring refld sselii ) ABCDEF $. + ( cfield cdr crefld bj-flddrng refld sselii ) ABCDEF $. + + ${ + bj-isclm.scal $e |- ( ph -> F = ( Scalar ` W ) ) $. + bj-isclm.base $e |- ( ph -> K = ( Base ` F ) ) $. + $( The predicate "is a subcomplex module". (Contributed by BJ, + 6-Jan-2024.) $) + bj-isclm $p |- ( ph -> ( W e. CMod <-> + ( W e. LMod /\ F = ( CCfld |`s K ) /\ K e. ( SubRing ` CCfld ) ) ) ) $= + ( cclm wcel clmod csca cfv ccnfld cbs cress wceq csubrg w3a eqid eqcomd + co isclm fveq2 wa eqtr syl2im mpd oveq2d eqeq12d eleq1d 3anbi23d syl5bb + ex ) DGHDIHZDJKZLUNMKZNTZOZUOLPKZHZQAUMBLCNTZOZCURHZQUNUODUNRUORUAAUQVAUS + VBUMAUNBUPUTABUNESAUOCLNABUNOZUOCOZEACBMKZOZVCVEUOOZVDFBUNMUBVFVGVDVFVGUC + CUOCVEUOUDSULUEUFZUGUHAUOCURVHUIUJUK $. + $} $( Symbol for the class of real vector spaces. $) $c RRVec $. @@ -537857,7 +537801,9 @@ singleton on a couple (with disjoint domain) at a point in the domain $( Definition of the class of real vector spaces. The previous definition, ` |- RRVec = { x e. LMod | ( Scalar `` x ) = RRfld } ` , can be recovered using ~ bj-isrvec . The present one is preferred since it does not use - any dummy variable. (Contributed by BJ, 9-Jun-2019.) $) + any dummy variable. That ` RRVec ` could be defined with ` LVec ` in + place of ` LMod ` is a consequence of ~ bj-isrvec2 . (Contributed by BJ, + 9-Jun-2019.) $) df-bj-rvec $a |- RRVec = ( LMod i^i ( `' Scalar " { RRfld } ) ) $. $( The predicate "is a real vector space". (Contributed by BJ, @@ -537880,7 +537826,7 @@ singleton on a couple (with disjoint domain) at a point in the domain bj-rvecssmod $p |- RRVec C_ LMod $= ( vx crrvec clmod cv bj-rvecmod ssriv ) ABCADEF $. - $( The field of scalars of a rela vector space is the field of real numbers. + $( The field of scalars of a real vector space is the field of real numbers. (Contributed by BJ, 6-Jan-2024.) $) bj-rvecrr $p |- ( V e. RRVec -> ( Scalar ` V ) = RRfld ) $= ( crrvec wcel clmod csca cfv crefld wceq bj-isrvec simprbi ) ABCADCAEFGHAIJ @@ -537888,54 +537834,41 @@ singleton on a couple (with disjoint domain) at a point in the domain ${ bj-isrvecd.scal $e |- ( ph -> ( Scalar ` V ) = K ) $. - $( The predicate "is a real vector space"; deduction form. (Contributed by - BJ, 6-Jan-2024.) $) + $( The predicate "is a real vector space". (Contributed by BJ, + 6-Jan-2024.) $) bj-isrvecd $p |- ( ph -> ( V e. RRVec <-> ( V e. LMod /\ K = RRfld ) ) ) $= ( crrvec wcel clmod csca cfv crefld wceq bj-isrvec eqeq1d anbi2d syl5bb wa ) CEFCGFZCHIZJKZPAQBJKZPCLASTQARBJDMNO $. $} - ${ - bj-islvecd.scal $e |- ( ph -> K = ( Scalar ` V ) ) $. - $( The predicate "is a vector space"; deduction form. (Contributed by BJ, - 6-Jan-2024.) $) - bj-islvecd $p |- - ( ph -> ( V e. LVec <-> ( V e. LMod /\ K e. DivRing ) ) ) $= - ( clvec wcel clmod csca cfv cdr eqid islvec eqcomd eleq1d anbi2d syl5bb - wa ) CEFCGFZCHIZJFZQARBJFZQSCSKLATUARASBJABSDMNOP $. - $} - $( Real vector spaces are vector spaces (elemental version). (Contributed by BJ, 6-Jan-2024.) $) bj-rvecvec $p |- ( V e. RRVec -> V e. LVec ) $= ( crrvec wcel clvec clmod crefld cdr bj-rvecmod bj-rrdrg a1i csca bj-rvecrr - cfv eqcomd bj-islvecd mpbir2and ) ABCZADCAECFGCZAHRQIJQFAQAKMFALNOP $. + cfv eqcomd bj-isvec mpbir2and ) ABCZADCAECFGCZAHRQIJQFAQAKMFALNOP $. + + ${ + bj-isrvec2.scal $e |- ( ph -> ( Scalar ` V ) = K ) $. + $( The predicate "is a real vector space". (Contributed by BJ, + 6-Jan-2024.) $) + bj-isrvec2 $p |- ( ph -> ( V e. RRVec <-> ( V e. LVec /\ K = RRfld ) ) ) $= + ( crrvec wcel clvec crefld wceq wa wi bj-rvecvec a1i cfv bj-rvecrr eqeq1d + csca syl5ib jcad clmod bj-vecssmodel anim1i bj-isrvecd syl5ibr impbid ) A + CEFZCGFZBHIZJZAUFUGUHUFUGKACLMUFCQNZHIAUHCOAUJBHDPRSUIUFACTFZUHJUGUKUHCUA + UBABCDUCUDUE $. + $} $( Real vector spaces are vector spaces. (Contributed by BJ, 6-Jan-2024.) $) bj-rvecssvec $p |- RRVec C_ LVec $= ( vx crrvec clvec cv bj-rvecvec ssriv ) ABCADEF $. - ${ - bj-isclmd.scal $e |- ( ph -> F = ( Scalar ` W ) ) $. - bj-isclmd.base $e |- ( ph -> K = ( Base ` F ) ) $. - $( The predicate "is a subcomplex module"; deduction form. (Contributed by - BJ, 6-Jan-2024.) $) - bj-isclmd $p |- ( ph -> ( W e. CMod <-> - ( W e. LMod /\ F = ( CCfld |`s K ) /\ K e. ( SubRing ` CCfld ) ) ) ) $= - ( cclm wcel clmod csca cfv ccnfld cbs cress wceq csubrg w3a eqid eqcomd - co isclm fveq2 wa eqtr syl2im mpd oveq2d eqeq12d eleq1d 3anbi23d syl5bb - ex ) DGHDIHZDJKZLUNMKZNTZOZUOLPKZHZQAUMBLCNTZOZCURHZQUNUODUNRUORUAAUQVAUS - VBUMAUNBUPUTABUNESAUOCLNABUNOZUOCOZEACBMKZOZVCVEUOOZVDFBUNMUBVFVGVDVFVGUC - CUOCVEUOUDSULUEUFZUGUHAUOCURVHUIUJUK $. - $} - $( Real vector spaces are subcomplex modules (elemental version). (Contributed by BJ, 6-Jan-2024.) $) bj-rveccmod $p |- ( V e. RRVec -> V e. CMod ) $= ( crrvec wcel cclm clmod crefld ccnfld cr cress co wceq csubrg cfv df-refld - bj-rvecmod a1i cdr resubdrg simpli csca bj-rvecrr eqcomd rebase bj-isclmd - cbs mpbir3and ) ABCZADCAECFGHIJKZHGLMCZAOUHUGNPUIUGUIFQCRSPUGFHAUGATMFAUAUB - HFUEMKUGUCPUDUF $. + bj-rvecmod a1i cdr resubdrg simpli csca bj-rvecrr eqcomd bj-isclm mpbir3and + cbs rebase ) ABCZADCAECFGHIJKZHGLMCZAOUHUGNPUIUGUIFQCRSPUGFHAUGATMFAUAUBHFU + EMKUGUFPUCUD $. $( Real vector spaces are subcomplex modules. (Contributed by BJ, 6-Jan-2024.) $) @@ -537945,7 +537878,7 @@ singleton on a couple (with disjoint domain) at a point in the domain $( Real vector spaces are subcomplex vector spaces. (Contributed by BJ, 6-Jan-2024.) $) bj-rvecsscvec $p |- RRVec C_ CVec $= - ( crrvec cclm clvec ccvs bj-rvecsscmod bj-rvecssvec ssini df-cvs sseqtr4i + ( crrvec cclm clvec ccvs bj-rvecsscmod bj-rvecssvec ssini df-cvs sseqtrri cin ) ABCJDABCEFGHI $. $( Real vector spaces are subcomplex vector spaces (elemental version). @@ -538783,7 +538716,7 @@ coordinates of a barycenter of two points in one dimension (complex icoreelrn $p |- ( ( A e. RR /\ B e. RR ) -> { z e. RR | ( A <_ z /\ z < B ) } e. I ) $= ( va vb cr wcel wa cico co cv cle wbr clt crab icoreval cxp cxr simpl cpw - cima simpr wf wfun df-ico ixxf ffun mp1i cdm wss rexpssxrxp fdmi sseqtr4i + cima simpr wf wfun df-ico ixxf ffun mp1i cdm wss rexpssxrxp fdmi sseqtrri a1i elovimad syl6eleqr eqeltrrd ) BHIZCHIZJZBCKLZBAMZNOVDCPOJAHQDABCRVBVC KHHSZUCDVBBCHHKUTVAUAUTVAUDTTSZTUBZKUEKUFVBFGANPKFGAUGUHZVFVGKUIUJVEKUKZU LVBVEVFVIUMVFVGKVHUNUOUPUQEURUS $. @@ -539055,10 +538988,10 @@ coordinates of a barycenter of two points in one dimension (complex quantifier, using implicit substitution. (Contributed by ML, 27-Mar-2021.) $) cbveud $p |- ( ph -> ( E! x ps <-> E! y ch ) ) $= - ( vz cv wceq wb wal wex weu nfvd nfbid eu6 wa simpr equequ1 adantr sylcom - bibi12d ex cbv2 exbidv 3bitr4g ) ABDLZKLZMZNZDOZKPCELZULMZNZEOZKPBDQCEQAU - OUSKAUNURDEFGABUMEHAUMERSACUQDIAUQDRSAUKUPMZBCNZUNURNZJUTVAVBUTVAUABCUMUQ - UTVAUBUTUMUQNVADEKUCUDUFUGUEUHUIBDKTCEKTUJ $. + ( vz weq wb wal wex weu nfvd nfbid wa eu6 simpr equequ1 adantr bibi12d ex + sylcom cbv2w exbidv 3bitr4g ) ABDKLZMZDNZKOCEKLZMZENZKOBDPCEPAULUOKAUKUND + EFGABUJEHAUJEQRACUMDIAUMDQRADELZBCMZUKUNMZJUPUQURUPUQSBCUJUMUPUQUAUPUJUMM + UQDEKUBUCUDUEUFUGUHBDKTCEKTUI $. $} ${ @@ -539415,7 +539348,7 @@ coordinates of a barycenter of two points in one dimension (complex ( F ` <. N , X >. ) ) $= ( com wcel c2o wss wa cvv cop cfv c1st c1o wceq c0 cif co cuni df-ov cmpo cxp cv eqeq1 eleq1 bi2anan9 wb adantl unieq adantr fveq2 opeq12 ifbieq12d - a1i opeq12d ifbieq2d wpss wne sssucid df-2o sseqtr4i 1on sucneqoni necomi + a1i opeq12d ifbieq2d wpss wne sssucid df-2o sseqtrri 1on sucneqoni necomi csuc df-pss mpbir2an ssnpss sseq2 mtbiri con2i intnanrd iffalsed sylan9eq mt2 iftrue sylan9eqr adantlll simpll elex opex ovmpod syl5reqr ) EHIZJEKZ LZFMBUEZIZLZEFNZDOEFDUAEUBZFPOZNZEFDUCWLCAEFHMCUFZQRZAUFZBIZLZSWSWJIZWQUB @@ -541742,7 +541675,7 @@ in disguise (see ~ wl-dfralsb ). $( Restrict an existential quantifier to a class ` A ` . This version does not interpret elementhood verbatim as ` E. x e. A ph ` does. Assuming a real elementhood can lead to awkward consequences should the class ` A ` - depend on ` x ` . Instead we base the definiton on ~ df-wl-ral , where + depend on ` x ` . Instead we base the definition on ~ df-wl-ral , where this is ruled out. Other definitions are ~ wl-dfrexsb and ~ wl-dfrexex . If ` x ` is not free in ` A ` , the defining expression can be simplified (see ~ wl-dfrexf , ~ wl-dfrexv ). @@ -541810,7 +541743,7 @@ can be simplified (see ~ wl-dfrexf , ~ wl-dfrexv ). $( Restrict "at most one" to a given class ` A ` . This version does not interpret elementhood verbatim like ` E* x e. A ph ` does. Assuming a real elementhood can lead to awkward consequences should the class ` A ` - depend on ` x ` . Instead we base the definiton on ~ df-wl-ral , where + depend on ` x ` . Instead we base the definition on ~ df-wl-ral , where this is already ruled out. This definition lets ` x ` appear as a formal parameter with no @@ -541865,7 +541798,7 @@ can be simplified (see ~ wl-dfrexf , ~ wl-dfrexv ). $( Restrict existential uniqueness to a given class ` A ` . This version does not interpret elementhood verbatim like ` E! x e. A ph ` does. Assuming a real elementhood can lead to awkward consequences should the - class ` A ` depend on ` x ` . Instead we base the definiton on + class ` A ` depend on ` x ` . Instead we base the definition on ~ df-wl-ral , where this is ruled out. This definition lets ` x ` appear as a formal parameter with no connection @@ -542001,7 +541934,7 @@ can be simplified (see ~ wl-dfrexf , ~ wl-dfrexv ). $( The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020.) $) imadifss $p |- ( ( F " A ) \ ( F " B ) ) C_ ( F " ( A \ B ) ) $= - ( cima cdif cun wss ssun2 undif2 sseqtr4i imass2 ax-mp imaundi sseqtri mpbi + ( cima cdif cun wss ssun2 undif2 sseqtrri imass2 ax-mp imaundi sseqtri mpbi ssundif ) CADZCBDZCABEZDZFZGQRETGQCBSFZDZUAAUBGQUCGABAFUBABHBAIJAUBCKLCBSMN QRTPO $. @@ -543418,7 +543351,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= reseq2d uneq12 adantrr adantrl 3adantr1 anbi12d rspcev syl12anc anim12i elrabi eqtr2 difeq1 3eqtr3g sylbi eqeqan12d anandis xpopth bitrd syl5ib opth sylan2 ralrimivva jctird reu4 rexrab anbi1i f1ompt sylanbrc ss2abi - eqid f1of mapval sseqtr4i ssexi xpex rabex f1oen ) ADUDZJIUDZUEOZVXNUFO + eqid f1of mapval sseqtrri ssexi xpex rabex f1oen ) ADUDZJIUDZUEOZVXNUFO ZUGEUDZUHUIZUJZUGUKZULZVXPVXQUGUMUIZGUHUIZUJZPUKZULZQZUMUUAZUIZGUGUMUIZ HUHUIZVYEULZQZBUNZRZEPGUHUIZUOZDVYPUPZIPFUQUIZUGGUHUIZURUIZVYTVYTCUDZUS ZCUTZULZVAZVXMJVXOVYAVXPVYBVYJUHUIZUJZVYEULZQZVYHUIZVYJUGUMUIZHUHUIZVYE @@ -545813,7 +545746,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cmul vex adantr nfan bitri eleq1 notbid eqeq2 syl5ibcom expimpd sylancr wb syl2anc eqtrd adantlr eqeq1 rspcva cen cdom wfo copab anbi12d sylan2 syl2an eqcomd adantl syl5bb wex anbi2d csbeq1 anim1i wreu sylbi anass - cin cfzo cmap csu fzofi f1of ss2abi ovex mapval sseqtr4i ssfi pm3.2i 2z + cin cfzo cmap csu fzofi f1of ss2abi ovex mapval sseqtrri ssfi pm3.2i 2z mapfi mp1i snfi hashcl nn0zi cn cof oveq2 imaeq2d oveq1 uneq12d imaeq1d oveq2d chvar ad4ant14 xp1st elmapi ad2antlr fvex f1oeq1 elab nfcsb nfne oveq12d cbvcsb syl5eq rspc adantll 1st2nd2 ad3antlr neeqtrd poimirlem25 @@ -546027,7 +545960,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= simpl ralimi neeq1d rexbidv syl6bb sylan sylib breqtrrid sylibr cuz syl wb wo bitri syl6bbr rexeqdv adantr adantlr xp1st ralbidv syldan fveqeq2 3syl eqeq1d fveq1d 3anbi123d anbi12d cof crn cfzo csu fzofi f1of ss2abi - ovex mapval sseqtr4i ssfi xpfi hashcl nn0zd dfrex2 nfrab1 nffv nfbr wex + ovex mapval sseqtrri ssfi xpfi hashcl nn0zd dfrex2 nfrab1 nffv nfbr wex neq0 iddvds hashsng oveq2i df-2 eqtr4i breqtrri diffi mp2b snfi disjdif vex incom eqtri hashun mp3an difsnid syl5eqr wreu ad3antrrr ifbid eqtrd csbeq2dv mpteq2dv breq1 id ifbieq12d oveq2 imaeq2d oveq1d oveq2d syl6eq @@ -547249,7 +547182,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= crn ctg cn0 cexp cop cmpo crab cima cuni cpw wral weq fveq2 sseq1d simprr cz elrab fvex elpw sylibr sylan2b ralrimiva wfun cdm wb cxr cxp iccf ffun wf ax-mp ssrab2 cle oveq1 oveq1d opeq12d oveq2 oveq2d cbvmpov dyadf inss2 - cin frn rexpssxrxp sstri fdmi sseqtr4i funimass4 mp2an sspwuni sylib cabs + cin frn rexpssxrxp sstri fdmi sseqtrri funimass4 mp2an sspwuni sylib cabs cmin ccom cres cbl crp wrex cxmet eqid rexmet cmopn mopni2 mp3an1 elssuni tgioo wceq uniretop syl6sseqr sselda rpre bl2ioo syl2an 2re 1lt2 expnlbnd cn mp3an23 ad2antrl cmul cfl ad2antrr 2nn nnnn0 nnexpcl sylancr nnred syl @@ -547591,7 +547524,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= mp1i 3jca vtoclg mpcom rpred suprlub ad2antll expr sylbid rexlimiva sylbi exlimiv simplrr anim12d reeanv syl6ibr cuni cxp cmap cabs cseq eqid 3expa ovolgelb an32s wf elmapi ssid ovollb mpan2 cpnf ovolsf frn icossxr syl6ss - supxrcl readdcl rexrd rncoss unissi unirnioo sseqtr4i xrletr mpand anim2d + supxrcl readdcl rexrd rncoss unissi unirnioo sseqtrri xrletr mpand anim2d cico 3syl reximdva mpd rexex indif2 df-ss biimpi difeq1d syl5eq ctb bastg retopbas uniopn opncld incld simprll simplll eqcoms biantrud bitr3d simpl ssdif2d adantlll ad4antlr cun difdif2 fveq2i unssi ovolge0 xrrege0 syl2an @@ -547737,7 +547670,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= csup ovolss wb ovolcl adantl syl2anc ancoms sylan2 simprrr uniretop cldss sylib opnmbl difmbl eqeltrrd mblvol ad2antrl ad2antrr wex ccom cuni c1 cn cin cmin cseq crp eqid ovolgelb wf elmapi ssid ovollb cc0 cpnf ovolsf frn - mpan2 syl6ss supxrcl rexrd rncoss unissi unirnioo sseqtr4i xrletr syl2anr + mpan2 syl6ss supxrcl rexrd rncoss unissi unirnioo sseqtrri xrletr syl2anr ax-mp mp3an1 mpand adantll anim2d reximdva mpd rexex difss sstri ad4antlr 3syl jctil mp2an mp3an12 ovolge0 xrrege0 adantrr readdcld ad3antlr syl2an impr sylan resubcld retop ctb uniopn ovolsscl ad5antr inss1 mp3an2 eqcomd @@ -547852,7 +547785,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= eqid eqeq2d anbi12d rspcev mp2an mpbir ne0ii suprcl syl2anc mp3an12 sstri c0ex elab simpll abbii supeq1i a1i vex adantr breq2 mpbid wex ctb cbvrexv id ex sylan cdm ctg ccld mblfinlem4 cpw elpwi cabs cmin c1 cseq cmap crab - cxp cinf elmapi rexrd rncoss unissi unirnioo sseqtr4i cico ovolsf icossxr + cxp cinf elmapi rexrd rncoss unissi unirnioo sseqtrri cico ovolsf icossxr frn syl6ss supxrcl 3syl pnfge nltpnft necon2abii ovolge0 0re mpanl12 mpan xrre3 sylbir dfss4 rembl cldopn opnmbl eqeltrrd sstr ctop 0cld 0mbl ovol0 difmbl eqtr2i difss2 ssrin ssdif dfin4 fveq2i oveq1i sseq2i anbi1i rexbii @@ -548115,7 +548048,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= wi syl6req a1d wfo wex csdm cvv reldom brrelex1i 0sdomg biimparc sylancom wb fodomr unissb anbi1i r19.26 bitr4i nulmbl eqtr expcom syl5 adantl jcai ovolctb2 syldan ralimi sylbi ancoms cfz co ciun cmpt crn cima cxr csup wf - clt caddc cfn fzfi cuz fzssuz nnuz sseqtr4i ffvelrnda eleq1 anbi12d sylan + clt caddc