diff --git a/changes-set.txt b/changes-set.txt index ebe3cc3bc..e0b7d75aa 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 sseqtr4d sseqtrrd proposed syl6sseqr sseqtrrdi proposed syl6eqss eqsstrdi compare to eqsstri or eqsstrd proposed syl6eqssr eqsstrrdi compare to eqsstrri or eqsstrrd @@ -80,6 +79,7 @@ make a github issue.) DONE: Date Old New Notes +15-Jan-24 sseqtr4d sseqtrrd 14-Jan-24 sseqtr4i sseqtrri 5-Jan-24 --- --- sections "Groups" and "Simple groups" moved from RR's mathbox to main set.mm diff --git a/iset.mm b/iset.mm index afae85e99..6091c901b 100644 --- a/iset.mm +++ b/iset.mm @@ -27369,11 +27369,11 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $} ${ - sseqtr4d.1 $e |- ( ph -> A C_ B ) $. - sseqtr4d.2 $e |- ( ph -> C = B ) $. + sseqtrrd.1 $e |- ( ph -> A C_ B ) $. + sseqtrrd.2 $e |- ( ph -> C = B ) $. $( Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) $) - sseqtr4d $p |- ( ph -> A C_ C ) $= + sseqtrrd $p |- ( ph -> A C_ C ) $= ( eqcomd sseqtrd ) ABCDEADCFGH $. $} @@ -27492,7 +27492,7 @@ practical reasons (to avoid having to prove sethood of ` A ` in every use $( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $) sseqtrrid $p |- ( ph -> B C_ C ) $= - ( wss a1i sseqtr4d ) ACBDCBGAEHFI $. + ( wss a1i sseqtrrd ) ACBDCBGAEHFI $. $} ${ @@ -50745,7 +50745,7 @@ We use their notation ("onto" under the arrow). (Contributed by NM, that set, using the ` Fn ` abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.) $) fnfvima $p |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) ) $= - ( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtr4d + ( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtrrd cfv jca simp3 funfvima2 sylc ) CAEZBAFZDBGZHZCIZBCJZFZKUFDCSCBLGUGUHUJUDUEU HUFACMNUGBAUIUDUEUFOUDUEUIAPUFACQNRTUDUEUFUABDCUBUC $. @@ -57955,7 +57955,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( cfv wceq wcel cv cuni cres wral crecs cdm wfn tfrlemibfn fndm syl wss tfrlemibacc unissd recsfval syl6sseqr dmss eqsstrrd sselda tfrlem9 wfun wa tfrlem7 a1i adantr eleq2d biimpar funssfv syl3anc word eloni ordelss - sylan sseqtr4d fun2ssres fveq2d 3eqtr3d ralrimiva fveq2 eqeq12d cbvralv + sylan sseqtrrd fun2ssres fveq2d 3eqtr3d ralrimiva fveq2 eqeq12d cbvralv con0 reseq2 sylibr ) AEUAZHUBZRZWEWDUCZLRZSZEBUAZUDFUAZWERZWEWKUCZLRZSZ FWJUDAWIEWJAWDWJTZVAZWDLUEZRZWRWDUCZLRZWFWHWQWDWRUFZTWSXASAWJXBWDAWJWEU FZXBAWEWJUGXCWJSZABCDEGHIJKLMNOPQUHWJWEUIUJZAWEWRUKZXCXBUKAWEGUBWRAHGAB @@ -58338,7 +58338,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( cv cuni cfv cres wceq wral wcel wa