cfn fzfi cuz fzssuz nnuz sseqtrri ffvelrnda eleq1 anbi12d sylan fof an32s ralrimiva ssralv mpsyl sylancr adantr oveq2 iuneq1d iunex fvmpt weq ovex sseq12d fveq1 unieqd supeq1d sylancl cab df-iun fveq2 rspcev wfn wrex eqtrd syl5eqr cpnf volf cle csu mblss 0re 0xr fveqeq2 rspccva simpld @@ -548193,66 +548126,66 @@ curry M LIndF ( R freeLMod I ) ) ) $= cuni cfn syl eqeltrrd fex ex jcai feq2 anbi1d eqeq2 anbi12d imbi1d imbi2d feq1 reseq1 eleq1d ralbidv eleq1 imbi12d wal csn cun rzal biantrud bicomd c0 unieq syl6eq eqeq1d 2albidv weq raleq anbi2d simpl simpr feq12d adantr - uni0 adantl cbval2v syl6bb wrel cdm frel fdm eqcom biimpi sylan9eq reldm0 - wb biimpar mbf0 syl2anc cre ccom cim ref fco mpan 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$} ${ @@ -552168,7 +552101,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= mpbir2and uzind4 eleq2s impcom dmmptg mprg eqeq1d syl6bi syl5 equcoms dmeqd feq2d biimpcd sylcom rexlimdvw simpld ffvelrnd cbvmptv sbceq1dd fmptdf mpteq1 dfsbcq 3syl cbvralv sylibr mptex vex resex sbcie fzssuz - sseqtr4i sbceq1d syl5bbr ralbidv spcev syl2anc ) ATIUPTUPUQZUYSPURZUR + sseqtrri sbceq1d syl5bbr ralbidv spcev syl2anc ) ATIUPTUPUQZUYSPURZUR ZUSZUTZBKUPRNUQZVAVBZVUAUSZVCZNTVDZTIJUQZUTZCNTVDZVEZJVFAMTMUQZVUMPUR ZURZIVUBULAVUMTVGZVEZRVUMVAVBZIVUMVUNVUQVURIVUNUTZDKVUNVCZVUQVUEIVUNU TZBKVUNVCZVEZNTVKZVUSVUTVEZVUQVUNQVGZVVDATQVUMPUMVHVVFVUNVUEIKUQZUTZB @@ -554190,7 +554123,7 @@ counterexample is the discrete extended metric (assigning distinct crp syl3anc sseli caddc ad3antrrr eluznn ad2ant2lr ffvelrnd syl22anc cr metcl rpred mpan2d anassrs ccl cab ccmp ctop cuni cxmet metxmet mopntop clm cmet frnd mopnuni clscld rexlimdvw abssdv elpw2 sylibr elin ssabral - velpw anbi1i csn cun ffn 0z ssv int0 sseqtr4i ssun1 anim2i imim1i ssun2 + velpw anbi1i csn cun ffn 0z ssv int0 sseqtrri ssun1 anim2i imim1i ssun2 ralsn sscls sseq2 syl5ibrcom anim12i intunsn syl6sseqr rexlimdvv reeanv ssin ss2in cbvrexv 3imtr3g expd findcard2 com12 impr ffnd eqeltri mpan2 sylcom 1z uzn0 wfun fnfun fndm sseqtrrid funfvima2 necomd neneqd nrexdv @@ -563951,7 +563884,7 @@ respect to the class of reflexive relations (see ~ dfrefrels2 ) if the refrelsredund4 $p |- { r e. Rels | ( _I |` dom r ) C_ r } Redund <. RefRels , ( RefRels i^i SymRels ) >. $= ( cid cv cdm cres wss crels crab crefrels csymrels cin wredund wceq crn cxp - wi inxpssres sstr2 in32 inass ssrabi dfrefrels2 sseqtr4i ccnv wa dfsymrels2 + wi inxpssres sstr2 in32 inass ssrabi dfrefrels2 sseqtrri ccnv wa dfsymrels2 ax-mp ineq2i refsymrels2 3eqtr4i ineq1i 3eqtr3ri 3eqtri df-redund mpbir2an inrab ) BACZDZEZUQFZAGHZIIJKZLVAIFVAVBKZIVBKZMVABURUQNZOKZUQFZAGHIUTVGAGVFU SFUTVGPURVEBQVFUSUQRUGUAAUBUCVCVBIKZIIKJKVDVAJKZIKVAIKJKVHVCVAJISVIVBIVAUQU @@ -600250,7 +600183,7 @@ says that (our) ~ cdlemg10 "implies (2)" (of p. 116). No details are NVOVP $. $d h B $. $d h G $. - $( Elmininate ` h ` from ~ cdlemg47 . (Contributed by NM, 5-Jun-2013.) $) + $( Eliminate ` h ` from ~ cdlemg47 . (Contributed by NM, 5-Jun-2013.) $) cdlemg48 $p |- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( F =/= ( _I |` B ) /\ ( R ` F ) = ( R ` G ) ) ) -> ( F o. G ) = ( G o. F ) ) $= @@ -609290,7 +609223,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, simpr breq1 notbid elrab sylibr eqeq2 riotabidv fveq2d eqeq2d anbi1d eqid cfv opabbidv cuni crn cpw cxp ctendo fvexi uniex rnex pwex xpex simpl wss fvssunirn elssuni adantl rnss uniss 3syl elpw2 eqeltrd jca ssopab2i df-xp - sstrid sseqtr4i ssexi fvmpt syl eqtrd ) JLUDMHUDUEZCAUDCMKUFZUGZUEZUEZCIV + sstrid sseqtrri ssexi fvmpt syl eqtrd ) JLUDMHUDUEZCAUDCMKUFZUGZUEZUEZCIV HCUBUCUHZMKUFZUGZUCAUIZEUHZBFUHVHZUBUHZUJZFDUKZNUHZVHZUJZXMGUDZUEZENULZUM ZVHZXHXICUJZFDUKZXMVHZUJZXPUEZENULZXCCIXSWSIXSUJXBABDEFGHIJKLMNUCUBOPQRST UAUNUOUPXCCXGUDZXTYFUJXCXBYGWSXBUQXFXAUCCAXDCUJXEWTXDCMKURUSUTVAUBCXRYFXG @@ -615284,7 +615217,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= ~ dvh4dimlem directly? (Contributed by NM, 24-Apr-2015.) $) dvhdimlem $p |- ( ph -> E. z e. V -. z e. ( N ` { X , Y } ) ) $= ( cv ctp cfv wcel wrex cpr dvh4dimlem clmod wss dvhlmod csn df-tp prssi - wn cun syl2anc snssd unssd eqsstrid ssun1 sseqtr4i lspss syl3anc ssneld + wn cun syl2anc snssd unssd eqsstrid ssun1 sseqtrri lspss syl3anc ssneld a1i reximdv mpd ) ABUBZIJJUCZFUDZUEUOZBGUFVIIJUGZFUDZUEUOZBGUFABCDEFGHI JKJLMNOPQRRSTUAUAUHAVLVOBGAVNVKVIACUIUEVJGUJVMVJUJZVNVKUJACDEHLMPUKAVJV MJULZUPZGIJJUMZAVMVQGAIGUEJGUEVMGUJQRIJGUNUQAJGRURUSUTVPAVMVRVJVMVQVAVS @@ -627339,6 +627272,7 @@ fixed reference functional determined by this vector (corresponding to #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $) + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= Utility theorems @@ -627630,13 +627564,13 @@ fixed reference functional determined by this vector (corresponding to ccatcan2d.c $e |- ( ph -> C e. Word V ) $. $( Cancellation law for concatenation. (Contributed by SN, 6-Sep-2023.) $) ccatcan2d $p |- ( ph -> ( ( A ++ C ) = ( B ++ C ) <-> A = B ) ) $= - ( cconcat co wceq chash cfv cpfx cc wcel cn0 lencl syl adantr simpr cword - wa nn0cnd caddc fveq2d ccatlen syl2an2r 3eqtr3d addcan2ad oveq12d syl2anc - ex pfxccat1 eqeq12d sylibd oveq1 impbid1 ) ABDIJZCDIJZKZBCKZAVAUSBLMZNJZU - TCLMZNJZKZVBAVAVGAVAUCZUSUTVCVENAVAUAZVHVCVEDLMZAVCOPVAAVCABEUBZPZVCQPFEB - RSUDTAVEOPVAAVEACVKPZVEQPGECRSUDTAVJOPVAAVJADVKPZVJQPHEDRSUDTVHUSLMZUTLMZ - VCVJUEJZVEVJUEJZVHUSUTLVIUFAVLVAVNVOVQKFAVNVAHTZEBDUGUHAVMVAVNVPVRKGVSECD - UGUHUIUJUKUMAVDBVFCAVLVNVDBKFHEBDUNULAVMVNVFCKGHECDUNULUOUPBCDIUQUR $. + ( cconcat co wceq chash cfv cpfx cc wcel cn0 lencl adantr syl2anc ccatlen + wa simpr cword syl caddc fveq2 sylan9req eqtrd addcan2ad oveq12d pfxccat1 + nn0cnd ex eqeq12d sylibd oveq1 impbid1 ) ABDIJZCDIJZKZBCKZAVAUSBLMZNJZUTC + LMZNJZKZVBAVAVGAVAUBZUSUTVCVENAVAUCVHVCVEDLMZAVCOPVAAVCABEUDZPZVCQPFEBRUE + UMSAVEOPVAAVEACVJPZVEQPGECRUEUMSAVIOPVAAVIADVJPZVIQPHEDRUEUMSVHVCVIUFJZUT + LMZVEVIUFJZAVAVNUSLMZVOAVKVMVQVNKFHEEBDUATUSUTLUGUHAVOVPKZVAAVLVMVRGHEECD + UATSUIUJUKUNAVDBVFCAVKVMVDBKFHEBDULTAVLVMVFCKGHECDULTUOUPBCDIUQUR $. $} ${ @@ -627875,11 +627809,11 @@ fixed reference functional determined by this vector (corresponding to syl caddc ccatlen cn0 cc0 wf cvv ovexd frlmbasf fnfzo0hash oveq12d 3eqtrd cfzo crg wa wb nn0addcld eqeltrrd frlmfzowrdb mpbir2and ) AEJUFUGZBUHZVRF UIUJZUKZUHZVRULUJZGUMZAEWAUHZJWAUHZWBAECUHZWEUDCVTFHLEORVTUNZUOURZAJDUHZW - FUEDVTFIMJPSWHUOURZVTEJUPUQAWCEULUJZJULUJZUSUGZHIUSUGZGAWEWFWCWNUMWIWKVTE - JUTUQAWLHWMIUSAHVAUHVBHVJUGZVTEVCZWLHUMUBAWPVDUHWGWQAVBHVJVEUDCFLWPVTVDEO - WHRVFUQVTEHVGUQAIVAUHVBIVJUGZVTJVCZWMIUMUCAWRVDUHWJWSAVBIVJVEUEDFMWRVTVDJ - PWHSVFUQVTJIVGUQVHUAVIAFVKUHGVAUHVSWBWDVLVMTAWOGVAUAAHIUBUCVNVOBVTFGVKKVR - NQWHVPUQVQ $. + FUEDVTFIMJPSWHUOURZVTEJUPUQAWCEULUJZJULUJZUSUGZHIUSUGZGAWEWFWCWNUMWIWKVTV + TEJUTUQAWLHWMIUSAHVAUHVBHVJUGZVTEVCZWLHUMUBAWPVDUHWGWQAVBHVJVEUDCFLWPVTVD + EOWHRVFUQVTEHVGUQAIVAUHVBIVJUGZVTJVCZWMIUMUCAWRVDUHWJWSAVBIVJVEUEDFMWRVTV + DJPWHSVFUQVTJIVGUQVHUAVIAFVKUHGVAUHVSWBWDVLVMTAWOGVAUAAHIUBUCVNVOBVTFGVKK + VRNQWHVPUQVQ $. $d x ph $. $d x A $. $d x L $. $d x M $. $d x N $. frlmvscadiccat.o $e |- O = ( .s ` W ) $. @@ -628569,7 +628503,7 @@ number axioms (add ~ ax-10 , ~ ax-11 , ~ ax-13 , ~ ax-nul , and remove exp11d.3 $e |- ( ph -> N e. ZZ ) $. exp11d.4 $e |- ( ph -> N =/= 0 ) $. exp11d.5 $e |- ( ph -> ( A ^ N ) = ( B ^ N ) ) $. - $( ~ sq11d for positive real bases and non-zero exponents. (Contributed by + $( ~ sq11d for positive real bases and nonzero exponents. (Contributed by Steven Nguyen, 6-Apr-2023.) $) exp11d $p |- ( ph -> A = B ) $= ( ccxp co c1 cexp rpcnd rpne0d cxpexpzd oveq2d cxpmuld cxp1d 3eqtr3d cdiv @@ -629141,8 +629075,8 @@ through the origin and points in the projective plane (see section ${ $d .0. m $. prjspreln0.z $e |- .0. = ( 0g ` S ) $. - $( Two non-zero vectors are equivalent by a non-zero scalar. - (Contributed by Steven Nguyen, 31-May-2023.) $) + $( Two nonzero vectors are equivalent by a nonzero scalar. (Contributed + by Steven Nguyen, 31-May-2023.) $) prjspreln0 $p |- ( V e. LVec -> ( X .~ Y <-> ( ( X e. B /\ Y e. B ) /\ E. m e. ( K \ { .0. } ) X = ( m .x. Y ) ) ) ) $= ( wcel wbr wa cv co wceq wrex clvec csn cdif prjsprel simprl wne wn c0g @@ -629158,7 +629092,7 @@ through the origin and points in the projective plane (see section WBWCWDWEWF $. $d l m x y N $. - $( A non-zero multiple of a vector is equivalent to the vector. + $( A nonzero multiple of a vector is equivalent to the vector. (Contributed by Steven Nguyen, 6-Jun-2023.) $) prjspvs $p |- ( ( V e. LVec /\ X e. B /\ N e. ( K \ { .0. } ) ) -> ( N .x. X ) .~ X ) $= @@ -629188,7 +629122,7 @@ through the origin and points in the projective plane (see section JVQVSVANVEVBVIKVQSZVGVHWAKVTCKVQVSVANVEVCVDVF $. $d l x V $. $d l x N $. $d l S $. $d l B $. - $( The vectors equivalent to a vector ` X ` are the non-zero vectors in + $( The vectors equivalent to a vector ` X ` are the nonzero vectors in the span of ` X ` . (Contributed by Steven Nguyen, 6-Jun-2023.) $) prjspeclsp $p |- ( ( V e. LVec /\ X e. B ) -> [ X ] .~ = ( ( N ` { X } ) \ { ( 0g ` V ) } ) ) $= @@ -629288,7 +629222,7 @@ with respect to a linear (1-dimensional) relation. (Contributed by BJ 0prjspnlem.b $e |- B = ( ( Base ` W ) \ { ( 0g ` W ) } ) $. 0prjspnlem.w $e |- W = ( K freeLMod ( 0 ... 0 ) ) $. 0prjspnlem.1 $e |- .1. = ( ( K unitVec ( 0 ... 0 ) ) ` 0 ) $. - $( Lemma for ~ 0prjspn . The given unit vector is a non-zero vector. + $( Lemma for ~ 0prjspn . The given unit vector is a nonzero vector. (Contributed by Steven Nguyen, 16-Jul-2023.) $) 0prjspnlem $p |- ( K e. DivRing -> .1. e. B ) $= ( cdr wcel cc0 cfz co cuvc cfv cbs c0g csn cdif cvv eqid drngnzr eleqtrri @@ -629430,7 +629364,7 @@ standardize vectors to a length (norm) of one, but such definitions require ${ $d n a b c x y z $. - $( Fermat's Last Theorem (FLT) for non-zero integers is equivalent to the + $( Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes ` ( a ^ n ) + ( b ^ n ) = ( c ^ n ) ` , and adds the negative of any negative term to both sides, thus creating the corresponding equation @@ -629710,9 +629644,820 @@ standardize vectors to a length (norm) of one, but such definitions require $. $} +$( (End of Steven Nguyen's mathbox.) $) $( End $[ set-mbox-sn.mm $] $) +$( Begin $[ set-mbox-gg.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Gino Giotto +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Minimizing ax-13 usage in a systematic way +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + This section contains proofs of existing theorems in main without requiring + ~ ax-13 . Similarly to ~ ax13w , the theorems proved here are weakened by + introducing one or more additional dv conditions, hence labeled as xxxw + (the pattern xxxv creates naming conflicts with main theorems, despite having + different statements. If xxxv is preferred, then the versions in main should + be renamed as xxxvv since they all have more dv conditions). These weaker + versions can replace the proof steps occupied by their original counterparts + with the command "MINIMIZE_WITH xxxw /MAY_GROW". Since these theorems + follow the already present proof chain, automatic minimizations should not + cause significant proof changes or lengthenings, hence OLD proofs are not + necessary (and they would also be unpractical). + + This list is not yet complete; the full one can be found at + https://github.com/GinoGiotto/set.mm/tree/ax-13-complete, where these + theorems are distinguishable from the others by searching for the + "(Contributed by Gino Giotto, 30-Dec-2023.)" tag. + + To achieve the expected results, the following substitutions should be + additionally performed whenever possible: + + - ~ spim -> ~ spimfv + - ~ spimev -> ~ spimevw + - ~ spv -> ~ spvv + - ~ spei -> ~ speivw + - ~ chvar -> ~ chvarfv + - ~ chvarv -> ~ chvarvv + - ~ cbvexv -> ~ cbvexvw + - ~ sbie -> ~ sbiev + - ~ sbid2v -> ~ sbid2vw + - ~ sb8 -> ~ sb8v + - ~ sb8e -> ~ sb8ev + - ~ nfsb -> ~ nfsbv + - ~ cbval2 -> ~ cbval2v + - ~ cbvex2 -> ~ cbvex2v + - ~ nfmod -> ~ nfmodv + - ~ nfmo -> ~ nfmov + - ~ sb8eu -> ~ sb8euv + - ~ moexexv -> ~ moexexvw + - ~ rgen2a -> ~ rgen2 + + If everything goes well, at the end of the process, we should observe a + reduction of ~ ax-13 usage from more than 30,000 theorems, as shown in + https://github.com/GinoGiotto/set.mm/tree/ax-13-complete. + + Additional information can be found in the mailing list at + https://groups.google.com/g/metamath/c/OB2_9sYgDfA. + +$) + + ${ + $d x y $. + $( Version of ~ nfnae with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfnaew $p |- F/ z -. A. x x = y $= + ( weq wal hbaev nf5i nfn ) ABDAEZCICABCFGH $. + $} + + ${ + $d x y $. + sbiedw.1 $e |- F/ x ph $. + sbiedw.2 $e |- ( ph -> F/ x ch ) $. + sbiedw.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $. + $( Version of ~ sbied with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + sbiedw $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $= + ( wsb wi sbrim nfim1 weq wb com12 pm5.74d sbiev bitr3i pm5.74ri ) ABDEIZC + ATJABJZDEIACJZABDEFKUAUBDEACDFGLDEMZABCAUCBCNHOPQRS $. + $} + + ${ + $d x ph $. $d x ch $. $d x y $. + sbiedvw.