crecs cdm wfn tfr1onlembfn syl fndm wss tfr1onlembacc unissd tfr1onlemssrecs sstrd eqsstrrd sselda dmss con0 wrex eqid tfrlem9 tfrfun eleq2d biimpar funssfv mp3an2ani - cab wfun word ordelon syl2anc eloni sylan adantr sseqtr4d fun2ssres + cab wfun word ordelon syl2anc eloni sylan adantr sseqtrrd fun2ssres ordelss fveq2d 3eqtr3d ralrimiva reseq2 eqeq12d cbvralv sylibr fveq2 ) AEUEZHUFZUGZWPWOUHZNUGZUIZEIUJFUEZWPUGZWPXAUHZNUGZUIZFIUJAW TEIAWOIUKZULZWONUMZUGZXHWOUHZNUGZWQWSXGWOXHUNZUKXIXKUIAIXLWOAIWPUNZ @@ -58729,7 +58729,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( vv vt ve cv cuni cfv cres wceq wral wcel wa crecs cdm tfrcllembfn wf fdm syl wss tfrcllembacc unissd tfrcllemssrecs sstrd dmss sselda eqsstrrd wfn con0 wrex cab eqid tfrlem9 wfun tfrfun biimpar funssfv - eleq2d mp3an2ani word ordelon syl2anc eloni ordelss adantr sseqtr4d + eleq2d mp3an2ani word ordelon syl2anc eloni ordelss adantr sseqtrrd sylan fun2ssres fveq2d 3eqtr3d fveq2 reseq2 eqeq12d cbvralv sylibr ralrimiva ) AEUIZHUJZUKZXAWTULZOUKZUMZEIUNFUIZXAUKZXAXFULZOUKZUMZFI UNAXEEIAWTIUOZUPZWTOUQZUKZXMWTULZOUKZXBXDXLWTXMURZUOXNXPUMAIXQWTAIX @@ -59173,7 +59173,7 @@ defined for all sets (being defined for all ordinals might be enough if ( A u. U_ x e. B ( F ` ( ( rec ( F , A ) |` B ) ` x ) ) ) ) $= ( vg vy cvv wfn wcel con0 cfv cv cdm ciun cun wceq wa wfun wss w3a tfri2d crdg cres cmpt df-irdg rdgruledefgg alrimiv 3impa eqidd 3ad2ant3 rdgifnon - dmeq onss fndm syl 3adant3 sseqtr4d ssdmres sylib sylan9eqr fveq2d adantl + dmeq onss fndm syl 3adant3 sseqtrrd ssdmres sylib sylan9eqr fveq2d adantl fveq1 iuneq12d uneq2d rdgfun resfunexg mpan simpr fvexg sylancl ralrimivw wral vex wi funfvex funfni ex ralimdv adantr iunexg syl2anc 3adant2 unexg mpd 3ad2ant2 fvmptd eqtrd ) DHIZBEJZCKJZUAZCDBUCZLZWNCUDZFHBAFMZNZAMZWQLZ @@ -69088,7 +69088,7 @@ elements or fails to hold (is equal to ` (/) ` ) for some element. cmpt cmap co csuc wral xnninf 1oex sucid df-2o eleqtrri 2on0 2onn nn0eln0 wf wne ax-mp mpbir nndcel ancoms ifcldcd eqid fmptd elexi elmap sylibr wn omex ssid iftrue sseq1d mpbiri iffalse wo peano2 syl2anc exmiddc mpjaodan - 0ss simpl adantr sseqtr4d wi wtr word nnord ordtr trsuc sylan ex con3dimp + 0ss simpl adantr sseqtrrd wi wtr word nnord ordtr trsuc sylan ex con3dimp 3sstr4d eleq1 ifbid fvmptg simpr ralrimiva fveq1 sseq12d ralbidv df-nninf elrab2 sylanbrc ) BEFZAEAGZBFZHIJZUAZKEUBUCZFZCGZUDZXGLZXJXGLZMZCEUEZXGUF FXCEKXGUNXIXCAEXFKXGXCXDEFZNZXEHIKHKFZXQHHUDKHUGUHUIUJZOIKFZXQXTKIUOZUKKE @@ -128997,7 +128997,7 @@ reduced fraction representation (no common factors, denominator sylib cn cuz wo cv elnn1uz2 csn phi1 0z hashsng ax-mp rabid2 elsni oveq1d syl6eq mprgbir fveq2i 3eqtr2i fveq2 oveq2 eqeq1d rabeqbidv fveq2d 3eqtr4a gcd1 fzo01 cfz eluz2nn phival fzossfz a1i sseqin2 fzo0ss1 syl6eqr rabeqdv - mpbi inrab2 3eqtr4g cmin phibndlem eluzelz fzoval sseqtr4d df-ss wi wa wn + mpbi inrab2 3eqtr4g cmin phibndlem eluzelz fzoval sseqtrrd df-ss wi wa wn wral cabs gcd0id eluzelre eluzge2nn0 nn0ge0d absidd eqtrd eluz2b3 simprbi eqnetrd adantr eleq2s neeq1d syl5ibrcom necon2bd 1z fzospliti sylancl ord simpr syld ralrimiva rabss sylibr 3eqtr3d jaoi sylbi ) BUACZBDEZBFUBGCZUC @@ -130339,7 +130339,7 @@ then the Limited Principle of Omniscience (LPO) implies excluded middle. syl dcbid cbvralv sylib simpl3 wss wf fof crn imassrn frn sstrid ad2antrr 3adantl1 simpl2 equequ2 cbvral2v ssralv ralimdv sylsyld syl5bi sylc simpr cfn cres wfun cdm fofun word ordom ordtr ax-mp trss mpsyl wceq fdmd fores - wtr sseqtr4d syl2anc vex resex foeq1 spcev foeq2 exbidv fidcenum sylanbrc + wtr sseqtrrd syl2anc vex resex foeq1 spcev foeq2 exbidv fidcenum sylanbrc rspcev inffinp1 simprl syl2an2r eqneltrrd ex reximdva rexlimddv ralrimiva foelrn simprr jca 3com23 3expia impbii ) CUALMZABNZOZBCPZACPZQCDRZUBZDUCZ QCUEMZUDZYMYQYTUUAYMYQYSERZYRUFZYRFRZUGZSUHZEQUIZFQPZTZDUCZABCDEFUJZUKYMU @@ -132888,7 +132888,7 @@ _Introduction to General Topology_ (1983), p. 114) and it is convenient $( A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) $) toponss $p |- ( ( J e. ( TopOn ` X ) /\ A e. J ) -> A C_ X ) $= - ( ctopon cfv wcel wa cuni wss elssuni adantl wceq toponuni adantr sseqtr4d + ( ctopon cfv wcel wa cuni wss elssuni adantl wceq toponuni adantr sseqtrrd ) BCDEFZABFZGABHZCQARIPABJKPCRLQCBMNO $. $( If ` K ` is a topology on the base set of topology ` J ` , then ` J ` is a @@ -133865,7 +133865,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ssntr $p |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) ) $= ( ctop wcel wss wa cpw cin cuni cnt cfv elin elpwg pm5.32i bitr2i elssuni - sylbi adantl wceq ntrval adantr sseqtr4d ) BFGADHIZCBGZCAHZIZICBAJZKZLZAB + sylbi adantl wceq ntrval adantr sseqtrrd ) BFGADHIZCBGZCAHZIZICBAJZKZLZAB MNNZUICULHZUFUICUKGZUNUOUGCUJGZIUICBUJOUGUPUHCABPQRCUKSTUAUFUMULUBUIABDEU CUDUE $. @@ -134776,7 +134776,7 @@ converges to zero (in the standard topology on the reals) with this ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) ) $= ( ctopon cfv wcel cv cima wss wa wrex wi wral wb ad2antlr ccnp co wf ccnv w3a iscnp wfun cdm ffun toponss adantlr wceq