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. + $( Version of ~ sbiedv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + sbiedvw $p |- ( ph -> ( [ y / x ] ps <-> ch ) ) $= + ( nfv nfvd weq wb ex sbiedw ) ABCDEADGACDHADEIBCJFKL $. + $} + + ${ + $d x y ps $. $d x y t $. $d u y $. + 2sbievw.1 $e |- ( ( x = t /\ y = u ) -> ( ph <-> ps ) ) $. + $( Version of ~ 2sbiev with additional disjoint variable conditions, which + does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + 2sbievw $p |- ( [ t / x ] [ u / y ] ph <-> ps ) $= + ( wsb weq sbiedvw sbievw ) ADEHBCFCFIABDEGJK $. + $} + + ${ + $d x y z $. + dvelimfw.1 $e |- F/ x ph $. + dvelimfw.2 $e |- F/ z ps $. + dvelimfw.3 $e |- ( z = y -> ( ph <-> ps ) ) $. + $( Version of ~ dvelimf with additional disjoint variable conditions, which + does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + dvelimfw $p |- F/ x ps $= + ( weq wi wal equsalv bicomi nfv nfim nfal nfxfr ) BEDIZAJZEKZCTBABEDGHLMS + CERACRCNFOPQ $. + $} + + ${ + $d x z $. $d y z $. + hbsbw.1 $e |- ( ph -> A. z ph ) $. + $( Version of ~ hbsb with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + hbsbw $p |- ( [ y / x ] ph -> A. z [ y / x ] ph ) $= + ( wsb nf5i nfsbv nf5ri ) ABCFDABCDADEGHI $. + $} + + ${ + $d x ph $. $d x ch $. $d x y $. + cbvaldw.1 $e |- F/ y ph $. + cbvaldw.2 $e |- ( ph -> F/ y ps ) $. + cbvaldw.3 $e |- ( ph -> ( x = y -> ( ps <-> ch ) ) ) $. + $( Version of ~ cbvald with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvaldw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= + ( nfv nfvd cbv2w ) ABCDEADIFGACDJHK $. + + $( Version of ~ cbvexd with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvexdw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $= + ( wex wn wal nfnd weq wb notbi syl6ib cbvaldw alnex 3bitr3g con4bid ) ABD + IZCEIZABJZDKCJZEKUAJUBJAUCUDDEFABEGLADEMBCNUCUDNHBCOPQBDRCERST $. + $} + + ${ + $d ps y $. $d ch x $. $d ph x $. $d ph y $. $d x y $. + cbvaldvaw.1 $e |- ( ( ph /\ x = y ) -> ( ps <-> ch ) ) $. + $( Version of ~ cbvaldva with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvaldvaw $p |- ( ph -> ( A. x ps <-> A. y ch ) ) $= + ( nfv nfvd weq wb ex cbvaldw ) ABCDEAEGABEHADEIBCJFKL $. + + $( Version of ~ cbvexdva with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvexdvaw $p |- ( ph -> ( E. x ps <-> E. y ch ) ) $= + ( nfv nfvd weq wb ex cbvexdw ) ABCDEAEGABEHADEIBCJFKL $. + $} + + ${ + $d x y z w $. $d z w ph $. $d x y ps $. + cbval2vw.1 $e |- ( ( x = z /\ y = w ) -> ( ph <-> ps ) ) $. + $( Version of ~ cbval2v with additional disjoint variable conditions, which + does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + cbval2vw $p |- ( A. x A. y ph <-> A. z A. w ps ) $= + ( wal weq cbvaldvaw cbvalvw ) ADHBFHCECEIABDFGJK $. + + $( Version of ~ cbvex2v with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvex2vw $p |- ( E. x E. y ph <-> E. z E. w ps ) $= + ( wex weq cbvexdvaw cbvexvw ) ADHBFHCECEIABDFGJK $. + $} + + ${ + $( @v f @. + @v g @. + @( Define temporary individual variables. @) + cbvex4vw.vf @f setvar f @. + cbvex4vw.vg @f setvar g @. $) + $d w z ch $. $d u v ph $. $d x y ps $. $d f g ps $. $d f w $. + $d v x $. $d f z $. $d g z $. $d u v w z $. $d u w x z $. + $d v w y z $. $d w x y z $. $d u y $. $d g w $. + cbvex4vw.1 $e |- ( ( x = v /\ y = u ) -> ( ph <-> ps ) ) $. + cbvex4vw.2 $e |- ( ( z = f /\ w = g ) -> ( ps <-> ch ) ) $. + $( Version of ~ cbvex4v with additional disjoint variable conditions, which + does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + cbvex4vw $p |- ( E. x E. y E. z E. w ph <-> E. v E. u E. f E. g ch ) $= + ( wex weq wa 2exbidv cbvex2vw 2exbii bitri ) AGNFNZENDNBGNFNZINHNCKNJNZIN + HNUAUBDEHIDHOEIOPABFGLQRUBUCHIBCFGJKMRST $. + $} + + ${ + $d x y $. + exdistrfw.1 $e |- ( -. A. x x = y -> F/ y ph ) $. + $( Version of ~ exdistrf with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + exdistrfw $p |- ( E. x E. y ( ph /\ ps ) -> E. x ( ph /\ E. y ps ) ) $= + ( wa wex nfe1 weq wal wi 19.8a anim2i eximi biidd drex1v syl5ibr wn 19.40 + 19.9d anim1d syl56 pm2.61i exlimi ) ABFZDGZABDGZFZCGZCUHCHCDICJZUFUIKUFUI + UJUHDGUEUHDBUGABDLMNUHUHCDUJUHOPQUFADGZUGFUJRZUHUIABDSULUKAUGAULDETUAUHCL + UBUCUD $. + $} + + ${ + $d x y $. + nfeudw.1 $e |- F/ y ph $. + nfeudw.2 $e |- ( ph -> F/ x ps ) $. + $( Version of ~ nfeud with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfeudw $p |- ( ph -> F/ x E! y ps ) $= + ( weu wex wmo wa df-eu nfexd nfmodv nfand nfxfrd ) BDGBDHZBDIZJACBDKAPQCA + BCDEFLABCDEFMNO $. + $} + + ${ + $d x y $. + nfeuw.1 $e |- F/ x ph $. + $( Version of ~ nfeu with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfeuw $p |- F/ x E! y ph $= + ( weu wnf wtru nftru a1i nfeudw mptru ) ACEBFGABCCHABFGDIJK $. + $} + + ${ + $d x y $. + cbvmow.1 $e |- F/ y ph $. + cbvmow.2 $e |- F/ x ps $. + cbvmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvmo with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvmow $p |- ( E* x ph <-> E* y ps ) $= + ( wmo wsb wex weu wi sb8ev sb8euv imbi12i moeu 3bitr4i sbiev mobii bitri + ) ACHZACDIZDHZBDHACJZACKZLUBDJZUBDKZLUAUCUDUFUEUGACDEMACDENOACPUBDPQUBBDA + BCDFGRST $. + $} + + ${ + $d x y $. + cbveuw.1 $e |- F/ y ph $. + cbveuw.2 $e |- F/ x ps $. + cbveuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbveu with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbveuw $p |- ( E! x ph <-> E! y ps ) $= + ( weu wsb sb8euv sbiev eubii bitri ) ACHACDIZDHBDHACDEJNBDABCDFGKLM $. + $} + + ${ + $d x z $. $d x y $. + hbabw.1 $e |- ( ph -> A. x ph ) $. + $( Version of ~ hbab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + hbabw $p |- ( z e. { y | ph } -> A. x z e. { y | ph } ) $= + ( cv cab wcel wsb df-clab hbsbw hbxfrbi ) DFACGHACDIBADCJACDBEKL $. + $} + + ${ + $d x z $. $d x y $. + nfsabw.1 $e |- F/ x ph $. + $( Version of ~ nfsab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfsabw $p |- F/ x z e. { y | ph } $= + ( cv cab wcel nf5ri hbabw nf5i ) DFACGHBABCDABEIJK $. + $} + + ${ + $d y A $. $d x z $. $d x y $. + hblemw.1 $e |- ( y e. A -> A. x y e. A ) $. + $( Version of ~ hblem with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + hblemw $p |- ( z e. A -> A. x z e. A ) $= + ( cv wcel wsb wal hbsbw clelsb3 albii 3imtr3i ) BFDGZBCHZOAICFDGZPAINBCAE + JBCDKZOPAQLM $. + $} + + ${ + $d x y z $. $d ph z $. $d ps z $. + cbvabw.1 $e |- F/ y ph $. + cbvabw.2 $e |- F/ x ps $. + cbvabw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvabw $p |- { x | ph } = { y | ps } $= + ( vz cab wsb cv wcel sbco2v sbiev sbbii bitr3i df-clab 3bitr4i eqriv ) HA + CIZBDIZACHJZBDHJZHKZTLUDUALUBACDJZDHJUCACHDEMUEBDHABCDFGNOPAHCQBHDQRS $. + $} + + ${ + $d w x $. $d w A $. $d x y $. + clelsb3fw.1 $e |- F/_ x A $. + $( Version of ~ clelsb3f with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + clelsb3fw $p |- ( [ y / x ] x e. A <-> y e. A ) $= + ( vw cv wcel wsb nfcri sbco2v clelsb3 sbbii 3bitr3i ) EFCGZEAHZABHNEBHAFC + GZABHBFCGNEBAAECDIJOPABEACKLEBCKM $. + $} + + ${ + $d x y z $. $d z ph $. + nfabw.1 $e |- F/ x ph $. + $( Version of ~ nfab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfabw $p |- F/_ x { y | ph } $= + ( vz cab nfsabw nfci ) BEACFABCEDGH $. + $} + + ${ + $d x y $. + $( Version of ~ nfaba1 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfaba1w $p |- F/_ x { y | A. x ph } $= + ( wal nfa1 nfabw ) ABDBCABEF $. + $} + + ${ + $d x y z $. $d z ph $. $d z ps $. + nfabdw.1 $e |- F/ y ph $. + nfabdw.2 $e |- ( ph -> F/ x ps ) $. + $( Version of ~ nfabd with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfabdw $p |- ( ph -> F/_ x { y | ps } ) $= + ( vz cab nfv cv wcel wsb df-clab wnf wal alrimi wi nfnf1 nfal nfim1 sp wb + nfa1 weq sb6 a1i biimpri axc4i syl6bi nf5d sbequ12 imbi2d dvelimfw nfbidf + pm5.5 mpbii syl nfxfrd nfcd ) ACGBDHZAGIGJUTKBDGLZACBGDMABCNZDOZVACNZAVBD + EFPVCVCVAQZCNVDVCBQVECGDVCBCVBCDBCRSZVBDUATVCVADVBDUCZVCVADVGVCVADGUDZBQZ + DOZVADOVAVJUBVCBDGUEZUFVIVADVAVJVKUGUHUIUJTVHBVAVCBDGUKULUMVCVEVACVFVCVAU + OUNUPUQURUS $. + $} + + ${ + $d x y $. + nfraldw.1 $e |- F/ y ph $. + nfraldw.2 $e |- ( ph -> F/_ x A ) $. + nfraldw.3 $e |- ( ph -> F/ x ps ) $. + $( Version of ~ nfrald with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfraldw $p |- ( ph -> F/ x A. y e. A ps ) $= + ( wral cv wcel wi wal df-ral nfcvd nfeld nfimd nfald nfxfrd ) BDEIDJZEKZB + LZDMACBDENAUBCDFAUABCACTEACTOGPHQRS $. + $} + + ${ + $d x y $. + nfralw.1 $e |- F/_ x A $. + nfralw.2 $e |- F/ x ph $. + $( Version of ~ nfral with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfralw $p |- F/ x A. y e. A ph $= + ( wral wnf wtru nftru wnfc a1i nfraldw mptru ) ACDGBHIABCDCJBDKIELABHIFLM + N $. + $} + + ${ + $d A y $. $d x y $. + $( Version of ~ nfra2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfra2w $p |- F/ y A. x e. A A. y e. B ph $= + ( wral nfcv nfra1 nfralw ) ACEFCBDCDGACEHI $. + $} + + ${ + $d x y $. + nfrexdw.1 $e |- F/ y ph $. + nfrexdw.2 $e |- ( ph -> F/_ x A ) $. + nfrexdw.3 $e |- ( ph -> F/ x ps ) $. + $( Version of ~ nfrexd with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfrexdw $p |- ( ph -> F/ x E. y e. A ps ) $= + ( wrex wn wral dfrex2 nfnd nfraldw nfxfrd ) BDEIBJZDEKZJACBDELAQCAPCDEFGA + BCHMNMO $. + $} + + ${ + $d x y $. + nfrexw.1 $e |- F/_ x A $. + nfrexw.2 $e |- F/ x ph $. + $( Version of ~ nfrex with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfrexw $p |- F/ x E. y e. A ph $= + ( wrex wnf wtru nftru wnfc a1i nfrexdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM + N $. + $} + + ${ + $d x y A $. + $( Version of ~ ralcom2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + ralcom2w $p |- ( A. x e. A A. y e. A ph -> A. y e. A A. x e. A ph ) $= + ( weq wal wral wi cv wcel sp eleq1d dral1v df-ral 3bitr4g wa nfan ralrimi + nfnaew ex imbi1d bicomd imbi12d biimpd wn nfra2w nfra1 nfvd nfan1 ancomsd + rsp2 expdimp adantll pm2.61i ) BCEZBFZACDGZBDGZABDGZCDGZHUPURUTUPBIZDJZUQ + HZBFCIZDJZUSHZCFURUTVCVFBCUPVBVEUQUSUPVAVDDUOBKLZUPVEAHZCFZVBAHZBFZUQUSUP + VKVIVJVHBCUPVBVEAVGUAMUBACDNABDNOUCMUQBDNUSCDNOUDUPUEZURUTVLURPZUSCDVLURC + BCCSABCDDUFQVMVEUSVMVEPABDVMVEBVLURBBCBSUQBDUGQVMVEBUHUIURVEVJVLURVEVBAUR + VBVEAABCDDUKUJULUMRTRTUN $. + $} + + ${ + $d x y $. + nfreuw.1 $e |- F/_ x A $. + nfreuw.2 $e |- F/ x ph $. + $( Version of ~ nfreu with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfreuw $p |- F/ x E! y e. A ph $= + ( wreu wnf cv wcel wa weu wtru df-reu nftru nfcvd wnfc a1i nfeld nfand + nfeudw nfxfrd mptru ) ACDGZBHUDCIZDJZAKZCLMBACDNMUGBCCOMUFABMBUEDMBUEPBDQ + MERSABHMFRTUAUBUC $. + + $( Version of ~ nfrmo with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfrmow $p |- F/ x E* y e. A ph $= + ( wrmo cv wcel wa wmo df-rmo wnf wtru nftru nfcvd wnfc a1i nfeld nfand + nfmodv mptru nfxfr ) ACDGCHZDIZAJZCKZBACDLUGBMNUFBCCONUEABNBUDDNBUDPBDQNE + RSABMNFRTUAUBUC $. + $} + + ${ + $d x y z $. $d z A $. + nfrabw.1 $e |- F/ x ph $. + nfrabw.2 $e |- F/_ x A $. + $( Version of ~ nfrab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfrabw $p |- F/_ x { y e. A | ph } $= + ( crab cv wcel wa cab df-rab wnfc wtru nftru wnf nfcri a1i nfand nfabdw + mptru nfcxfr ) BACDGCHDIZAJZCKZACDLBUEMNUDBCCONUCABUCBPNBCDFQRABPNERSTUAU + B $. + $} + + ${ + $d x y z $. $d z A $. $d z ps $. $d z ph $. + cbvralfw.1 $e |- F/_ x A $. + cbvralfw.2 $e |- F/_ y A $. + cbvralfw.3 $e |- F/ y ph $. + cbvralfw.4 $e |- F/ x ps $. + cbvralfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvralf with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvralfw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( vz cv wcel wi wal wral wsb nfv nfcri nfim nfs1v sbequ12 imbi12d cbvalv1 + weq eleq1w nfsbv sbequ sbiev syl6bb bitri df-ral 3bitr4i ) CLEMZANZCOZDLE + MZBNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUATCKUEUNUTAVACK + EUFACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUEZUTUQVABKDEUFVCVAACDQBAKDCUHA + BCDIJUIUJUCUDUKACEULBDEULUM $. + $} + + ${ + $d x y $. + cbvrexfw.1 $e |- F/_ x A $. + cbvrexfw.2 $e |- F/_ y A $. + cbvrexfw.3 $e |- F/ y ph $. + cbvrexfw.4 $e |- F/ x ps $. + cbvrexfw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvrexf with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrexfw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= + ( wn wral wrex nfn weq notbid cbvralfw notbii dfrex2 3bitr4i ) AKZCELZKBK + ZDELZKACEMBDEMUBUDUAUCCDEFGADHNBCINCDOABJPQRACESBDEST $. + $} + + ${ + $d x y A $. + cbvralw.1 $e |- F/ y ph $. + cbvralw.2 $e |- F/ x ps $. + cbvralw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvral with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvralw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( nfcv cbvralfw ) ABCDECEIDEIFGHJ $. + + $( Version of ~ cbvrex with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrexw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= + ( nfcv cbvrexfw ) ABCDECEIDEIFGHJ $. + $} + + ${ + $d A x y z $. $d ph z $. $d ps z $. + cbvreuw.1 $e |- F/ y ph $. + cbvreuw.2 $e |- F/ x ps $. + cbvreuw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvreu with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvreuw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( vz cv wcel wa weu wreu wsb nfv sb8euv sban eubii df-reu anbi1i nfan weq + clelsb3 nfsbv eleq1w sbequ sbiev syl6bb anbi12d cbveuw 3bitri 3bitr4i + bitri ) CJEKZALZCMZDJEKZBLZDMZACENBDENUQUPCIOZIMUOCIOZACIOZLZIMZUTUPCIUPI + PQVAVDIUOACIRSVEIJEKZVCLZIMUTVDVGIVBVFVCCIEUDUASVGUSIDVFVCDVFDPACIDFUEUBU + SIPIDUCZVFURVCBIDEUFVHVCACDOBAIDCUGABCDGHUHUIUJUKUNULACETBDETUM $. + $} + + ${ + $d x y A $. + cbvrmow.1 $e |- F/ y ph $. + cbvrmow.2 $e |- F/ x ps $. + cbvrmow.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvrmo with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrmow $p |- ( E* x e. A ph <-> E* y e. A ps ) $= + ( wrex wreu wi wrmo cbvrexw cbvreuw imbi12i rmo5 3bitr4i ) ACEIZACEJZKBDE + IZBDEJZKACELBDELRTSUAABCDEFGHMABCDEFGHNOACEPBDEPQ $. + $} + + ${ + $d x y A $. $d y ph $. $d x ps $. + cbvralvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvralv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvralvw $p |- ( A. x e. A ph <-> A. y e. A ps ) $= + ( nfv cbvralw ) ABCDEADGBCGFH $. + + $( Version of ~ cbvrexv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrexvw $p |- ( E. x e. A ph <-> E. y e. A ps ) $= + ( nfv cbvrexw ) ABCDEADGBCGFH $. + + $( Version of ~ cbvreuv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvreuvw $p |- ( E! x e. A ph <-> E! y e. A ps ) $= + ( nfv cbvreuw ) ABCDEADGBCGFH $. + $} + + ${ + $d x z $. $d w y $. $d x A $. $d z A $. $d x y B $. $d z y B $. + $d w B $. $d z ph $. $d y ps $. $d x ch $. $d w ch $. + cbvral2vw.1 $e |- ( x = z -> ( ph <-> ch ) ) $. + cbvral2vw.2 $e |- ( y = w -> ( ch <-> ps ) ) $. + $( Version of ~ cbvral2v with additional disjoint variable conditions, + which does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + cbvral2vw $p |- ( A. x e. A A. y e. B ph <-> A. z e. A A. w e. B ps ) $= + ( wral weq ralbidv cbvralvw ralbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD + FMACEIJNOSTFHCBEGIKOPQ $. + $} + + ${ + $d x z $. $d w y $. $d A x $. $d A z $. $d B w $. $d B x y $. + $d B z y $. $d ch w $. $d ch x $. $d ph z $. $d ps y $. + cbvrex2vw.1 $e |- ( x = z -> ( ph <-> ch ) ) $. + cbvrex2vw.2 $e |- ( y = w -> ( ch <-> ps ) ) $. + $( Version of ~ cbvrex2v with additional disjoint variable conditions, + which does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + cbvrex2vw $p |- ( E. x e. A E. y e. B ph <-> E. z e. A E. w e. B ps ) $= + ( wrex weq rexbidv cbvrexvw rexbii bitri ) AEILZDHLCEILZFHLBGILZFHLRSDFHD + FMACEIJNOSTFHCBEGIKOPQ $. + $} + + ${ + $d w ph $. $d z ps $. $d x ch $. $d v ch $. $d y th $. $d u th $. + $d x A $. $d u z $. $d w A $. $d x y B $. $d w y B $. $d v B $. + $d x y z C $. $d w x $. $d w y z C $. $d v z C $. $d z y C $. + $d z C $. $d u C $. $d v y $. + cbvral3vw.1 $e |- ( x = w -> ( ph <-> ch ) ) $. + cbvral3vw.2 $e |- ( y = v -> ( ch <-> th ) ) $. + cbvral3vw.3 $e |- ( z = u -> ( th <-> ps ) ) $. + $( Version of ~ cbvral3v with additional disjoint variable conditions, + which does not require ~ ax-13 . (Contributed by Gino Giotto, + 10-Jan-2024.) $) + cbvral3vw $p |- ( A. x e. A A. y e. B A. z e. C ph <-> + A. w e. A A. v e. B A. u e. C ps ) $= + ( wral weq 2ralbidv cbvralvw cbvral2vw ralbii bitri ) AGMQFLQZEKQCGMQFLQZ + HKQBJMQILQZHKQUDUEEHKEHRACFGLMNSTUEUFHKCBDFGIJLMOPUAUBUC $. + $} + + ${ + $d z x A $. $d y A $. $d z y ph $. $d x y $. + $( Version of ~ cbvralsv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvralsvw $p |- ( A. x e. A ph <-> A. y e. A [ y / x ] ph ) $= + ( vz wral wsb nfv nfs1v sbequ12 cbvralw sbequ bitri ) ABDFABEGZEDFABCGZCD + FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. + $} + + ${ + $d z x A $. $d y z ph $. $d y A $. $d x y $. + $( Version of ~ cbvrexsv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrexsvw $p |- ( E. x e. A ph <-> E. y e. A [ y / x ] ph ) $= + ( vz wrex wsb nfv nfs1v sbequ12 cbvrexw sbequ bitri ) ABDFABEGZEDFABCGZCD + FANBEDAEHABEIABEJKNOECDNCHOEHAECBLKM $. + $} + + ${ + $d x y z $. $d A z $. $d ph z $. $d ps z $. + cbvrabw.1 $e |- F/_ x A $. + cbvrabw.2 $e |- F/_ y A $. + cbvrabw.3 $e |- F/ y ph $. + cbvrabw.4 $e |- F/ x ps $. + cbvrabw.5 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvrab with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvrabw $p |- { x e. A | ph } = { y e. A | ps } $= + ( vz cv wcel wa cab crab wsb nfv nfcri nfan eleq1w sbequ12 anbi12d cbvabw + nfs1v weq nfsbv sbequ sbiev syl6bb eqtri df-rab 3eqtr4i ) CLEMZANZCOZDLEM + ZBNZDOZACEPBDEPUPKLEMZACKQZNZKOUSUOVBCKUOKRUTVACCKEFSACKUETCKUFUNUTAVACKE + UAACKUBUCUDVBURKDUTVADDKEGSACKDHUGTURKRKDUFZUTUQVABKDEUAVCVAACDQBAKDCUHAB + CDIJUIUJUCUDUKACEULBDEULUM $. + $} + + ${ + $d x ph $. $d x A $. $d x y $. + euxfr2w.1 $e |- A e. _V $. + euxfr2w.2 $e |- E* y x = A $. + $( Version of ~ euxfr2 with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + euxfr2w $p |- ( E! x E. y ( x = A /\ ph ) <-> E! y ph ) $= + ( cv wceq wa wex weu wmo wi 2euswapv moani ancom mobii mpbi mpg moeq + impbii biidd ceqsexv eubii bitri ) BGDHZAIZCJBKZUGBJZCKZACKUHUJUGCLZUHUJM + BUGBCNAUFIZCLUKUFACFOULUGCAUFPZQRSUGBLZUJUHMCUGCBNULBLUNUFABBDTOULUGBUMQR + SUAUIACAABDEUFAUBUCUDUE $. + $} + + ${ + $d x ps $. $d y ph $. $d x A $. $d x y $. + euxfrw.1 $e |- A e. _V $. + euxfrw.2 $e |- E! y x = A $. + euxfrw.3 $e |- ( x = A -> ( ph <-> ps ) ) $. + $( Version of ~ euxfr with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + euxfrw $p |- ( E! x ph <-> E! y ps ) $= + ( weu cv wceq wa wex euex ax-mp biantrur 19.41v pm5.32i exbii 3bitr2i + eubii eumoi euxfr2w bitri ) ACICJEKZBLZDMZCIBDIAUGCAUEDMZALUEALZDMUGUHAUE + DIUHGUEDNOPUEADQUIUFDUEABHRSTUABCDEFUEDGUBUCUD $. + $} + + ${ + $d x y $. + nfsbcdw.1 $e |- F/ y ph $. + nfsbcdw.2 $e |- ( ph -> F/_ x A ) $. + nfsbcdw.3 $e |- ( ph -> F/ x ps ) $. + $( Version of ~ nfsbcd with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfsbcdw $p |- ( ph -> F/ x [. A / y ]. ps ) $= + ( wsbc cab wcel df-sbc nfabdw nfeld nfxfrd ) BDEIEBDJZKACBDELACEPGABCDFHM + NO $. + $} + + ${ + $d x y $. + nfsbcw.1 $e |- F/_ x A $. + nfsbcw.2 $e |- F/ x ph $. + $( Version of ~ nfsbc with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfsbcw $p |- F/ x [. A / y ]. ph $= + ( wsbc wnf wtru nftru wnfc a1i nfsbcdw mptru ) ACDGBHIABCDCJBDKIELABHIFLM + N $. + $} + + ${ + $d x z $. $d z A $. $d y z ph $. $d x y $. + $( Version of ~ sbcco with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + sbccow $p |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph ) $= + ( vz cv wsbc cvv wcel sbcex dfsbcq wsb sbsbc sbbii sbco2vv 3bitr3ri bitri + vtoclbg pm5.21nii ) ABCFGZCDGZDHIABDGZTCDJABDJTCEFZGZABUCGZUAUBEDHTCUCDKA + BUCDKUDABELZUEABCLZCELTCELUFUDUGTCEABCMNABECOTCEMPABEMQRS $. + $} + + ${ + $d x y $. + cbvsbcw.1 $e |- F/ y ph $. + cbvsbcw.2 $e |- F/ x ps $. + cbvsbcw.3 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvsbc with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvsbcw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $= + ( cab wcel wsbc cbvabw eleq2i df-sbc 3bitr4i ) EACIZJEBDIZJACEKBDEKPQEABC + DFGHLMACENBDENO $. + $} + + ${ + $d y ph $. $d x ps $. $d x y $. + cbvsbcvw.1 $e |- ( x = y -> ( ph <-> ps ) ) $. + $( Version of ~ cbvsbcv with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvsbcvw $p |- ( [. A / x ]. ph <-> [. A / y ]. ps ) $= + ( nfv cbvsbcw ) ABCDEADGBCGFH $. + $} + + ${ + $d x y z $. $d z A $. $d z C $. $d z D $. + cbvcsbw.1 $e |- F/_ y C $. + cbvcsbw.2 $e |- F/_ x D $. + cbvcsbw.3 $e |- ( x = y -> C = D ) $. + $( Version of ~ cbvcsb with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + cbvcsbw $p |- [_ A / x ]_ C = [_ A / y ]_ D $= + ( vz cv wcel wsbc cab csb nfcri weq eleq2d cbvsbcw abbii df-csb 3eqtr4i ) + IJZDKZACLZIMUBEKZBCLZIMACDNBCENUDUFIUCUEABCBIDFOAIEGOABPDEUBHQRSAICDTBICE + TUA $. + $} + + ${ + $d z A $. $d x y z $. $d y z B $. + $( Version of ~ csbco with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + csbcow $p |- [_ A / y ]_ [_ y / x ]_ B = [_ A / x ]_ B $= + ( vz cv csb wcel wsbc cab df-csb abeq2i sbcbii sbccow bitri abbii 3eqtr4i + ) EFZABFZDGZHZBCIZEJRDHZACIZEJBCTGACDGUBUDEUBUCASIZBCIUDUAUEBCUEETAESDKLM + UCABCNOPBECTKAECDKQ $. + $} + + ${ + $d x y z $. $d A z $. $d B z $. + nfcsbw.1 $e |- F/_ x A $. + nfcsbw.2 $e |- F/_ x B $. + $( Version of ~ nfcsb with a disjoint variable condition, which does not + require ~ ax-13 . (Contributed by Gino Giotto, 10-Jan-2024.) $) + nfcsbw $p |- F/_ x [_ A / y ]_ B $= + ( vz csb wnfc wtru cv wcel wsbc cab df-csb nftru a1i nfcrd nfsbcdw nfabdw + nfcxfrd mptru ) ABCDHZIJAUCGKDLZBCMZGNBGCDOJUEAGGPJUDABCBPACIJEQJAGDADIJF + QRSTUAUB $. + $} + +$( (End of Gino Giotto's mathbox.) $) +$( End $[ set-mbox-gg.mm $] $) + + +$( Begin $[ set-mbox-ii.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Igor Ieskov +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + ${ + binom2d.1 $e |- ( ph -> A e. CC ) $. + binom2d.2 $e |- ( ph -> B e. CC ) $. + $( Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.) $) + binom2d $p |- ( ph -> ( ( A + B ) ^ 2 ) = + ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) ) $= + ( cc wcel caddc co c2 cexp cmul wceq binom2 syl2anc ) ABFZGCPGBCHZIJZKZIB + RSIRBCLZITIQICRSIQIMDEBCNO $. + $} + + ${ + cu3addd.1 $e |- ( ph -> A e. CC ) $. + cu3addd.2 $e |- ( ph -> B e. CC ) $. + cu3addd.3 $e |- ( ph -> C e. CC ) $. + $( Cube of sum of three numbers. (Contributed by Igor Ieskov, + 14-Dec-2023.) $) + cu3addd $p |- ( ph -> ( ( ( A + B ) + C ) ^ 3 ) = + ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) + + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) + + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) + + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) + + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) + + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) ) $= + ( co wcel wceq addcld binom3 a1i oveq1d eqtrd oveq2d sqcld mulcld adddird + adddid caddc c3 cexp c2 cmul cc wa jca wi mpd syl2anc binom2d 3cn mulassd + 2cnd eqcomd ) ABCUAZHZDUQHUBZUCZHZBUSUTHUSBUDZUTHZCUEZHVDHUQHUSBCVBUTHZVD + HVDHCUSUTHUQHUQHZUSVCDVDHZVDHZUSVBVDHBCVDHZVDHZDVDHZUQHZUSVEDVDHZVDHZUQHZ + UQHZUSBDVBUTHZVDHZCVQVDHZUQHZVDHZDUSUTHZUQHZUQHZVPUSVRVDHUSVSVDHUQHZWBUQH + ZUQHAVAVPUSURVQVDHZVDHZWBUQHZUQHZWDAVAVFVHUSVBVIVDHZDVDHZVDHZUQHZVNUQHZUQ + HZWIUQHZWJAVAVFUSVGWLUQHZVDHZVNUQHZUQHZWIUQHZWQAVAVFUSWRVMUQHZVDHZUQHZWIU + QHZXBAVAVFUSVCWKUQHZDVDHZVMUQHZVDHZUQHZWIUQHZXFAVAVFUSXGVEUQHZDVDHZVDHZUQ + HZWIUQHZXLAVAVFUSURVBUTHZDVDHZVDHZUQHZWIUQHZXQAVAURUSUTHZXTUQHZWIUQHZYBAU + RUFZIZDYFIZUGZVAYEJZAYGYHABCEFKGUHYIYJUIAURDLMUJAYDYAWIUQAYCVFXTUQABYFICY + FIYCVFJEFBCLUKNNOAYAXPWIUQAXTXOVFUQAXSXNUSVDAXRXMDVDABCEFULNPPNOAXPXKWIUQ + AXOXJVFUQAXNXIUSVDAXGVEDAVCWKABEQZAVBVIAUOZABCEFRZRZKACFQZGSPPNOAXKXEWIUQ + AXJXDVFUQAXIXCUSVDAXHWRVMUQAVCWKDYKYNGSNPPNOAXEXAWIUQAXDWTVFUQAUSWRVMUSYF + IAUMMZAVGWLAVCDYKGRZAWKDYNGRZKAVEDYOGRTPNOAXAWPWIUQAWTWOVFUQAWSWNVNUQAUSV + GWLYPYQYRTNPNOAWPVPWIUQAVPWPAVOWOVFUQAVLWNVNUQAVKWMVHUQAVKUSWKVDHZDVDHWMA + VJYSDVDAUSVBVIYPYLYMUNNAUSWKDYPYNGUNOPNPUPNOAWIWCVPUQAWHWAWBUQAWGVTUSVDAB + CVQEFADGQZSPNPOAWCWFVPUQAWAWEWBUQAUSVRVSYPABVQEYTRACVQFYTRTNPO $. + $} + + ${ + cu3addi.1 $e |- A e. CC $. + cu3addi.2 $e |- B e. CC $. + cu3addi.3 $e |- C e. CC $. + $( Cube of sum of three numbers. (Contributed by Igor Ieskov, + 14-Dec-2023.) $) + cu3addi $p |- ( ( ( A + B ) + C ) ^ 3 ) = + ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) + + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) + + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) + + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) + + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) + + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) + $= + ( caddc co c3 cexp c2 cmul wceq wtru cc wcel a1i cu3addd mptru ) ABGZHCTH + IZJZHAUAUBHUAAKZUBHZBLZHUEHTHUAABUCUBHZUEHUEHBUAUBHTHTHUAUDCUEHUEHUAUCUEH + ABUEHUEHCUEHTHUAUFCUEHUEHTHTHUAACUCUBHZUEHUEHUABUGUEHUEHTHCUAUBHTHTHMNZAB + CAOZPUHDQBUIPUHEQCUIPUHFQRS $. + $} + + $( Cube of sum of three numbers. (Contributed by Igor Ieskov, + 14-Dec-2023.) $) + cu3add $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) + C ) ^ 3 ) + = ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) + + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) + + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) + + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) + + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) ) $= + ( cc wcel w3a simp1 simp2 simp3 cu3addd ) ADZEZBKEZCKEZFABCLMNGLMNHLMNIJ $. + +$( (End of Igor Ieskov's mathbox.) $) +$( End $[ set-mbox-ii.mm $] $) + + +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Larry Lesyna +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + +$( (End of Larry Lesyna's mathbox.) $) + + $( Begin $[ set-mbox-oai.mm $] $) $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# @@ -629999,7 +630744,7 @@ family of sets (implicit). (Contributed by Stefan O'Rear, ex fveq1 sseq2d sseq12d fveq12d eqeq12d 3anbi123d imbi2d 2albidv 3anbi23d id syl5ibcom rexlimiv cid cin simp1 simp2 ssid 3simpb imim2i 2alimi sseq1 cdm weq adantr sseq12 ancoms anbi12d fveq2 2fveq3 imbi12d spc2gv 3ad2ant3 - el2v mpan2i simpld syl2anr 3impib ismrcd2 ismrcd1 fvssunirn fndm sseqtr4i + el2v mpan2i simpld syl2anr 3impib ismrcd2 ismrcd1 fvssunirn fndm sseqtrri imp simprd funfvima2 mp2an eqeltrd impbii ) DGCHIZUAZJZCKJZCUCZYADUDZALZC MZBLZYCMZNZYCYCDIZMZYEDIZYHMZYHDIZYHOZPZQZBRARZPZXSELZGIZDOZEXQUBZYQGUEZX SUUAGHUFUGZUHZUUBUIUUCGUJUKZEDXQGULUMYTYQEXQYRXQJZXTYAYAYSUDZYGYCYCYSIZMZ @@ -630289,7 +631034,7 @@ family of sets (implicit). (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.) $) mapfzcons1cl $p |- ( A e. ( B ^m ( 1 ... M ) ) -> ( A |` ( 1 ... N ) ) e. ( B ^m ( 1 ... N ) ) ) $= - ( c1 cfz cmap wcel wss cres caddc fzssp1 oveq2i sseqtr4i elmapssres mpan2 + ( c1 cfz cmap wcel wss cres caddc fzssp1 oveq2i sseqtrri elmapssres mpan2 co ) ABFCGRZHRIFDGRZSJATKBTHRITFDFLRZGRSFDMCUAFGENOABSTPQ $. $( Recover added element from an extended mapping. (Contributed by Stefan @@ -631763,7 +632508,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, NN0 ph } e. ( Dioph ` N ) ) $= ( va cn0 wcel cfv wsbc c1 cfz co cres crab cv cmap cdioph wrex wa 2sbcrex rabbii peano2nn0 eqeltrid adantr sbcrot3 sbcbii reseq1 sbccomieg wss wceq - caddc wb fzssp1 oveq2i sseqtr4i resabs1 dfsbcq mp2b cvv vex resex sbcco3g + caddc wb fzssp1 oveq2i sseqtrri resabs1 dfsbcq mp2b cvv vex resex sbcco3g fveq1 ax-mp cn nn0p1nn elfz1end sylib 3syl syl5bb sbcbidv syl5rbb rabbidv fvres eleq1d biimpa rexfrabdioph syl2anc syldan ) HLMZABFEUAZNZOZCGWGNZOZ DWGPHQRZSZOZELPFQRUBRZTZFUCNZMZABLUDZCGKUAZNZODWTWLSZOZKLPGQRZUBRZTZGUCNZ @@ -631787,7 +632532,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, | E. v e. NN0 E. w e. NN0 E. x e. NN0 ph } e. ( Dioph ` N ) ) $= ( va cn0 wcel cfv wsbc c1 co cv cfz cres cmap crab cdioph wrex wa sbc2rex sbcbii bitri rabbii caddc nn0p1nn eqeltrid nnnn0d adantr reseq1 sbccomieg - cn sbcrot3 wss wceq wb fzssp1 oveq2i sseqtr4i resabs1 dfsbcq mp2b cvv vex + cn sbcrot3 wss wceq wb fzssp1 oveq2i sseqtrri resabs1 dfsbcq mp2b cvv vex resex fveq1 sbcco3g ax-mp sylib fvres 3syl syl5bb sbcbidv syl5bbr rabbidv elfz1end eleq1d biimpar 2rexfrabdioph syl2anc rexfrabdioph syldan ) JOPZA BGFUAZQZRCHWLQZRZDIWLQZRZEWLSJUBTZUCZRZFOSGUBTUDTZUEZGUFQZPZABOUGCOUGZDIN @@ -631812,7 +632557,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, E. y e. NN0 ph } e. ( Dioph ` N ) ) $= ( va cn0 wcel wsbc cv cfv c1 cfz co cres cmap crab cdioph wrex wa 2sbcrex rexbii bitri sbcbii sbc2rex rabbii caddc eqeltrid peano2nnd nnnn0d adantr - cn nn0p1nn sbcrot3 bitr3i reseq1 sbccomieg wceq wb fzssp1 oveq2i sseqtr4i + cn nn0p1nn sbcrot3 bitr3i reseq1 sbccomieg wceq wb fzssp1 oveq2i sseqtrri wss sstri resabs1 dfsbcq mp2b fveq1 elfz1end sylib sseldi fvres cvv resex 3syl vex sbcco3g ax-mp syl5bb sbcbidv bitrd rabbidv biimpar 2rexfrabdioph eleq1d syl2anc syldan ) LRSZACHGUAZUBZTZBIWTUBZTZDJWTUBZTZEKWTUBZTZFWTUCL @@ -631845,7 +632590,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, E. y e. NN0 E. z e. NN0 E. p e. NN0 ph } e. ( Dioph ` N ) ) $= ( va cn0 wcel cv cfv wsbc c1 cfz co cres cmap crab cdioph wrex wa sbc4rex sbcbii bitri rabbii caddc nn0p1nn eqeltrid peano2nnd nnnn0d adantr reseq1 - cn sbcrot5 sbccomieg wss wceq fzssp1 oveq2i sseqtr4i sstri resabs1 dfsbcq + cn sbcrot5 sbccomieg wss wceq fzssp1 oveq2i sseqtrri sstri resabs1 dfsbcq wb mp2b fveq1 elfz1end sylib sseldi fvres 3syl cvv vex resex ax-mp syl5bb sbcco3g sbcbidv bitrd syl5bbr rabbidv biimpar 4rexfrabdioph 2rexfrabdioph eleq1d syl2anc syldan ) OUDUEZAPIHUFZUGZUHDJXEUGZUHCKXEUGZUHBLXEUGZUHZEMX @@ -631880,7 +632625,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, E. y e. NN0 E. z e. NN0 E. p e. NN0 E. q e. NN0 ph } e. ( Dioph ` N ) ) $= ( va cn0 wcel cv cfv wsbc c1 cfz co cres cmap crab cdioph wrex wa sbc2rex sbc4rex 2rexbii bitri sbcbii 3bitri rabbii caddc cn nn0p1nn nnnn0d adantr - eqeltrid sbcrot3 sbcrot5 reseq1 sbccomieg wss wceq fzssp1 oveq2i sseqtr4i + eqeltrid sbcrot3 sbcrot5 reseq1 sbccomieg wss wceq fzssp1 oveq2i sseqtrri wb resabs1 dfsbcq mp2b cvv resex fveq1 sbcco3g ax-mp elfz1end sylib fvres vex 3syl syl5bb sbcbidv syl5bbr rabbidv eleq1d 6rexfrabdioph rexfrabdioph biimpar syl2anc syldan ) PUGUHZAQIHUIZUJZUKRJXHUJZUKDKXHUJZUKCLXHUJZUKZBM @@ -631933,7 +632678,7 @@ to be empty ( ` ( 1 ... 0 ) ` ). (Contributed by Stefan O'Rear, ( va wcel cz c1 cfz co cmap cmpt cmzp cfv wa cv csb nfcsb1v cn0 cres nfcv csbeq1a cbvmpt fveq1i eqid csbeq1 mapfzcons1cl adantl wral wf mzpf sylibr fmpt ad2antlr nfel1 wceq eleq1d rspc sylc fvmptd3 syl5req mpteq2dva ovexd - cvv wss caddc fzssp1 oveq2i sseqtr4i simpr mzpresrename syl3anc eqeltrd + cvv wss caddc fzssp1 oveq2i sseqtrri simpr mzpresrename syl3anc eqeltrd a1i ) EUAHZAIJEKLZMLZCNZVROPHZQZBIJDKLZMLZABRZVRUBZCSZNBWDWFVTPZNZWCOPZWB BWDWGWHWBWEWDHZQZWHWFGVSAGRZCSZNZPWGWFVTWOAGVSCWNGCUCAWMCTAWMCUDUEUFWLGWF WNWGVSWOIWOUGAWMWFCUHWKWFVSHZWBWEIDEFUIUJZWLWPCIHZAVSUKZWGIHZWQWAWSVQWKWA @@ -636546,7 +637291,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by midpoints of range; ~ jm2.27dlem2 is deprecated. (Contributed by Stefan O'Rear, 11-Oct-2014.) $) jm2.27dlem5 $p |- ( 1 ... A ) C_ ( 1 ... C ) $= - ( c1 cfz co caddc fzssp1 oveq2i sseqtr4i sstri ) FAGHZFBGHZFCGHNFAFIHZGHO + ( c1 cfz co caddc fzssp1 oveq2i sseqtrri sstri ) FAGHZFBGHZFCGHNFAFIHZGHO FAJBPFGDKLEM $. $} @@ -640135,14 +640880,14 @@ is in the span of P(i)(X), so there is an R-linear combination of 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) $) algbase $p |- ( B e. V -> B = ( Base ` A ) ) $= ( cbs c1 c6 cop algstr baseid cnx cfv csn cplusg cmulr ctp csca cvsca cpr - snsstp1 cun ssun1 sseqtr4i sstri strfv ) BAIGJKLABCDEFHMNOIPBLZQUJORPCLZO + snsstp1 cun ssun1 sseqtrri sstri strfv ) BAIGJKLABCDEFHMNOIPBLZQUJORPCLZO SPFLZTZAUJUKULUDUMUMOUAPDLOUBPELUCZUEAUMUNUFHUGUHUI $. $( The additive operation of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) $) algaddg $p |- ( .