fdm funimass3 syl2anc anbi2d - sseqtr4d rexbidva imbi2d ralbidv pm5.32da 3ad2ant1 bitrd ) EGIJKZFHIJKZCG + sseqtrrd rexbidva imbi2d ralbidv pm5.32da 3ad2ant1 bitrd ) EGIJKZFHIJKZCG KZUEDCEFUAUBJKGHDUCZCDJBLZKZCALZKZDVJMVHNZOZAEPZQZBFRZOZVGVIVKVJDUDVHMNZO ZAEPZQZBFRZOZABCDEFGHUFVDVEVQWCSVFVDVGVPWBVDVGOZVOWABFWDVNVTVIWDVMVSAEWDV JEKZOZVLVRVKWFDUGZVJDUHZNVLVRSVGWGVDWEGHDUITWFVJGWHVDWEVJGNVGVJEGUJUKVGWH @@ -134811,7 +134811,7 @@ converges to zero (in the standard topology on the reals) with this ( vx vf vy vg vj vk cfv wcel ctop co cuni cv cmap cvv wceq ctopon w3a cdm ccnp cima wa wrex wi wral crab cmpt topontop 3ad2ant1 simp2 uniexg mptexg wss unieq oveq2d rexeq imbi2d ralbidv rabeqbidv mpteq12dv oveq1d mpteq2dv - raleq df-cnp ovmpog syl3anc dmeqd eqid dmmptss syl6eqss toponuni sseqtr4d + raleq df-cnp ovmpog syl3anc dmeqd eqid dmmptss syl6eqss toponuni sseqtrrd syl wrel mptrel releqd mpbiri simp3 relelfvdm syl2anc sseldd ) CEUALZMZDN MZBACDUDOZLMZUBZWIUCZEAWKWLCPZEWKWLFWMFQZGQZLHQZMZWNIQZMWOWRUEWPUQUFZICUG ZUHZHDUIZGDPZWMROZUJZUKZUCWMWKWIXFWKCNMZWHXFSMZWIXFTWGWHXGWJECULUMWGWHWJU @@ -134843,7 +134843,7 @@ converges to zero (in the standard topology on the reals) with this tgcn $p |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. B ( `' F " y ) e. J ) ) ) $= ( vx vz wcel cv wral wa wi ctop syl ccn co wf ccnv cima ctopon wb syl2anc - cfv iscn wss ctg ctb topontop eqeltrrd tgclb sylibr bastg sseqtr4d ssralv + cfv iscn wss ctg ctb topontop eqeltrrd tgclb sylibr bastg sseqtrrd ssralv cuni wceq wex eleq2d eltg3 bitrd ciun iunopn sylan9r imaeq2 imauni syl6eq eleq1d imbi2d syl5ibrcom expimpd exlimdv sylbid imp cbvralv syl6ib impbid ex ralrimdva anbi2d ) ADEFUAUBNZGHDUCZDUDZBOZUEZENZBFPZQZWGWKBCPZQAEGUFUI @@ -134862,7 +134862,7 @@ converges to zero (in the standard topology on the reals) with this ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. B ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) ) $= ( cfv wcel wss wa wi syl vz ccnp co wf cv cima wrex wral wb iscnp syl3anc - ctopon ctg ctb ctop topontop eqeltrrd tgclb sylibr sseqtr4d ssralv anim2d + ctopon ctg ctb ctop topontop eqeltrrd tgclb sylibr sseqtrrd ssralv anim2d bastg sylbid eleq2d biimpa tg2 r19.29 sstr expcom reximdv com12 rexlimivw imim2i imp32 ex com23 ralrimdva sylibrd impbid ) AFEGHUBUCOPZIJFUDZEFOZCU EZPZEBUEZPZFWFUFZWDQZRZBGUGZSZCDUHZRZAWAWBWLCHUHZRZWNAGIULOPZHJULOPZEIPZW @@ -134884,7 +134884,7 @@ converges to zero (in the standard topology on the reals) with this ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) ) $= ( vx ctopon cfv wcel wa cid cres ccn co ccnv cv cima wral wf syl6bbr wceq wss iscn wf1o f1oi f1of biantrur cnvresid imaeq1i elssuni adantl toponuni - ax-mp cuni ad2antlr sseqtr4d resiima syl5eq eleq1d ralbidva dfss3 bitrd + ax-mp cuni ad2antlr sseqtrrd resiima syl5eq eleq1d ralbidva dfss3 bitrd syl ) ACEFZGZBVBGZHZICJZABKLGZVFMZDNZOZAGZDBPZBATZVEVGCCVFQZVLHVLDVFABCCU AVNVLCCVFUBVNCUCCCVFUDUKUERVEVLVIAGZDBPVMVEVKVODBVEVIBGZHZVJVIAVQVJVFVIOZ VIVHVFVICUFUGVQVICTVRVISVQVIBULZCVPVIVSTVEVIBUHUIVDCVSSVCVPCBUJUMUNCVIUOV @@ -135270,7 +135270,7 @@ converges to zero (in the standard topology on the reals) with this ad2antrr jca simprl wrex cdm cnvimass fdm sseqtrid ssralv simp-4l simp-4r ctop topontop cnprcl2k syl3anc simpllr wfn ffn ad2antlr elpreima simplbda simprr syl2anc icnpimaex syl33anc wfun ffund toponss sylan fdmd funimass3 - wi sseqtr4d anbi2d rexbidva mpbid expr ralimdva syld impr an32s ad3antrrr + wi sseqtrrd anbi2d rexbidva mpbid expr ralimdva syld impr an32s ad3antrrr eltop2 adantr mpbir2and impbida ) CEIJKZDFIJKZLZBCDUCMKZEFBUDZBANZCDUAMJK ZAEOZLZXCXDLZXEXHXCXDXEBUBGNZPZCKZGDOZGBCDEFUEZUFXJXHXGACUGZOZXDXQXCXDXGA XPXFBCDXPXPUHUIQUJXJXGAEXPXAEXPUKXBXDECULUOUMUNUPXCXILZXDXEXNXCXEXHUQXRXM @@ -136060,7 +136060,7 @@ F C_ ( CC X. X ) ) $= cvv nfeq fveq2 eqeq12d rspc sylc eleq1d adantr ad2antrr simplrl icnpimaex simprl simplrr simprr jca ex opelxp reeanv 3imtr4g sylbid cin an4 biimpri syl33anc elin a1i simpl toponss syl2an ssinss1 adantl sselda elin1d ffund - wfun cdm fdmd sseqtr4d sseldd funfvima mpd elin2d funimass4 mpbird syldan + wfun cdm fdmd sseqtrrd sseldd funfvima mpd elin2d funimass4 mpbird syldan eqeltrd adantlr xpss12 sstr2 syl2im anim12d ctop topontop syl inopn 3expb syl5bi sylan eleq2 vex jctild expimpd imaeq2 sseq1d rspcev syl6 rexlimdvv anbi12d expd syld ralrimivva rgen2w sseq2 anbi2d rexbidv imbi12d ralrnmpo @@ -136825,7 +136825,7 @@ F C_ ( CC X. X ) ) $= A e. X ) ) -> ( R " { A } ) e. K ) $= ( vy ctop wcel wa cima cuni wss adantr eqid syl2anc wceq ad2antll ctopon co ctx csn cv cop cmpt ccnv crab nfv nfcv nfrab1 cxp txtop simprl eltopss - txuni sseqtr4d imass1 syl xpimasn sseqtrd sseld pm4.71rd wb elimasng elvd + txuni sseqtrrd imass1 syl xpimasn sseqtrd sseld pm4.71rd wb elimasng elvd cvv anbi2d bitrd rabid syl6bbr eqrd mptpreima syl6eqr ccn toptopon biimpi cfv ad2antlr ad2antrr simprr cnmptc cnmptid cnmpt1t cnima eqeltrd ) CHIZD HIZJZBCDUATZIZAEIZJZJZBAUBZKZGDLZAGUCZUDZUEZUFBKZDWMWOWRBIZGWPUGZWTWMGWOX @@ -136962,7 +136962,7 @@ F C_ ( CC X. X ) ) $= ( wcel wss cima cnt cfv ccnv ccn adantr wceq cnntri