+ e. V -> .+ = ( +g ` A ) ) $= ( cplusg c1 c6 cop algstr plusgid cnx cfv csn cbs cmulr ctp snsstp2 cvsca - csca cpr cun ssun1 sseqtr4i sstri strfv ) CAIGJKLABCDEFHMNOIPCLZQORPBLZUJ + csca cpr cun ssun1 sseqtrri sstri strfv ) CAIGJKLABCDEFHMNOIPCLZQORPBLZUJ OSPFLZTZAUKUJULUAUMUMOUCPDLOUBPELUDZUEAUMUNUFHUGUHUI $. $( The multiplicative operation of a constructed algebra. (Contributed by @@ -640150,14 +640895,14 @@ is in the span of P(i)(X), so there is an R-linear combination of 29-Aug-2015.) $) algmulr $p |- ( .X. e. V -> .X. = ( .r ` A ) ) $= ( cmulr c1 c6 cop algstr mulrid cnx cfv csn cbs cplusg ctp csca cvsca cpr - snsstp3 cun ssun1 sseqtr4i sstri strfv ) FAIGJKLABCDEFHMNOIPFLZQORPBLZOSP + snsstp3 cun ssun1 sseqtrri sstri strfv ) FAIGJKLABCDEFHMNOIPFLZQORPBLZOSP CLZUJTZAUKULUJUDUMUMOUAPDLOUBPELUCZUEAUMUNUFHUGUHUI $. $( The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 29-Aug-2015.) $) algsca $p |- ( S e. V -> S = ( Scalar ` A ) ) $= ( csca c1 c6 cop algstr scaid cnx cfv csn cvsca cpr snsspr1 cbs cmulr ctp - cplusg cun ssun2 sseqtr4i sstri strfv ) DAIGJKLABCDEFHMNOIPDLZQUJORPELZSZ + cplusg cun ssun2 sseqtrri sstri strfv ) DAIGJKLABCDEFHMNOIPDLZQUJORPELZSZ AUJUKTULOUAPBLOUDPCLOUBPFLUCZULUEAULUMUFHUGUHUI $. $( The scalar product operation of a constructed algebra. (Contributed by @@ -640165,7 +640910,7 @@ is in the span of P(i)(X), so there is an R-linear combination of 29-Aug-2015.) $) algvsca $p |- ( .x. e. V -> .x. = ( .s ` A ) ) $= ( cvsca c1 c6 cop algstr vscaid cnx cfv csn csca cpr snsspr2 cplusg cmulr - cbs ctp cun ssun2 sseqtr4i sstri strfv ) EAIGJKLABCDEFHMNOIPELZQORPDLZUJS + cbs ctp cun ssun2 sseqtrri sstri strfv ) EAIGJKLABCDEFHMNOIPELZQORPDLZUJS ZAUKUJTULOUCPBLOUAPCLOUBPFLUDZULUEAULUMUFHUGUHUI $. $} @@ -641416,6 +642161,7 @@ is in the span of P(i)(X), so there is an R-linear combination of $( End $[ set-mbox-jp.mm $] $) +$( Begin $[ set-mbox-rp.mm $] $) $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# Mathbox for Richard Penner @@ -643566,7 +644312,7 @@ of all sets ( ~ inex1g ). $( The image under the intersection of relations is a subset of the intersection of the images. (Contributed by RP, 13-Apr-2020.) $) intimass2 $p |- ( |^| A " B ) C_ |^|_ x e. A ( x " B ) $= - ( vy cint cima cv wceq wrex cab ciin intimass intima0 sseqtr4i ) BECFDGAG + ( vy cint cima cv wceq wrex cab ciin intimass intima0 sseqtrri ) BECFDGAG CFZHABIDJEABOKDBCALDBCAMN $. $} @@ -644390,7 +645136,7 @@ over the natural numbers (including zero) is equivalent to the (Contributed by RP, 10-Jun-2020.) $) comptiunov2i $p |- ( X o. Y ) = Z $= ( ccom wfun wceq cvv cv ciun funmpt2 funco mp2an cdm cfv wral crn wss ssv - co wa ovex iunex dmmpti sseqtr4i dmcosseq ax-mp eqtri unex eqeltri eqtr4i + co wa ovex iunex dmmpti sseqtrri dmcosseq ax-mp eqtri unex eqeltri eqtr4i cun wcel vex eleqtrri fvco weq oveq1 iuneq2d fvmpt fveq2i 3eqtri raleqbii elv eqeq12i iunxun unssi eqsstri eqssi iuneq1 a1i mprgbir eqfunfv biimprd mp2ani ) HIUDZUEZJUEZWOJUFZHUEIUEZWPKUGAEKUHZAUHZDUSZUIZHOUJLUGBFLUHZBUHZ @@ -644610,7 +645356,7 @@ over the natural numbers (including zero) is equivalent to the ( wcel crelexp co cc0 wss wceq wi oveq2 oveq1d sseq1d imbi2d cid cun cres c1 cvv ax-mp vx vy cn0 cn wo elnn0 cv weq relexp1g ssid syl6eqss w3a ccom caddc simp2 simp1 wa relexpsucnnr syl2anc cdm crn ovex coexg relexp0g syl - dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 sseqtrrid adantr sseqtr4i + dmcoss rncoss unss12 mp2an ssres2 resundi ssun1 sseqtrrid adantr sseqtrri mpan ssun2 simpr unssd eqsstrid 3adant1 sstrd eqsstrd 3exp a2d relexp0idm sstrid nnind sylan9eq eqimss ex jaoi sylbi impcom ) BUCDZACDZABEFZGEFZAGE FZHZWOBUDDZBGIZUEWPWTJZBUFXAXCXBWPAUAUGZEFZGEFZWSHZJWPAREFZGEFZWSHZJWPAUB @@ -651570,8 +652316,10 @@ base set if and only if the neighborhoods (convergents) of every point $} $( (End of Richard Penner's mathbox.) $) +$( End $[ set-mbox-rp.mm $] $) +$( Begin $[ set-mbox-sp.mm $] $) $( #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# Mathbox for Stanislas Polu @@ -652312,6 +653060,7 @@ base set if and only if the neighborhoods (convergents) of every point $} $( (End of Stanislas Polu's mathbox.) $) +$( End $[ set-mbox-sp.mm $] $) $( @@ -654278,7 +655027,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing cn 3t1e3 syl6eq dvmptsub 1re mp2an w3a 3pm3.2i dvcn rescncf eqeltrri cibl mp3an oveq1 oveq1d fvmpt c4 leidi elicc2i mpbir3an oveq2 oveq12d c8 3t2e6 mp2 oveq1i oveq12i 2cn 6cn addcomli eqtr4i 4cn subaddrii divsubdir eqtr3i - caddc sseqtr4i dvres3 reseq1i sqcl divcan3 eqtr3d 3ex dvmptid iccssre cnt + caddc sseqtrri dvres3 reseq1i sqcl divcan3 eqtr3d 3ex dvmptid iccssre cnt tgioo2 iccntr dvmptres2 ioossicc sstri subcl cnelprrecn 2nn c0ex syl6eqel dvmptc cvol ioombl cniccibl iblss eqeltrd ftc2 itgeq2 cu2 divdiri divmuli mprg 6p2e8 mpbir subsub3 4p2e6 cz 3z 1exp sub4 pm3.2i 2m1e1 dividi divneg @@ -654917,9 +655666,9 @@ collection and union ( ~ mnuop3d ), from which closure under pairing derivative of ` ( ( 1 + b ) ^c -u C ) ` , with respect to ` b ` in the disk of convergence ` D ` . We later multiply the derivative in the later ~ binomcxplemdvsum by this derivative to show that - ` ( ( 1 + b ) ^c C ) ` (with a non-negated ` C ` ) and the later - sum, since both at ` b = 0 ` equal one, are the same. (Contributed - by Steve Rodriguez, 22-Apr-2020.) $) + ` ( ( 1 + b ) ^c C ) ` (with a nonnegated ` C ` ) and the later sum, + since both at ` b = 0 ` equal one, are the same. (Contributed by + Steve Rodriguez, 22-Apr-2020.) $) binomcxplemdvbinom $p |- ( ( ph /\ -. C e. NN0 ) -> ( CC _D ( b e. D |-> ( ( 1 + b ) ^c -u C ) ) ) = ( b e. D |-> ( -u C x. ( ( 1 + b ) ^c ( -u C - 1 ) ) ) ) ) $= @@ -673086,7 +673835,7 @@ not even needed (it can be any class). (Contributed by Glauco $( A nonempty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) $) iocnct $p |- ( ph -> -. C ~<_ _om ) $= - ( cioo co eqid ioonct wss cioc ioossioc sseqtr4i a1i ssnct ) ABCIJZDABCSE + ( cioo co eqid ioonct wss cioc ioossioc sseqtrri a1i ssnct ) ABCIJZDABCSE FGSKLSDMASBCNJDBCOHPQR $. $} @@ -673098,7 +673847,7 @@ not even needed (it can be any class). (Contributed by Glauco $( A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) $) iccnct $p |- ( ph -> -. C ~<_ _om ) $= - ( cioo co eqid ioonct wss cicc ioossicc sseqtr4i a1i ssnct ) ABCIJZDABCSE + ( cioo co eqid ioonct wss cicc ioossicc sseqtrri a1i ssnct ) ABCIJZDABCSE FGSKLSDMASBCNJDBCOHPQR $. $} @@ -682435,7 +683184,7 @@ distinct definitions for the same symbol (limit of a sequence). ( x e. RR |-> -u ( sin ` x ) ) $= ( cr ccos cres cdv co cc cfv cmpt csin wcel wss wceq cosf reseq1i ax-resscn cv ax-mp resmpt eqtri cneg cpr wf cdm reelprrecn ssid crab wi nfrab1 dfss2f - cvv nfcv recn sincld negcld elex rabid sylanbrc mpgbir dvcos dmmpt sseqtr4i + cvv nfcv recn sincld negcld elex rabid sylanbrc mpgbir dvcos dmmpt sseqtrri syl dvres3 mp4an wfn ffn dffn5 mpbi oveq2i 3eqtr3i ) BCBDZEFZGCEFZBDZBABAQZ CHZIZEFABVPJHZUAZIZBBGUBKGGCUCZGGLBVNUDZLVMVOMUENGUFBVTUKKZAGUGZWCBWELVPBKZ VPWEKZUHAABWEABULWDAGUIUJWFVPGKWDWGVPUMZWFVTGKWDWFVSWFVPWHUNUOVTGUPVCWDAGUQ @@ -694789,7 +695538,7 @@ approximated is nonnegative (this assumption is removed in a later cle neqne adantl leneltd eliood pm2.61dan rescncf sylc leidd elicod fvres resabs1d oveq1d ne0d cpnf pnfxr xrltled sylancr limcresi rexri jca anbi2d eleq1 reseq2d oveq12d neeq1d imbi12d vtoclg ssn0 eqnetrd cioc eliocd cmnf - wi id mnfxr unssd eqsstrid cfz cfzo elfzofz fzofzp1 crn sseqtr4i ioossicc + wi id mnfxr unssd eqsstrid cfz cfzo elfzofz fzofzp1 crn sseqtrri ioossicc ssun2 syl6ss sseldi eleqtrd wrex wfn wf ffn fvelrnb mpbid elfzelz simplll ad2antlr elfzoelz ioogtlb iooltub btwnnz syl21anc icossre icogelb icoltub wiso nrexdv condan icocncflimc 3eltr4d ltpnfd iooss2 fourierdlem10 simpld @@ -695988,7 +696737,7 @@ approximated is nonnegative (this assumption is removed in a later pm2.65da neqned fourierdlem44 syl2anc mulne0d cz 2z expne0d redivcld eqid fmptd oveq2d cpr reelprrecn fourierdlem28 0red ccnfld ctopn crest crn ctg eqtrd c1 fveq2d halfcn wss resmpt ax-mp oveq2i ax-resscn fmpti ssid mp4an - mpteq2ia cdm 2cn sseqtr4i reseq1i 3eqtri iooretop tgioo2 eleqtri dvmptsub + mpteq2ia cdm 2cn sseqtrri reseq1i 3eqtri iooretop tgioo2 eleqtri dvmptsub dvmptconst subid1d mpteq2dva csn cdif eldifsn sylanbrc divrec2d eqcomd id recn coscld 3syl eqeltrrd cnt ioossre eqcomi dvres eqtr2i ioontr reseq12i dmmptg mulcli dvasinbx mp2an dvres3 resabs1 ioosscn recidi oveq1i mulid2d @@ -696048,7 +696797,7 @@ approximated is nonnegative (this assumption is removed in a later ssid ccos cres cnt resmptd eqcomd recn fmpti dvres syl22anc ctop uniretop retop isopn3 mpbid resmpt ax-mp id fveq2d mpteq2ia halfcn 2cn eqtrd eqtri reseq2d reseq1i 3eqtrd divrec2d eqtr2i oveq2i mp2an recidi oveq12d halfcl - dvasinbx coscld mulid2d dmeqi dmmptg mprg sseqtr4i dvres3 3eqtri resabs1d + dvasinbx coscld mulid2d dmeqi dmmptg mprg sseqtrri dvres3 3eqtri resabs1d mp4an coscn idcncfg eldifsn sylanbrc difssd cncfmpt1f dvdivcncf cncff fdm divcncf 3syl feq2d cncffvrn mpbird ) AICJKZBIUAKLZBIUVMUFZAUVMUBZIUVMUFZU VOABICUFBIUCZUVQADBDUGZMUVSMUDKZUENZOKZUDKZICAUVSBLZUHZUVSUWBUWEUIUJZUIUK @@ -696304,7 +697053,7 @@ approximated is nonnegative (this assumption is removed in a later w3a elioo2 mpbir3an 1ex dmmpti cpr reelprrecn 1red dvmptid tgioo2 sncldre ccld toponunii difopn dvmptres eqtr3i eqimssi fvex divrec2d mulcli dmmptd dvcnre reseq1i recidi eqcomd mulid2d eqtrd idcncfg cnlimc eleq12d rspccva - sseqtr4i dvasinbx sincn divccncf cncfmpt1f div0i sin0 2t0e0 wrex ioossicc + sseqtrri dvasinbx sincn divccncf cncfmpt1f div0i sin0 2t0e0 wrex ioossicc 3eltr3i eldifsni fourierdlem44 eqnetrd nrex fnmpt mprg fvelimab cre rered picn divneg crp 2rp ioogtlb eqbrtrid iooltub eliood cosne0 fnmpti imaeq1i eleq2i renegcld recoscld elsng ad2antrl cosf ffvelrnda cnmptlimc syl6eleq @@ -701168,7 +701917,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u wrex sselii jca cres cc ccncf climc c0 oveq1d adantlr eqnetrd ctp cdm cpr tpfi renegcli negpilt0 pipos lttri ltleii prunioo difeq1i difundir eqtr3i 0re prfi diffi hashcl nn0zd ltneii hashprg mpbi difexg ax-mp unex eqeltri - negex tpid1 tpid2 prssi ssun1 sseqtr4i hashss eqbrtrrid syl3anbrc uz2m1nn + negex tpid1 tpid2 prssi ssun1 sseqtrri hashss eqbrtrrid syl3anbrc uz2m1nn sstri cmap cfzo wral wiso wf1o cioc negpitopissre 2timesi subnegi 3eqtr4i eluz2 cmul fourierdlem4 3jca fvex tpss sylib iccssre fourierdlem36 isof1o ssdifss f1of elmap sylibr ffvelrnda leidd elfzelz elfzle1 ne0gt0d nnssnn0 @@ -701330,7 +702079,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u rexri 0xr mp3an12 pipos lttrd ltled iccss syl22anc feq1d mpbird cn0 chash wne c0 cpr crn cin cun elexi prid1 elun1 ax-mp eleqtrri ne0ii cfn wb prfi cfz fzfi rnmptfi infi unfi eqeltrid hashnncl nnm1nn0 1red nn0red 0lt1 2re - nnred ioogtlb ltned hashprg mpbid ssun1 sseqtr4i hashssle sylancl eqbrtrd + nnred ioogtlb ltned hashprg mpbid ssun1 sseqtrri hashssle sylancl eqbrtrd eqcomd lesub1dd 1e2m1 3brtr4g ltletrd gt0ne0d jca elnnne0 sylibr vex prss eliccd sylib inss2 ioossicc syl6ss eqsstrid fourierdlem52 simpld elfzoelz unssd prid2 simprd zred ltp1d wiso fourierdlem36 elfzofz fzofzp1 syl12anc @@ -701865,7 +702614,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u 0xr rexri mp3an12 lttrd ltled leidd iccss syl22anc feq1d mpbird cn0 chash wne c0 cpr crn cin cun elexi prid2 elun1 ax-mp eleqtrri cfn prfi cfz fzfi ne0ii rnmptfi infi mp1i unfi eqeltrid hashnncl 1red nn0red 0lt1 2re nnred - nnm1nn0 iooltub ltned hashprg sylancl mpbid eqcomd ssun1 sseqtr4i eqbrtrd + nnm1nn0 iooltub ltned hashprg sylancl mpbid eqcomd ssun1 sseqtrri eqbrtrd hashssle lesub1dd 3brtr4g ltletrd gt0ne0d elnnne0 sylanbrc eliccd jca vex 1e2m1 prss sylib inss2 ioossicc syl6ss unssd eqsstrid prid1 fourierdlem52 simplld simplrd simprd elfzoelz ltp1d sstri fourierdlem36 elfzofz fzofzp1 @@ -703531,7 +704280,7 @@ u C_ ( -u _pi [,] _pi ) /\ ( vol ` u ) <_ d ) -> A. k e. NN ( abs ` S. u syldan rexrd syl3anc letri3d mpbir2and wrex sselii eleqtrrid cres climc c0 cneg ctp cdm tpfi cpr renegcli negpilt0 pipos 0re lttri ltleii prunioo difeq1i difundir eqtr3i prfi diffi hashcl nn0zd ltneii hashprg mpbi ax-mp - 2z difexg eqeltri negex tpid1 tpid2 prssi ssun1 sseqtr4i hashss eqbrtrrid + 2z difexg eqeltri negex tpid1 tpid2 prssi ssun1 sseqtrri hashss eqbrtrrid unex sstri eluz2 syl3anbrc uz2m1nn cmap cfzo wral wiso wf1o negpitopissre cioc cmul 2timesi subnegi 3eqtr4i fourierdlem4 3jca sylib iccssre ssdifss fvex tpss fourierdlem36 isof1o f1of reex sylibr ffvelrnda elfzelz elfzle1 @@ -709309,7 +710058,7 @@ the supremum (in the real numbers) of finite subsums. Similar to ( vy wcel wceq wa adantr cpnf cxr cle vx va vb vc vd vz csumge0 cres cxad cfv cr co caddc cin c0 cc0 cicc wf simpr sge0resplit cvv syl2anc eqeltrid unexg sge0ssre rexadd eqtrd wn simpl wb sge0repnf mtbid notnotrd sge0xrcl - cun eqcomd wss ssun1 sseqtr4i fssresd iccssxr ssun2 sge0cl sseldi xaddcld + cun eqcomd wss ssun1 sseqtrri fssresd iccssxr ssun2 sge0cl sseldi xaddcld a1i wbr crn pnfxr eqid xreqle mp2an rnresss adantl sge0pnfval xrge0neqmnf sseli cmnf wne syl xaddpnf2 oveq1d 3eqtr4d eqtr3d breq12d mpbird xaddpnf1 oveq2d adantlr cpw cfn cv csu cmpt clt csup wral wrex elrnmpt inss2 sstri @@ -710435,7 +711184,7 @@ the supremum (in the real numbers) of finite subsums. Similar to ( vx wbr cfn wcel wa cr cc0 cv csu clt cpw cin wrex cfz cvv cuz fvexi a1i co sge0gtfsumgt w3a wss cz 3ad2ant1 elpwinss 3ad2ant2 elinel2 uzfissfz wi ad2antrr nfv nfan fzfid simpr simpll sselda cpnf cico rge0ssre cfv fzssuz - ssfid sseqtr4i sseldi sylan2 syl2anc fsumreclf adantlr simplr cle cxr 0xr + ssfid sseqtrri sseldi sylan2 syl2anc fsumreclf adantlr simplr cle cxr 0xr pnfxr icogelb syl3anc fsumlessf ltletrd adantr 3adantl2 reximdva mpd 3exp id ex rexlimdv ) ACNUAZBDUBZUCOZNGUDZPUEZUFCFEUAZUGULZBDUBZUCOZEGUFZANGBC DUHHGUHQAGFUIJUJUKKLMUMAXAXHNXCAWSXCQZXAXHAXIXAUNZWSXEUOZEGUFXHXJWSEFGAXI @@ -714238,7 +714987,7 @@ This is the proof of the statement in the middle of Step (e) in the ( cfv co cico cvol cprod csn cdif cmul cun cfn wcel snfi a1i unfi syl2anc cv eqeltrid snidg syl elun2 syl6eleqr ne0d hoidmvn0val ffvelrnda volicore wa cr recnd wceq fveq2 oveq12d fveq2d adantl fprodsplit1 difeq1i wn difsn - difun2 3eqtrd prodeq1d eqcomi eqtrd oveq2d ffvelrnd adantr ssun1 sseqtr4i + difun2 3eqtrd prodeq1d eqcomi eqtrd oveq2d ffvelrnd adantr ssun1 sseqtrri wf id sseldi fprodrecl mulcomd ) ACDIGUBUCIEUQZCUBZWNDUBZUDUCZUEUBZEUFKCU