syl2anc syl imacnvcnv co chmeo wa cuni hmeocn crn imassrn wf1o wfo eqid hmeof1o f1ofo forn 3syl sseqtrid wf1 f1of1 f1imacnv sylancom fveq2d sseqtrd wfun wi f1ofun cntop2 - ctop ntrss3 sseqtr4d funimass1 mpd hmeocnvcn sylan fveq2i 3sstr3g eqssd ) + ctop ntrss3 sseqtrrd funimass1 mpd hmeocnvcn sylan fveq2i 3sstr3g eqssd ) BCDUATGZAEHZUBZBAIZDJKZKZBACJKZKZIZVQBLZVTIZWBHZVTWCHZVQWEWDVRIZWAKZWBVQB CDMTGZVRDUCZHZWEWIHVOWJVPBCDUDNZVQBUEZVRWKBAUFVQEWKBUGZEWKBUHWNWKOVOWOVPB CDEWKFWKUIZUJNZEWKBUKEWKBULUMZUNZVRBCDWKWPPQVQWHAWAVOVPEWKBUOZWHAOVQWOWTW @@ -138526,7 +138526,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric blssec $p |- ( ( D e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( P ( ball ` D ) S ) C_ [ P ] .~ ) $= ( cxmet cfv wcel cxr w3a cbl co cpnf cec wss wa cle wbr pnfge adantl ssbl - wi pnfxr 3expia mpanr2 mpd 3impa wceq xmetec 3adant3 sseqtr4d ) AEGHIZBEI + wi pnfxr 3expia mpanr2 mpd 3impa wceq xmetec 3adant3 sseqtrrd ) AEGHIZBEI ZDJIZKBDALHZMZBNUPMZBCOZUMUNUOUQURPZUMUNQZUOQDNRSZUTUOVBVADTUAVAUONJIZVBU TUCUDVAUOVCQVBUTABDNEUBUEUFUGUHUMUNUSURUIUOABCEFUJUKUL $. $} @@ -138986,7 +138986,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric $( The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.) $) blssopn $p |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ J ) $= - ( cxmet cfv wcel cbl crn ctg ctb wss blbas bastg syl mopnval sseqtr4d ) A + ( cxmet cfv wcel cbl crn ctg ctb wss blbas bastg syl mopnval sseqtrrd ) A CEFGZAHFIZSJFZBRSKGSTLACMSKNOABCDPQ $. $( The union of a collection of open sets of a metric space is open. @@ -139535,7 +139535,7 @@ S C_ ( P ( ball ` D ) T ) ) $= crn ctg cfv cbl wceq wrex wb eqid elrnmpog biimpi adantl xpeq1 eqeq2d cvv elv xpeq2 cbvrex2v sylib simplll simplrl simplrr cuni wral ctopon simpr cxmet mopntopon syl adantr simprl toponss syl2anc simprr xpss12 - xmetxp unirnbl sseqtr4d c1st ad2antrr xp1st mopni2 syl3anc c2nd xp2nd + xmetxp unirnbl sseqtrrd c1st ad2antrr xp1st mopni2 syl3anc c2nd xp2nd crp cpr cxr clt cinf wfn cpw wf blf ffnd ad4antr sselda rpxr ad2antrl xrmincl fnovrn eleq2 anbi12d xrminrpcl blcntr xmetxpbl cle wbr sseldd sseq1 xrmin1inf ssbl syl221anc sstrd xrmin2inf jca rspcedvd rexlimddv @@ -139635,7 +139635,7 @@ S C_ ( P ( ball ` D ) T ) ) $= A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) ) $= ( cfv wcel co crp wral wa wi cxmet w3a ccnp cbl cima wss wrex clt metcnp3 wf cv wbr wb wfun cdm ffun ad2antlr simpll1 simpll3 rpxr ad2antll syl3anc - cxr blssm wceq fdm sseqtr4d funimass4 syl2anc elbl imbi1d impexp ad2antrr + cxr blssm wceq