BZKDUBZUDUCZUEUBZIKUGZUHZWREUFZUIUCZFXBUIUCZABCDEGILMNAIJXCUJZUKRAJUKULXC UKULZXHUKULOXIAKUMUNJXCUOUPURZAIKAKXHIAKXCULZKXHULAKHULXKPKHUSUTKXCJVAUTR @@ -714725,7 +715474,7 @@ This is the proof of the statement in the middle of Step (e) in the eqeltrd wral w3a syl3anc vex rexbidv elab biimpi adantl nfcv nfel nfan wi eqeq1 3adant3 eqcomd 3exp mpd ralrimiva simp3 rexlimdv remulcld adantlr ex caddc cicc crab csn cun cdif snidg elun2 ffvelrnd ssrab2 eqsstri leidd - ltled eliccd recnd subidd cico rge0ssre cres ssfid ssun1 sseqtr4i fssresd + ltled eliccd recnd subidd cico rge0ssre cres ssfid ssun1 sseqtrri fssresd eqeltrid eqtrd 1red rpred readdcld 0red 0lt1 ltaddrpd lttrd nnex icossicc mul01d snfi unfi cmap wf ffvelrnda elmapi eleq1w breq1d ifbieq12d cbvmptv ifbieq1d mpteq2i eqtri fmpttd sge0cl sge0xrcl pnfxr eldifbd eldifd ltpnfd @@ -714888,7 +715637,7 @@ This is the proof of the statement in the middle of Step (e) in the iccgelb simpll syl6eleq elsni con3i elunnel1 adantll wb vex elrnmpt sylib wral w3a eleq2d elrab biimpi simprd icoltub 3adant3 3ad2ant3 rexlimdv mpd 3exp ralrimiva breq2 rspcva lelttrd ltled fiminre lbinfle eqbrtrid eliccd - npncand cpnf ssfid sseqtr4i fssresd hoidmvcl subcld adddid mulcld addcomd + npncand cpnf ssfid sseqtrri fssresd hoidmvcl subcld adddid mulcld addcomd cres ssun1 resubcl remulcl readdcl eqeltrrd rpred readdcld eldifbd eldifd 1red sge0hsphoire fzfid elfznn ovexd fvmpt2 0red fmptd resex ssexd mptexg iffalse ifcld eqeltrd fsumrecl mpteq2dv breq12d posdifd mulassd fsummulc1 @@ -715141,7 +715890,7 @@ This is the proof of the statement in the middle of Step (e) in the simpl fveq1d rspcva mpd ralbidv 3ad2ant1 caddc cfz csn 0le0 res0 oveq123d reseq2 f0 hoidmv0val 3eqtrd nfcvd 1red rge0ssre elexi eleq1 anbi2d eleq1d nfv ovexd cun eldifad snssi unssd eqsstrd ssfid ssun1 eqcomi sseqtri cmap - iftrue ffvelrnda elmapi sseqtr4i fssresd reex ssexd wn iffalse 0red fmptd + iftrue ffvelrnda elmapi sseqtrri fssresd reex ssexd wn iffalse 0red fmptd hoidmvcl vtocl recnd sumsnd ovex fvmpt ax-mp oveqd fvex resex syl6eqel wb elexd eqcomd feq2d rpred readdcld mul01d eqidd breq12d oveq2 cz nnzi fzsn sumeq1d breq2d rspcev wne neqne cdiv eqeltrid 1rp jca rpaddcl rpgt0 gtned @@ -724922,6 +725671,328 @@ exists a proof for (a and b). (Contributed by Jarvin Udandy, $( (End of Jarvin Udandy's mathbox.) $) +$( Begin $[ set-mbox-adh.mm $] $) +$( +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# + Mathbox for Adhemar +#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# +$) + + $( Replacement of a nested antecedent with an outer antecedent. Commuted + simplificated form of elimination of a nested antecedent. Also holds + intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed + by ADH, 10-Nov-2023.) (Proof modification is discouraged.) $) + adh-jarrsc $p |- ( ( ( ph -> ps ) -> ch ) -> ( th -> ( ps -> ch ) ) ) $= + ( wi jarr ax-1 ax-mp pm2.04 ) DABECEZBCEZEZEZJDKEELMABCFLDGHDJKIH $. + $( $j usage 'adh-jarrsc' avoids 'ax-3'; $) + + +$( +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + Minimal implicational calculus +=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= + + Minimal implicational calculus, or intuitionistic implicational calculus, or + positive implicational calculus, is the implicational fragment of minimal + calculus (which is also the implicational fragment of intuitionistic calculus + and of positive calculus). It is sometimes called "C-pure intuitionism" + since the letter C is used to denote implication in Polish prefix notation. + It can be axiomatized by the inference rule of modus ponens ~ ax-mp together + with the axioms { ~ ax-1 , ~ ax-2 } (sometimes written KS), or with + { ~ imim1 , ~ ax-1 , ~ pm2.43 } (written B'KW), or with { ~ imim2 , + ~ pm2.04 , ~ ax-1 , ~ pm2.43 } (written BCKW), or with the single axiom + ~ adh-minim , or with the single axiom ~ adh-minimp . This section proves + first ~ adh-minim from { ~ ax-1 , ~ ax-2 }, followed by the converse, due to + Ivo Thomas; and then it proves ~ adh-minimp from { ~ ax-1 , ~ ax-2 }, also + followed by the converse, also due to Ivo Thomas. + + Sources for this section are + * Carew Arthur Meredith, _A single axiom of positive logic_, The Journal of + Computing Systems, volume 1, issue 3, July 1953, pages 169--170; + * Ivo Thomas, _On Meredith's sole positive axiom_, Notre Dame Journal of + Formal Logic, volume XV, number 3, July 1974, page 477, in which the + derivations of { ~ ax-1 , ~ ax-2 } from ~ adh-minim are shortened (compared + to Meredith's derivations in the aforementioned paper); + * Carew Arthur Meredith and Arthur Norman Prior, _Notes on the axiomatics of + the propositional calculus_, Notre Dame Journal of Formal Logic, volume IV, + number 3, July 1963, pages 171--187; and + * the webpage + ~ https://web.ics.purdue.edu/~~dulrich/C-pure-intuitionism-page.htm + on Dolph Edward "Ted" Ulrich's website, where these and other single + axioms for the minimal implicational calculus are listed. + + This entire section also holds intuitionistically. + + Users of the Polish prefix notation also often use a compact notation for + proof derivations known as the D-notation where "D" stands for "condensed + Detachment". For instance, "D21" means detaching ~ ax-1 from ~ ax-2 , that + is, using modus ponens ~ ax-mp with ~ ax-1 as minor premise and ~ ax-2 as + major premise. When the numbered lemmas surpass 10, dots are added between + the numbers. D-strings are accepted by the grammar + Dundotted := digit | "D" Dundotted Dundotted ; + Ddotted := digit + | "D" Ddotted "." Ddotted ; + Dstr := Dundotted | Ddotted . + + (Contributed by BJ, 11-Apr-2021.) + (Revised by ADH, 10-Nov-2023.) + +$) + + $( A single axiom for minimal implicational calculus, due to Meredith. Other + single axioms of the same length are known, but it is thought to be the + minimal length. This is the axiom from Carew Arthur Meredith, _A single + axiom of positive logic_, The Journal of Computing Systems, volume 1, + issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of + this article can be found in The Journal of Symbolic Logic, volume 19, + issue 2, June 1954, page 144, ~ https://doi.org/10.2307/2268914 . Known + as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas + and 3 theorems, ~ ax-1 and ~ ax-2 are derived from this single axiom in 16 + detachments (instances of ~ ax-mp ) in total. Polish prefix notation: + CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023.) $) + adh-minim $p |- ( ( ( ph -> ps ) -> ch ) -> + ( th -> ( ( ps -> ( ch -> ta ) ) -> ( ps -> ta ) ) ) ) $= + ( wi pm2.04 adh-jarrsc ax-2 imim2 ax-mp ax-1 ) DABFCFZBCEFFZBEFZFZFZFZMDPFF + QRNMOFZFZQNCOFZFZTBCEGUASFZUBTFMUAOFZFZUCMUABCFZFZFZUEABCUAHUGUDFZUHUEFUAUF + OFZFZUIUANFZUKCBEGNUJFULUKFBCEINUJUAJKKUAUFOIKUGUDMJKKMUAOGKUASNJKKNMOGKQDL + KDMPGK $. + $( $j usage 'adh-minim' avoids 'ax-3'; $) + + $( First lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and + ~ ax-mp . Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed + by ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax1-ax2-lem1 $p |- ( ph -> ( ( ps -> ( ( ch -> ( ( th -> + ( ps -> ta ) ) -> ( th -> ta ) ) ) -> et ) ) -> ( ps -> et ) ) ) $= + ( wze wi adh-minim ax-mp ) GDHZBHCDBEHHDEHHHZHABLFHHBFHHHGDBCEIKBLAFIJ $. + + $( Second lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and + ~ ax-mp . Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by + ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax1-ax2-lem2 $p |- ( ( ph -> ( ( ps -> + ( ( ch -> ( ph -> th ) ) -> ( ch -> th ) ) ) -> ta ) ) -> ( ph -> ta ) ) $= + ( wet wze wsi wrh wmu wla wi adh-minim-ax1-ax2-lem1 ax-mp ) FGHIGJLLIJLLLKL + LGKLLLZABCADLLCDLLLELLAELLFGHIJKMOABCDEMN $. + + $( Third lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and + ~ ax-mp . Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax1-ax2-lem3 $p |- ( ( ph -> ( ps -> ch ) ) -> + ( ps -> ( th -> ( ph -> ch ) ) ) ) $= + ( wi adh-minim-ax1-ax2-lem1 adh-minim-ax1-ax2-lem2 ax-mp ) ABCEEZBDIACEZEED + JEZEEBKEZEEILEIBDACKFIBDJLGH $. + + $( Fourth lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and + ~ ax-mp . Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by + ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax1-ax2-lem4 $p |- ( ( ( ph -> ps ) -> ch ) -> + ( ( ps -> ( ch -> th ) ) -> ( ps -> th ) ) ) $= + ( wet wze wsi wi adh-minim adh-minim-ax1-ax2-lem2 ax-mp ) ABHCHZEFLGHHFGHHH + ZBCDHHBDHHZHHLNHABCMDILEFGNJK $. + + $( Derivation of ~ ax-1 from ~ adh-minim and ~ ax-mp . Carew Arthur Meredith + derived ~ ax-1 in _A single axiom of positive logic_, The Journal of + Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, + here we follow the shortened derivation by Ivo Thomas, _On Meredith's sole + positive axiom_, Notre Dame Journal of Formal Logic, volume XV, number 3, + July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by + ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax1 $p |- ( ph -> ( ps -> ph ) ) $= + ( wch wth wta wet wze wsi wi adh-minim-ax1-ax2-lem1 adh-minim-ax1-ax2-lem3 + adh-minim-ax1-ax2-lem4 ax-mp ) ABCDBEIIDEIIIZAIZIZBAIZIIZAQIZABCDEAJQPIZRSI + QBFGBHIIGHIIIZOIIZPIIZTQBFGHOJNQIUBIUCTINBAUAKNQUBPLMMBAPQLMM $. + + $( Fifth lemma for the derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . + Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax2-lem5 $p |- ( ph -> ( ( ( ps -> ch ) -> th ) -> + ( ( ch -> ( th -> ta ) ) -> ( ch -> ta ) ) ) ) $= + ( wi adh-minim-ax1-ax2-lem4 adh-minim-ax1 ax-mp ) BCFDFCDEFFCEFFFZAJFBCDEGJ + AHI $. + + $( Sixth lemma for the derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . + Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax2-lem6 $p |- ( ( ph -> ( ( ( ( ps -> ch ) -> th ) -> + ( ( ch -> ( th -> ta ) ) -> ( ch -> ta ) ) ) -> et ) ) -> ( ph -> et ) ) $= + ( wze wi adh-minim-ax2-lem5 adh-minim-ax1-ax2-lem4 ax-mp ) GAHZBCHDHCDEHHCE + HHHZHAMFHHAFHHLBCDEIGAMFJK $. + + $( Derivation of a commuted form of ~ ax-2 from ~ adh-minim and ~ ax-mp . + Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax2c $p |- ( ( ph -> ps ) -> + ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) $= + ( wth wta wet wze wsi wrh wmu wla adh-minim-ax2-lem5 adh-minim-ax1-ax2-lem4 + wi adh-minim-ax2-lem6 ax-mp ) ABNZDENFNEFGNNEGNNNZANZBNZABCNNACNNZNNZQUANZQ + RABCLSQNTNZUBUCNHINJNIJKNNIKNNNZSNZANZUDUFUEANNUGUEDEFGAOUFHIJKAOPUESABMPSQ + TUAMPP $. + + $( Derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . Carew Arthur Meredith + derived ~ ax-2 in _A single axiom of positive logic_, The Journal of + Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, + here we follow the shortened derivation by Ivo Thomas, _On Meredith's sole + positive axiom_, Notre Dame Journal of Formal Logic, volume XV, number 3, + July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . + (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> + ( ( ph -> ps ) -> ( ph -> ch ) ) ) $= + ( wth wta wi adh-minim-ax2c adh-minim-ax1-ax2-lem3 ax-mp adh-minim-ax2-lem6 + wet wze ) ABCFFZDEFKFEKLFFELFFFZABFZACFZFZFFZMQFOMPFFRABCGOMPNHIMDEKLQJI $. + + $( Derivation of ~ id (reflexivity of implication, PM *2.08 WhiteheadRussell + p. 101) from ~ adh-minim-ax1 , ~ adh-minim-ax2 , and ~ ax-mp . It uses + the derivation written DD211 in D-notation. (See head comment for an + explanation.) Polish prefix notation: Cpp . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minim-idALT $p |- ( ph -> ph ) $= + ( wps wi adh-minim-ax1 adh-minim-ax2 ax-mp ) ABACZCZAACZABDAGACCHICAGDAGAEF + F $. + + $( Derivation of ~ pm2.43 WhiteheadRussell p. 106 (also called "hilbert" or + "W") from ~ adh-minim-ax1 , ~ adh-minim-ax2 , and ~ ax-mp . It uses the + derivation written DD22D21 in D-notation. (See head comment for an + explanation.) (Contributed by ADH, 10-Nov-2023.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + adh-minim-pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $= + ( wi adh-minim-ax1 adh-minim-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFHIG + CCJKCAABEHIGEFF $. + + $( Another single axiom for minimal implicational calculus, due to Meredith. + Other single axioms of the same length are known, but it is thought to be + the minimal length. Among single axioms of this length, it is the one + with simplest antecedents (i.e., in the corresponding ordering of binary + trees which first compares left subtrees, it is the first one). Known as + "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and + 5 theorems, ~ ax-1 and ~ ax-2 are derived from this other single axiom in + 20 detachments (instances of ~ ax-mp ) in total. Polish prefix notation: + CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and + Arthur Norman Prior, _Notes on the axiomatics of the propositional + calculus_, Notre Dame Journal of Formal Logic, volume IV, number 3, July + 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) + (Revised by ADH, 10-Nov-2023.) $) + adh-minimp $p |- ( ph -> + ( ( ps -> ch ) -> ( ( ( th -> ps ) -> ( ch -> ta ) ) -> ( ps -> ta ) ) ) ) $= + ( wi jarr ax-2 imim2 ax-mp pm2.04 ax-1 ) BCFZDBFCEFZFZBEFZFFZAQFOMPFZFZQOBN + FZFZSDBNGTRFUASFBCEHTROIJJOMPKJQALJ $. + $( $j usage 'adh-minimp' avoids 'ax-3'; $) + + $( First lemma for the derivation of ~ jarr , ~ imim1 , and a commuted form + of ~ ax-2 , and indirectly ~ ax-1 and ~ ax-2 , from ~ adh-minimp and + ~ ax-mp . Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by + ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-jarr-imim1-ax2c-lem1 $p |- ( ( ph -> ps ) -> + ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) $= + ( wet wze wsi wrh wmu wi adh-minimp ax-mp ) EFGJHFJGIJJFIJJJJZABJCAJBDJJADJ + JJEFGHIKMABCDKL $. + + $( Second lemma for the derivation of ~ jarr , and indirectly ~ ax-1 , a + commuted form of ~ ax-2 , and ~ ax-2 proper, from ~ adh-minimp and + ~ ax-mp . Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . + (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-jarr-lem2 $p |- ( ( ( ph -> ps ) -> + ( ( ( ch -> th ) -> ( ( ( ta -> ch ) -> ( th -> et ) ) -> + ( ch -> et ) ) ) -> ze ) ) -> ( ps -> ze ) ) $= + ( wi adh-minimp adh-minimp-jarr-imim1-ax2c-lem1 ax-mp ) BCDHECHDFHHCFHHHZHA + BHLGHHBGHHBCDEFIBLAGJK $. + + $( Third lemma for the derivation of ~ jarr and a commuted form of ~ ax-2 , + and indirectly ~ ax-1 and ~ ax-2 proper , from ~ adh-minimp and ~ ax-mp . + Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-jarr-ax2c-lem3 $p |- ( ( ( ( ph -> ps ) -> + ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) -> ta ) $= + ( wet wze wsi wrh wmu wi adh-minimp-jarr-lem2 ax-mp ) FGHKIGKHJKKGJKKKZKZAB + KCAKBDKKADKKKEKZKNEKKPEKFNABCDELOPGHIJELM $. + + $( Derivation of ~ jarr (also called "syll-simp") from ~ minimp and ~ ax-mp . + Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) + (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-sylsimp $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $= + ( wi adh-minimp-jarr-ax2c-lem3 adh-minimp-jarr-imim1-ax2c-lem1 ax-mp + adh-minimp-jarr-lem2 ) ABDZICDZCDZDZJBCDZDZIIDZOODODDZJDJDZLIIIIJEIQDJIDQKD + DLDDQLDIQJKFIQJIPCLHGGJLDBJDLMDDNDDLNDJLBMFJLBJACNHGG $. + + $( Derivation of ~ ax-1 from ~ adh-minimp and ~ ax-mp . Polish prefix + notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-ax1 $p |- ( ph -> ( ps -> ph ) ) $= + ( wi adh-minimp-sylsimp ax-mp ) ABCZACBACZCAGCABADFAGDE $. + + $( Derivation of ~ imim1 ("left antimonotonicity of implication", theorem + *2.06 of [WhiteheadRussell] p. 100) from ~ adh-minimp and ~ ax-mp . + Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + adh-minimp-imim1 $p |- ( ( ph -> ps ) -> + ( ( ps -> ch ) -> ( ph -> ch ) ) ) $= + ( wth wrh wi adh-minimp-sylsimp adh-minimp-jarr-imim1-ax2c-lem1 ax-mp ) DAF + ZBCFZFACFZFZKLFZFZABFZNFZJKLGEPFZOFQFZOQFPMFSABDCHPMENHIROQGII $. + + $( Derivation of a commuted form of ~ ax-2 from ~ adh-minimp and ~ ax-mp . + Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) + (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-ax2c $p |- ( ( ph -> ps ) -> + ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) $= + ( wth wta wet wze adh-minimp-jarr-ax2c-lem3 adh-minimp-jarr-imim1-ax2c-lem1 + wi ax-mp adh-minimp-sylsimp adh-minimp-imim1 ) DEJFDJEGJJDGJJJZAJZBCJZJZACJ + ZJZAPJZRJZJZABJZUAJZTTJZSJUAJZUBTQJZUFOOJZTJQJZUGOAJUIDEFGAHOAOPIKUHTQLKTQT + RIKUESUALKUCSJUBUDJABNCIUCSUAMKK $. + + $( Fourth lemma for the derivation of ~ ax-2 from ~ adh-minimp and ~ ax-mp . + Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + adh-minimp-ax2-lem4 $p |- ( ph -> ( ( ps -> + ( ph -> ch ) ) -> ( ps -> ch ) ) ) $= + ( wi adh-minimp-ax2c adh-minimp-sylsimp ax-mp ) BADBACDDBCDDZDAHDBACEBAHFG + $. + + $( Derivation of ~ ax-2 from ~ adh-minimp and ~ ax-mp . Polish prefix + notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by + ADH, 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> + ( ( ph -> ps ) -> ( ph -> ch ) ) ) $= + ( wi adh-minimp-ax2-lem4 adh-minimp-ax2c ax-mp ) ABCDDZABDZHACDZDDZIJDZDDZH + LDZHIJEKMNDABCFKHLEGG $. + + $( Derivation of ~ id (reflexivity of implication, PM *2.08 WhiteheadRussell + p. 101) from ~ adh-minimp-ax1 , ~ adh-minimp-ax2 , and ~ ax-mp . It uses + the derivation written DD211 in D-notation. (See head comment for an + explanation.) Polish prefix notation: Cpp . (Contributed by ADH, + 10-Nov-2023.) (Proof modification is discouraged.) + (New usage is discouraged.) $) + adh-minimp-idALT $p |- ( ph -> ph ) $= + ( wps wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) ABACZCZAACZABDAGACCHICAGDAGA + EFF $. + + $( Derivation of ~ pm2.43 WhiteheadRussell p. 106 (also called "hilbert" or + "W") from ~ adh-minimp-ax1 , ~ adh-minimp-ax2 , and ~ ax-mp . It uses the + derivation written DD22D21 in D-notation. (See head comment for an + explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, + 31-May-2021.) (Revised by ADH, 10-Nov-2023.) + (Proof modification is discouraged.) (New usage is discouraged.) $) + adh-minimp-pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $= + ( wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFH + IGCCJKCAABEHIGEFF $. + +$( (End of Adhemar's mathbox.) $) +$( End $[ set-mbox-adh.mm $] $) + + $( Begin $[ set-mbox-av.mm $] $) $( Skip $[ set-main.mm $] $) $( @@ -724930,12 +726001,14 @@ exists a proof for (a and b). (Contributed by Jarvin Udandy, #*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# $) + $( =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= General auxiliary theorems (1) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= $) + $( -.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.- Unordered and ordered pairs - extension for singletons @@ -727644,12 +728717,12 @@ Negated membership (alternative) (Contributed by AV, 20-Sep-2020.) $) funop1 $p |- ( E. x E. y F = <. x , y >. -> ( Fun F <-> E. x E. y F = { <. x , y >. } ) ) $= - ( vv vw va cv cop wceq wex wfun csn wb weq opeq12 eqeq2d cbvex2v exlimivv - wa vex funopsn sneqd spc2ev adantl exlimiv syl expcom funsn funeq impbid1 - mpbiri sylbi ) CAGZBGZHZIZBJAJCDGZEGZHZIZEJDJCKZCUOLZIZBJAJZMZUPUTABDEADN - BENSUOUSCUMUNUQUROPQUTVEDEUTVAVDVAUTVDVAUTSUQFGZLIZCVFVFHZLZIZSZFJVDCUQUR - FDTETUAVKVDFVJVDVGVCVJABVFVFFTZVLAFNBFNSZVBVICVMUOVHUMUNVFVFOUBPUCUDUEUFU - GVCVAABVCVAVBKUMUNATBTUHCVBUIUKRUJRUL $. + ( vv vw va cv cop wceq wex wfun csn wb wa opeq12 eqeq2d cbvex2vv exlimivv + weq vex funopsn sneqd spc2ev adantl exlimiv syl expcom funsn funeq mpbiri + impbid1 sylbi ) CAGZBGZHZIZBJAJCDGZEGZHZIZEJDJCKZCUOLZIZBJAJZMZUPUTABDEAD + SBESNUOUSCUMUNUQUROPQUTVEDEUTVAVDVAUTVDVAUTNUQFGZLIZCVFVFHZLZIZNZFJVDCUQU + RFDTETUAVKVDFVJVDVGVCVJABVFVFFTZVLAFSBFSNZVBVICVMUOVHUMUNVFVFOUBPUCUDUEUF + UGVCVAABVCVAVBKUMUNATBTUHCVBUIUJRUKRUL $. $} ${ @@ -734752,7 +735825,7 @@ the ternary Goldbach conjecture (see section 1.2.2 in [Helfgott] p. 4) or mpbir simpl oveq1 oveq1d eqeq12d 2rexbidv eqeq2d rexbidv df-3 df-2 oveq1i oveq2 syl6reqr rspcedeq2vd rspcedvd syl 3p1e4 df-5 oveq12i 4z fzval3 fzsn eqtr4i bitri 2prm olci df-4 eqcomd eqtrd sylbi jaoi 3prm a1d sbgoldbm wss - rspa ssun2 sseqtr4i rexss simpr reximi ex com12 syl5bi ralrimi ) GBUDZUAU + rspa ssun2 sseqtrri rexss simpr reximi ex com12 syl5bi ralrimi ) GBUDZUAU BYLUCHUEZBUFUGZYLEUDZDUDZIJZCUDZIJZKZCALZDALZEALZBMUHUIZYMBUFUJYLUUDHZYLM NOUKJZULJZNUHUIZUMZHZYNUUCNUUDHZUUEUUJPUUKMUNHZNUNHMNUOUBUPNUQURMNUSVAVBU TMNVCVDUUKUUDUUIYLMNVEVFVGUUJYNUUCUUJYLUUGHZYLUUHHZVHYNUUCUEZYLUUGUUHVIUU @@ -736537,20 +737610,20 @@ it allows only walks consisting of not proper hyperedges (i.e. edges uspgrsprf1 $p |- F : G -1-1-> P $= ( va vw vf cv cfv wceq wi wa wex vb wf1 wf weq wral uspgrsprf wcel c2nd uspgrsprfv eqeqan12d cvtx cedg cuspgr copab eleq2i elopab opeq12 eqeq2d - cop wb eqeq1 adantr eqeq2 bi2anan9 rexbidv anbi12d cbvex2v 3bitri bitri - ex equcoms syl6bir ad2antrl com12 imp vex op2ndd eqeq12d eqeq12 3imtr4d - wrex exlimivv syl2anb sylbid rgen2a dff13 mpbir2an ) FBEUBFBEUCLOZEPZUA - OZEPZQZLUAUDZRZUAFUELFUEABCDEFGHIJKUFWNLUAFWHFUGZWJFUGZSWLWHUHPZWJUHPZQ - ZWMWOWPWIWQWKWRABCDEFGWHHIJKUIABCDEFGWJHIJKUIUJWOWHMOZNOZUSZQZWTGQZHOZU - KPZWTQZXEULPZXAQZSZHUMWAZSZSZNTMTZWJAOZCOZUSZQZXOGQZXFXOQZXHXPQZSZHUMWA - ZSZSZCTATZWSWMRZWPWOWHYDACUNZUGWHXQQZYDSZCTATXNFYHWHJUOYDACWHUPYJXMACMN - AMUDZCNUDZSZYIXCYDXLYMXQXBWHXOXPWTXAUQURYMXSXDYCXKYKXSXDUTYLXOWTGVAVBYM - YBXJHUMYKXTXGYLYAXIXOWTXFVCXPXAXHVCVDVEVFVFVGVHWPWJYHUGYFFYHWJJUOYDACWJ - UPVIXNYFYGXMYFYGRMNYFXMYGYEXMYGRACYEXMYGYEXMSZNCUDZXBXQQZWSWMYEXMYOYPRZ - XSXMYQRXRYCXMXSYQXDXSYQRXCXKXDXSYKYQWTGXOVCYQMAMAUDYOYPWTXAXOXPUQVJVKVL - VMVNVMVOYNWQXAWRXPXCWQXAQYEXLWTXAWHMVPNVPVQVMYEWRXPQZXMXRYRYDXOXPWJAVPC - VPVQVBVBVRYEXMWMYPUTZXRXMYSRYDXMXRYSXCXRYSRXLXCXRYSWHXBWJXQVSVJVBVNVBVO - VTVJWBVNWBVOWCWDWELUAFBEWFWG $. + cop wrex wb eqeq1 adantr eqeq2 bi2anan9 rexbidv anbi12d cbvex2vv 3bitri + bitri ex equcoms syl6bir ad2antrl imp vex op2ndd eqeq12d eqeq12 3imtr4d + com12 exlimivv syl2anb sylbid rgen2a dff13 mpbir2an ) FBEUBFBEUCLOZEPZU + AOZEPZQZLUAUDZRZUAFUELFUEABCDEFGHIJKUFWNLUAFWHFUGZWJFUGZSWLWHUHPZWJUHPZ + QZWMWOWPWIWQWKWRABCDEFGWHHIJKUIABCDEFGWJHIJKUIUJWOWHMOZNOZUSZQZWTGQZHOZ + UKPZWTQZXEULPZXAQZSZHUMUTZSZSZNTMTZWJAOZCOZUSZQZXOGQZXFXOQZXHXPQZSZHUMU + TZSZSZCTATZWSWMRZWPWOWHYDACUNZUGWHXQQZYDSZCTATXNFYHWHJUOYDACWHUPYJXMACM + NAMUDZCNUDZSZYIXCYDXLYMXQXBWHXOXPWTXAUQURYMXSXDYCXKYKXSXDVAYLXOWTGVBVCY + MYBXJHUMYKXTXGYLYAXIXOWTXFVDXPXAXHVDVEVFVGVGVHVIWPWJYHUGYFFYHWJJUOYDACW + JUPVJXNYFYGXMYFYGRMNYFXMYGYEXMYGRACYEXMYGYEXMSZNCUDZXBXQQZWSWMYEXMYOYPR + ZXSXMYQRXRYCXMXSYQXDXSYQRXCXKXDXSYKYQWTGXOVDYQMAMAUDYOYPWTXAXOXPUQVKVLV + MVNWAVNVOYNWQXAWRXPXCWQXAQYEXLWTXAWHMVPNVPVQVNYEWRXPQZXMXRYRYDXOXPWJAVP + CVPVQVCVCVRYEXMWMYPVAZXRXMYSRYDXMXRYSXCXRYSRXLXCXRYSWHXBWJXQVSVKVCWAVCV + OVTVKWBWAWBVOWCWDWELUAFBEWFWG $. $d V a f p $. $d W a b $. $d W p $. $d W q $. $d a q $. $( The mapping ` F ` is a function from the "simple pseudographs" with a @@ -742083,7 +743156,7 @@ Basic algebraic structures (extension) dmmpossx2 $p |- dom F C_ U_ y e. B ( A X. { y } ) $= ( vu vt vv cv csb csn cxp ciun cfv cmpo nfcv nfcsb1v csbeq1a cdm c2nd weq c1st cmpt nfcsb sylan9eqr cbvmpox2 cop vex op2ndd csbeq1d op1std csbeq2dv - wceq eqtrd mpomptx2 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtr4i ) F + wceq eqtrd mpomptx2 3eqtr4i dmmptss nfxp sneq xpeq12d cbviun sseqtrri ) F UAHDBHKZCLZVEMZNZOZBDCBKZMZNZOIVIBIKZUBPZAVMUDPZELZLZFABCDEQJHVFDBVEAJKZE LZLZQFIVIVQUEABHJCDEVFVTHCRBVECSZHERJERABVEVSAVERAVRESUFBVEVSSBVECTZAJUCB HUCZEVSVTAVRETBVEVSTUGUHGJHIVFDVQVTVMVRVEUIUOZVQBVEVPLVTWDBVNVEVPVRVEVMJU @@ -752904,7 +753977,7 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler, imbi1d dmmpti f1eq2 anbi1i bitri wnel con34b df-nel xchbinx imbi1i bitr4i velsn ralbii raldifb ralnex 3bitri 3bitr3g clss cmri cuvc cpw cmre lssmre clspn cmrc mrclsp clvec islvec mpbir2an cacs lssacsex frnd dif0 syl6sseqr - simprd uvcff frn sseqtr4i fveq2i clbs lbssp eqtri pm3.2i lindsind2 mp3an1 + simprd uvcff frn sseqtrri fveq2i clbs lbssp eqtri pm3.2i lindsind2 mp3an1 frlmlbs ralrimiva ismri2 sylancr biimpar eqeltrid wo mptfi rnfi mp2b orci mreexexd elpwi ssdomg mpsyl endomtr f1f1orn f1oen uvcendim ensymi domentr rnex wf1o chash hashdom hashfz0 hashfz1 breq12d syl5bbr syl5ib expd com23 @@ -753256,415 +754329,4 @@ to Davis and Putnam (1960). (Contributed by David A. Wheeler, $( (End of Kunhao Zheng's mathbox.) $) -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Larry Lesyna -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - -$( (End of Larry Lesyna's mathbox.) $) - -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Adhemar -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - - $( Replacement of a nested antecedent with an outer antecedent. Commuted - simplificated form of elimination of a nested antecedent. Also holds - intuitionistically. Polish prefix notation: CCCpqrCsCqr . (Contributed - by ADH, 10-Nov-2023.) (Proof modification is discouraged.) $) - adh-jarrsc $p |- ( ( ( ph -> ps ) -> ch ) -> ( th -> ( ps -> ch ) ) ) $= - ( wi jarr ax-1 ax-mp pm2.04 ) DABECEZBCEZEZEZJDKEELMABCFLDGHDJKIH $. - $( $j usage 'adh-jarrsc' avoids 'ax-3'; $) - - -$( -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - Minimal implicational calculus -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= - - Minimal implicational calculus, or intuitionistic implicational calculus, or - positive implicational calculus, is the implicational fragment of minimal - calculus (which is also the implicational fragment of intuitionistic calculus - and of positive calculus). It is sometimes called "C-pure intuitionism" - since the letter C is used to denote implication in Polish prefix notation. - It can be axiomatized by the inference rule of modus ponens ~ ax-mp together - with the axioms { ~ ax-1 , ~ ax-2 } (sometimes written KS), or with - { ~ imim1 , ~ ax-1 , ~ pm2.43 } (written B'KW), or with { ~ imim2 , - ~ pm2.04 , ~ ax-1 , ~ pm2.43 } (written BCKW), or with the single axiom - ~ adh-minim , or with the single axiom ~ adh-minimp . This section proves - first ~ adh-minim from { ~ ax-1 , ~ ax-2 }, followed by the converse, due to - Ivo Thomas; and then it proves ~ adh-minimp from { ~ ax-1 , ~ ax-2 }, also - followed by the converse, also due to Ivo Thomas. - - Sources for this section are - * Carew Arthur Meredith, _A single axiom of positive logic_, The Journal of - Computing Systems, volume 1, issue 3, July 1953, pages 169--170; - * Ivo Thomas, _On Meredith's sole positive axiom_, Notre Dame Journal of - Formal Logic, volume XV, number 3, July 1974, page 477, in which the - derivations of { ~ ax-1 , ~ ax-2 } from ~ adh-minim are shortened (compared - to Meredith's derivations in the aforementioned paper); - * Carew Arthur Meredith and Arthur Norman Prior, _Notes on the axiomatics of - the propositional calculus_, Notre Dame Journal of Formal Logic, volume IV, - number 3, July 1963, pages 171--187; and - * the webpage - ~ https://web.ics.purdue.edu/~~dulrich/C-pure-intuitionism-page.htm - on Dolph Edward "Ted" Ulrich's website, where these and other single - axioms for the minimal implicational calculus are listed. - - This entire section also holds intuitionistically. - - Users of the Polish prefix notation also often use a compact notation for - proof derivations known as the D-notation where "D" stands for "condensed - Detachment". For instance, "D21" means detaching ~ ax-1 from ~ ax-2 , that - is, using modus ponens ~ ax-mp with ~ ax-1 as minor premise and ~ ax-2 as - major premise. When the numbered lemmas surpass 10, dots are added between - the numbers. D-strings are accepted by the grammar - Dundotted := digit | "D" Dundotted Dundotted ; - Ddotted := digit + | "D" Ddotted "." Ddotted ; - Dstr := Dundotted | Ddotted . - - (Contributed by BJ, 11-Apr-2021.) - (Revised by ADH, 10-Nov-2023.) - -$) - - $( A single axiom for minimal implicational calculus, due to Meredith. Other - single axioms of the same length are known, but it is thought to be the - minimal length. This is the axiom from Carew Arthur Meredith, _A single - axiom of positive logic_, The Journal of Computing Systems, volume 1, - issue 3, July 1953, pages 169--170. A two-line review by Alonzo Church of - this article can be found in The Journal of Symbolic Logic, volume 19, - issue 2, June 1954, page 144, ~ https://doi.org/10.2307/2268914 . Known - as "HI-1" on Dolph Edward "Ted" Ulrich's web page. In the next 6 lemmas - and 3 theorems, ~ ax-1 and ~ ax-2 are derived from this single axiom in 16 - detachments (instances of ~ ax-mp ) in total. Polish prefix notation: - CCCpqrCsCCqCrtCqt . (Contributed by ADH, 10-Nov-2023.) $) - adh-minim $p |- ( ( ( ph -> ps ) -> ch ) -> - ( th -> ( ( ps -> ( ch -> ta ) ) -> ( ps -> ta ) ) ) ) $= - ( wi pm2.04 adh-jarrsc ax-2 imim2 ax-mp ax-1 ) DABFCFZBCEFFZBEFZFZFZFZMDPFF - QRNMOFZFZQNCOFZFZTBCEGUASFZUBTFMUAOFZFZUCMUABCFZFZFZUEABCUAHUGUDFZUHUEFUAUF - OFZFZUIUANFZUKCBEGNUJFULUKFBCEINUJUAJKKUAUFOIKUGUDMJKKMUAOGKUASNJKKNMOGKQDL - KDMPGK $. - $( $j usage 'adh-minim' avoids 'ax-3'; $) - - $( First lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and - ~ ax-mp . Polish prefix notation: CpCCqCCrCCsCqtCstuCqu . (Contributed - by ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax1-ax2-lem1 $p |- ( ph -> ( ( ps -> ( ( ch -> ( ( th -> - ( ps -> ta ) ) -> ( th -> ta ) ) ) -> et ) ) -> ( ps -> et ) ) ) $= - ( wze wi adh-minim ax-mp ) GDHZBHCDBEHHDEHHHZHABLFHHBFHHHGDBCEIKBLAFIJ $. - - $( Second lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and - ~ ax-mp . Polish prefix notation: CCpCCqCCrCpsCrstCpt . (Contributed by - ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax1-ax2-lem2 $p |- ( ( ph -> ( ( ps -> - ( ( ch -> ( ph -> th ) ) -> ( ch -> th ) ) ) -> ta ) ) -> ( ph -> ta ) ) $= - ( wet wze wsi wrh wmu wla wi adh-minim-ax1-ax2-lem1 ax-mp ) FGHIGJLLIJLLLKL - LGKLLLZABCADLLCDLLLELLAELLFGHIJKMOABCDEMN $. - - $( Third lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and - ~ ax-mp . Polish prefix notation: CCpCqrCqCsCpr . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax1-ax2-lem3 $p |- ( ( ph -> ( ps -> ch ) ) -> - ( ps -> ( th -> ( ph -> ch ) ) ) ) $= - ( wi adh-minim-ax1-ax2-lem1 adh-minim-ax1-ax2-lem2 ax-mp ) ABCEEZBDIACEZEED - JEZEEBKEZEEILEIBDACKFIBDJLGH $. - - $( Fourth lemma for the derivation of ~ ax-1 and ~ ax-2 from ~ adh-minim and - ~ ax-mp . Polish prefix notation: CCCpqrCCqCrsCqs . (Contributed by - ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax1-ax2-lem4 $p |- ( ( ( ph -> ps ) -> ch ) -> - ( ( ps -> ( ch -> th ) ) -> ( ps -> th ) ) ) $= - ( wet wze wsi wi adh-minim adh-minim-ax1-ax2-lem2 ax-mp ) ABHCHZEFLGHHFGHHH - ZBCDHHBDHHZHHLNHABCMDILEFGNJK $. - - $( Derivation of ~ ax-1 from ~ adh-minim and ~ ax-mp . Carew Arthur Meredith - derived ~ ax-1 in _A single axiom of positive logic_, The Journal of - Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, - here we follow the shortened derivation by Ivo Thomas, _On Meredith's sole - positive axiom_, Notre Dame Journal of Formal Logic, volume XV, number 3, - July 1974, page 477. Polish prefix notation: CpCqp . (Contributed by - ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax1 $p |- ( ph -> ( ps -> ph ) ) $= - ( wch wth wta wet wze wsi wi adh-minim-ax1-ax2-lem1 adh-minim-ax1-ax2-lem3 - adh-minim-ax1-ax2-lem4 ax-mp ) ABCDBEIIDEIIIZAIZIZBAIZIIZAQIZABCDEAJQPIZRSI - QBFGBHIIGHIIIZOIIZPIIZTQBFGHOJNQIUBIUCTINBAUAKNQUBPLMMBAPQLMM $. - - $( Fifth lemma for the derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . - Polish prefix notation: CpCCCqrsCCrCstCrt . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax2-lem5 $p |- ( ph -> ( ( ( ps -> ch ) -> th ) -> - ( ( ch -> ( th -> ta ) ) -> ( ch -> ta ) ) ) ) $= - ( wi adh-minim-ax1-ax2-lem4 adh-minim-ax1 ax-mp ) BCFDFCDEFFCEFFFZAJFBCDEGJ - AHI $. - - $( Sixth lemma for the derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . - Polish prefix notation: CCpCCCCqrsCCrCstCrtuCpu . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax2-lem6 $p |- ( ( ph -> ( ( ( ( ps -> ch ) -> th ) -> - ( ( ch -> ( th -> ta ) ) -> ( ch -> ta ) ) ) -> et ) ) -> ( ph -> et ) ) $= - ( wze wi adh-minim-ax2-lem5 adh-minim-ax1-ax2-lem4 ax-mp ) GAHZBCHDHCDEHHCE - HHHZHAMFHHAFHHLBCDEIGAMFJK $. - - $( Derivation of a commuted form of ~ ax-2 from ~ adh-minim and ~ ax-mp . - Polish prefix notation: CCpqCCpCqrCpr . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax2c $p |- ( ( ph -> ps ) -> - ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) $= - ( wth wta wet wze wsi wrh wmu wla adh-minim-ax2-lem5 adh-minim-ax1-ax2-lem4 - wi adh-minim-ax2-lem6 ax-mp ) ABNZDENFNEFGNNEGNNNZANZBNZABCNNACNNZNNZQUANZQ - RABCLSQNTNZUBUCNHINJNIJKNNIKNNNZSNZANZUDUFUEANNUGUEDEFGAOUFHIJKAOPUESABMPSQ - TUAMPP $. - - $( Derivation of ~ ax-2 from ~ adh-minim and ~ ax-mp . Carew Arthur Meredith - derived ~ ax-2 in _A single axiom of positive logic_, The Journal of - Computing Systems, volume 1, issue 3, July 1953, pages 169--170. However, - here we follow the shortened derivation by Ivo Thomas, _On Meredith's sole - positive axiom_, Notre Dame Journal of Formal Logic, volume XV, number 3, - July 1974, page 477. Polish prefix notation: CCpCqrCCpqCpr . - (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> - ( ( ph -> ps ) -> ( ph -> ch ) ) ) $= - ( wth wta wi adh-minim-ax2c adh-minim-ax1-ax2-lem3 ax-mp adh-minim-ax2-lem6 - wet wze ) ABCFFZDEFKFEKLFFELFFFZABFZACFZFZFFZMQFOMPFFRABCGOMPNHIMDEKLQJI $. - - $( Derivation of ~ id (reflexivity of implication, PM *2.08 WhiteheadRussell - p. 101) from ~ adh-minim-ax1 , ~ adh-minim-ax2 , and ~ ax-mp . It uses - the derivation written DD211 in D-notation. (See head comment for an - explanation.) Polish prefix notation: Cpp . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minim-idALT $p |- ( ph -> ph ) $= - ( wps wi adh-minim-ax1 adh-minim-ax2 ax-mp ) ABACZCZAACZABDAGACCHICAGDAGAEF - F $. - - $( Derivation of ~ pm2.43 WhiteheadRussell p. 106 (also called "hilbert" or - "W") from ~ adh-minim-ax1 , ~ adh-minim-ax2 , and ~ ax-mp . It uses the - derivation written DD22D21 in D-notation. (See head comment for an - explanation.) (Contributed by ADH, 10-Nov-2023.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - adh-minim-pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $= - ( wi adh-minim-ax1 adh-minim-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFHIG - CCJKCAABEHIGEFF $. - - $( Another single axiom for minimal implicational calculus, due to Meredith. - Other single axioms of the same length are known, but it is thought to be - the minimal length. Among single axioms of this length, it is the one - with simplest antecedents (i.e., in the corresponding ordering of binary - trees which first compares left subtrees, it is the first one). Known as - "HI-2" on Dolph Edward "Ted" Ulrich's web page. In the next 4 lemmas and - 5 theorems, ~ ax-1 and ~ ax-2 are derived from this other single axiom in - 20 detachments (instances of ~ ax-mp ) in total. Polish prefix notation: - CpCCqrCCCsqCrtCqt ; or CtCCpqCCCspCqrCpr in Carew Arthur Meredith and - Arthur Norman Prior, _Notes on the axiomatics of the propositional - calculus_, Notre Dame Journal of Formal Logic, volume IV, number 3, July - 1963, pages 171--187, on page 180. (Contributed by BJ, 4-Apr-2021.) - (Revised by ADH, 10-Nov-2023.) $) - adh-minimp $p |- ( ph -> - ( ( ps -> ch ) -> ( ( ( th -> ps ) -> ( ch -> ta ) ) -> ( ps -> ta ) ) ) ) $= - ( wi jarr ax-2 imim2 ax-mp pm2.04 ax-1 ) BCFZDBFCEFZFZBEFZFFZAQFOMPFZFZQOBN - FZFZSDBNGTRFUASFBCEHTROIJJOMPKJQALJ $. - $( $j usage 'adh-minimp' avoids 'ax-3'; $) - - $( First lemma for the derivation of ~ jarr , ~ imim1 , and a commuted form - of ~ ax-2 , and indirectly ~ ax-1 and ~ ax-2 , from ~ adh-minimp and - ~ ax-mp . Polish prefix notation: CCpqCCCrpCqsCps . (Contributed by - ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-jarr-imim1-ax2c-lem1 $p |- ( ( ph -> ps ) -> - ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) $= - ( wet wze wsi wrh wmu wi adh-minimp ax-mp ) EFGJHFJGIJJFIJJJJZABJCAJBDJJADJ - JJEFGHIKMABCDKL $. - - $( Second lemma for the derivation of ~ jarr , and indirectly ~ ax-1 , a - commuted form of ~ ax-2 , and ~ ax-2 proper, from ~ adh-minimp and - ~ ax-mp . Polish prefix notation: CCCpqCCCrsCCCtrCsuCruvCqv . - (Contributed by ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-jarr-lem2 $p |- ( ( ( ph -> ps ) -> - ( ( ( ch -> th ) -> ( ( ( ta -> ch ) -> ( th -> et ) ) -> - ( ch -> et ) ) ) -> ze ) ) -> ( ps -> ze ) ) $= - ( wi adh-minimp adh-minimp-jarr-imim1-ax2c-lem1 ax-mp ) BCDHECHDFHHCFHHHZHA - BHLGHHBGHHBCDEFIBLAGJK $. - - $( Third lemma for the derivation of ~ jarr and a commuted form of ~ ax-2 , - and indirectly ~ ax-1 and ~ ax-2 proper , from ~ adh-minimp and ~ ax-mp . - Polish prefix notation: CCCCpqCCCrpCqsCpstt . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-jarr-ax2c-lem3 $p |- ( ( ( ( ph -> ps ) -> - ( ( ( ch -> ph ) -> ( ps -> th ) ) -> ( ph -> th ) ) ) -> ta ) -> ta ) $= - ( wet wze wsi wrh wmu wi adh-minimp-jarr-lem2 ax-mp ) FGHKIGKHJKKGJKKKZKZAB - KCAKBDKKADKKKEKZKNEKKPEKFNABCDELOPGHIJELM $. - - $( Derivation of ~ jarr (also called "syll-simp") from ~ minimp and ~ ax-mp . - Polish prefix notation: CCCpqrCqr . (Contributed by BJ, 4-Apr-2021.) - (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-sylsimp $p |- ( ( ( ph -> ps ) -> ch ) -> ( ps -> ch ) ) $= - ( wi adh-minimp-jarr-ax2c-lem3 adh-minimp-jarr-imim1-ax2c-lem1 ax-mp - adh-minimp-jarr-lem2 ) ABDZICDZCDZDZJBCDZDZIIDZOODODDZJDJDZLIIIIJEIQDJIDQKD - DLDDQLDIQJKFIQJIPCLHGGJLDBJDLMDDNDDLNDJLBMFJLBJACNHGG $. - - $( Derivation of ~ ax-1 from ~ adh-minimp and ~ ax-mp . Polish prefix - notation: CpCqp . (Contributed by BJ, 4-Apr-2021.) (Revised by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-ax1 $p |- ( ph -> ( ps -> ph ) ) $= - ( wi adh-minimp-sylsimp ax-mp ) ABCZACBACZCAGCABADFAGDE $. - - $( Derivation of ~ imim1 ("left antimonotonicity of implication", theorem - *2.06 of [WhiteheadRussell] p. 100) from ~ adh-minimp and ~ ax-mp . - Polish prefix notation: CCpqCCqrCpr . (Contributed by ADH, 10-Nov-2023.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - adh-minimp-imim1 $p |- ( ( ph -> ps ) -> - ( ( ps -> ch ) -> ( ph -> ch ) ) ) $= - ( wth wrh wi adh-minimp-sylsimp adh-minimp-jarr-imim1-ax2c-lem1 ax-mp ) DAF - ZBCFZFACFZFZKLFZFZABFZNFZJKLGEPFZOFQFZOQFPMFSABDCHPMENHIROQGII $. - - $( Derivation of a commuted form of ~ ax-2 from ~ adh-minimp and ~ ax-mp . - Polish prefix notation: CCpqCCpCqrCpr . (Contributed by BJ, 4-Apr-2021.) - (Revised by ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-ax2c $p |- ( ( ph -> ps ) -> - ( ( ph -> ( ps -> ch ) ) -> ( ph -> ch ) ) ) $= - ( wth wta wet wze adh-minimp-jarr-ax2c-lem3 adh-minimp-jarr-imim1-ax2c-lem1 - wi ax-mp adh-minimp-sylsimp adh-minimp-imim1 ) DEJFDJEGJJDGJJJZAJZBCJZJZACJ - ZJZAPJZRJZJZABJZUAJZTTJZSJUAJZUBTQJZUFOOJZTJQJZUGOAJUIDEFGAHOAOPIKUHTQLKTQT - RIKUESUALKUCSJUBUDJABNCIUCSUAMKK $. - - $( Fourth lemma for the derivation of ~ ax-2 from ~ adh-minimp and ~ ax-mp . - Polish prefix notation: CpCCqCprCqr . (Contributed by ADH, 10-Nov-2023.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - adh-minimp-ax2-lem4 $p |- ( ph -> ( ( ps -> - ( ph -> ch ) ) -> ( ps -> ch ) ) ) $= - ( wi adh-minimp-ax2c adh-minimp-sylsimp ax-mp ) BADBACDDBCDDZDAHDBACEBAHFG - $. - - $( Derivation of ~ ax-2 from ~ adh-minimp and ~ ax-mp . Polish prefix - notation: CCpCqrCCpqCpr . (Contributed by BJ, 4-Apr-2021.) (Revised by - ADH, 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-ax2 $p |- ( ( ph -> ( ps -> ch ) ) -> - ( ( ph -> ps ) -> ( ph -> ch ) ) ) $= - ( wi adh-minimp-ax2-lem4 adh-minimp-ax2c ax-mp ) ABCDDZABDZHACDZDDZIJDZDDZH - LDZHIJEKMNDABCFKHLEGG $. - - $( Derivation of ~ id (reflexivity of implication, PM *2.08 WhiteheadRussell - p. 101) from ~ adh-minimp-ax1 , ~ adh-minimp-ax2 , and ~ ax-mp . It uses - the derivation written DD211 in D-notation. (See head comment for an - explanation.) Polish prefix notation: Cpp . (Contributed by ADH, - 10-Nov-2023.) (Proof modification is discouraged.) - (New usage is discouraged.) $) - adh-minimp-idALT $p |- ( ph -> ph ) $= - ( wps wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) ABACZCZAACZABDAGACCHICAGDAGA - EFF $. - - $( Derivation of ~ pm2.43 WhiteheadRussell p. 106 (also called "hilbert" or - "W") from ~ adh-minimp-ax1 , ~ adh-minimp-ax2 , and ~ ax-mp . It uses the - derivation written DD22D21 in D-notation. (See head comment for an - explanation.) Polish prefix notation: CCpCpqCpq . (Contributed by BJ, - 31-May-2021.) (Revised by ADH, 10-Nov-2023.) - (Proof modification is discouraged.) (New usage is discouraged.) $) - adh-minimp-pm2.43 $p |- ( ( ph -> ( ph -> ps ) ) -> ( ph -> ps ) ) $= - ( wi adh-minimp-ax1 adh-minimp-ax2 ax-mp ) AABCZCZAACZCZHGCZAGACCJAGDAGAEFH - IGCCJKCAABEHIGEFF $. - -$( (End of Adhemar's mathbox.) $) - - -$( -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# - Mathbox for Igor Ieskov -#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*# -$) - - ${ - binom2d.1 $e |- ( ph -> A e. CC ) $. - binom2d.2 $e |- ( ph -> B e. CC ) $. - $( Deduction form of binom2. (Contributed by Igor Ieskov, 14-Dec-2023.) $) - binom2d $p |- ( ph -> ( ( A + B ) ^ 2 ) = - ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) ) $= - ( cc wcel caddc co c2 cexp cmul wceq binom2 syl2anc ) ABFZGCPGBCHZIJZKZIB - RSIRBCLZITIQICRSIQIMDEBCNO $. - $} - - ${ - cu3addd.1 $e |- ( ph -> A e. CC ) $. - cu3addd.2 $e |- ( ph -> B e. CC ) $. - cu3addd.3 $e |- ( ph -> C e. CC ) $. - $( Cube of sum of three numbers. (Contributed by Igor Ieskov, - 14-Dec-2023.) $) - cu3addd $p |- ( ph -> ( ( ( A + B ) + C ) ^ 3 ) = - ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) - + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) - + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) - + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) - + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) - + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) - + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) ) $= - ( co wcel wceq addcld binom3 a1i oveq1d eqtrd oveq2d sqcld mulcld adddird - adddid caddc c3 cexp c2 cmul cc wa jca wi mpd syl2anc binom2d 3cn mulassd - 2cnd eqcomd ) ABCUAZHZDUQHUBZUCZHZBUSUTHUSBUDZUTHZCUEZHVDHUQHUSBCVBUTHZVD - HVDHCUSUTHUQHUQHZUSVCDVDHZVDHZUSVBVDHBCVDHZVDHZDVDHZUQHZUSVEDVDHZVDHZUQHZ - UQHZUSBDVBUTHZVDHZCVQVDHZUQHZVDHZDUSUTHZUQHZUQHZVPUSVRVDHUSVSVDHUQHZWBUQH - ZUQHAVAVPUSURVQVDHZVDHZWBUQHZUQHZWDAVAVFVHUSVBVIVDHZDVDHZVDHZUQHZVNUQHZUQ - HZWIUQHZWJAVAVFUSVGWLUQHZVDHZVNUQHZUQHZWIUQHZWQAVAVFUSWRVMUQHZVDHZUQHZWIU - QHZXBAVAVFUSVCWKUQHZDVDHZVMUQHZVDHZUQHZWIUQHZXFAVAVFUSXGVEUQHZDVDHZVDHZUQ - HZWIUQHZXLAVAVFUSURVBUTHZDVDHZVDHZUQHZWIUQHZXQAVAURUSUTHZXTUQHZWIUQHZYBAU - RUFZIZDYFIZUGZVAYEJZAYGYHABCEFKGUHYIYJUIAURDLMUJAYDYAWIUQAYCVFXTUQABYFICY - FIYCVFJEFBCLUKNNOAYAXPWIUQAXTXOVFUQAXSXNUSVDAXRXMDVDABCEFULNPPNOAXPXKWIUQ - AXOXJVFUQAXNXIUSVDAXGVEDAVCWKABEQZAVBVIAUOZABCEFRZRZKACFQZGSPPNOAXKXEWIUQ - AXJXDVFUQAXIXCUSVDAXHWRVMUQAVCWKDYKYNGSNPPNOAXEXAWIUQAXDWTVFUQAUSWRVMUSYF - IAUMMZAVGWLAVCDYKGRZAWKDYNGRZKAVEDYOGRTPNOAXAWPWIUQAWTWOVFUQAWSWNVNUQAUSV - GWLYPYQYRTNPNOAWPVPWIUQAVPWPAVOWOVFUQAVLWNVNUQAVKWMVHUQAVKUSWKVDHZDVDHWMA - VJYSDVDAUSVBVIYPYLYMUNNAUSWKDYPYNGUNOPNPUPNOAWIWCVPUQAWHWAWBUQAWGVTUSVDAB - CVQEFADGQZSPNPOAWCWFVPUQAWAWEWBUQAUSVRVSYPABVQEYTRACVQFYTRTNPO $. - $} - - ${ - cu3addi.1 $e |- A e. CC $. - cu3addi.2 $e |- B e. CC $. - cu3addi.3 $e |- C e. CC $. - $( Cube of sum of three numbers. (Contributed by Igor Ieskov, - 14-Dec-2023.) $) - cu3addi $p |- ( ( ( A + B ) + C ) ^ 3 ) = - ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) - + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) - + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) - + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) - + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) - + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) - + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) - $= - ( caddc co c3 cexp c2 cmul wceq wtru cc wcel a1i cu3addd mptru ) ABGZHCTH - IZJZHAUAUBHUAAKZUBHZBLZHUEHTHUAABUCUBHZUEHUEHBUAUBHTHTHUAUDCUEHUEHUAUCUEH - ABUEHUEHCUEHTHUAUFCUEHUEHTHTHUAACUCUBHZUEHUEHUABUGUEHUEHTHCUAUBHTHTHMNZAB - CAOZPUHDQBUIPUHEQCUIPUHFQRS $. - $} - - $( Cube of sum of three numbers. (Contributed by Igor Ieskov, - 14-Dec-2023.) $) - cu3add $p |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A + B ) + C ) ^ 3 ) - = ( ( ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) - + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) - + ( ( ( 3 x. ( ( A ^ 2 ) x. C ) ) - + ( ( ( 3 x. 2 ) x. ( A x. B ) ) x. C ) ) - + ( 3 x. ( ( B ^ 2 ) x. C ) ) ) ) + ( ( ( 3 x. ( A x. ( C ^ 2 ) ) ) - + ( 3 x. ( B x. ( C ^ 2 ) ) ) ) + ( C ^ 3 ) ) ) ) $= - ( cc wcel w3a simp1 simp2 simp3 cu3addd ) ADZEZBKEZCKEZFABCLMNGLMNHLMNIJ $. - -$( (End of Igor Ieskov's mathbox.) $) - $( End $[ set-mbox.mm $] $)