fdm sseqtrrd funimass4 syl2anc elbl imbi1d impexp ad2antrr simpl2 simplrl rpxrd simpllr adantr ffvelrnd simplr elbl2 syl22anc imbi2d ffvelrnda pm5.74da syl5bb ralbidv2 anassrs rexbidva ralbidva pm5.32da bitrd ) DJUANOZEKUANOZFJOZUBZGFHIUCPNOJKGUJZGFBUKZDUDNPZUEFGNZAUKZEUDNPZU @@ -147835,7 +147835,7 @@ a singleton (where the latter can also be thought of as representing ( vj vk vf xnninf com cv wceq c1o cfv wcel c2o wss adantr adantl fveq2d c0 cuni cif cmpt cmap co csuc wral wf wa a1i nninff nnpredcl ffvelrnd wdc 1lt2o nndceq0 ifcldcd eqid fmptd cvv wb 2onn omex elmapg mp2an sylibr wtr - wn 1on ontrci peano2 syl df-2o syl6eleq trsucss mpsyl sseqtr4d word simpr + wn 1on ontrci peano2 syl df-2o syl6eleq trsucss mpsyl sseqtrrd word simpr iftrue nnord ordtr unisucg mpbid wne neqned nnsucpred syl2anc suceq fveq2 3syl sseq12d fveq1 ralbidv df-nninf elrab2 simprbi cbvralv sylib ad2antrr rspcdva eqsstrrd eqsstrd peano3 neneqd ad2antlr iffalsed 3sstr4d mpjaodan @@ -148070,7 +148070,7 @@ C_ if ( A. k e. suc J ( Q ` ( i e. _om |-> if ( i e. k , 1o , (/) ) ) ) = 1o , 1o , (/) ) ) $= ( c2o xnninf cmap co wcel com wa c1o c0 cif wceq csuc wral wss syl adantl - wel cfv peano2 nninfsellemcl el2oss1o sylan2 adantr iftrue sseqtr4d simpl + wel cfv peano2 nninfsellemcl el2oss1o sylan2 adantr iftrue sseqtrrd simpl wn csn con3i cun df-suc raleqi ralunb bitri sylnibr iffalsed 0ss syl6eqss cmpt wdc wo nninfsellemdc exmiddc mpjaodan ) AEFGHIZDJIZKZBJBCUALMNVCAUBL OZCDPZQZVLCVMPZQZLMNZVNLMNZRZVNUKZVKVNKVQLVRVKVQLRZVNVJVIVMJIZWADUCVIWBKV diff --git a/set.mm b/set.mm index 229665a5f..fb7be82f9 100644 --- a/set.mm +++ b/set.mm @@ -36620,11 +36620,11 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $} ${ - sseqtr4d.1 $e |- ( ph -> A C_ B ) $. - sseqtr4d.2 $e |- ( ph -> C = B ) $. + sseqtrrd.1 $e |- ( ph -> A C_ B ) $. + sseqtrrd.2 $e |- ( ph -> C = B ) $. $( Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.) $) - sseqtr4d $p |- ( ph -> A C_ C ) $= + sseqtrrd $p |- ( ph -> A C_ C ) $= ( eqcomd sseqtrd ) ABCDEADCFGH $. $} @@ -36743,7 +36743,7 @@ technically classes despite morally (and provably) being sets, like ` 1 ` $( Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) $) sseqtrrid $p |- ( ph -> B C_ C ) $= - ( wss a1i sseqtr4d ) ACBDCBGAEHFI $. + ( wss a1i sseqtrrd ) ACBDCBGAEHFI $. $} ${ @@ -64399,7 +64399,7 @@ from the cartesian product of two singletons onto a singleton (case where that set, using the ` Fn ` abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.) $) fnfvima $p |- ( ( F Fn A /\ S C_ A /\ X e. S ) -> ( F ` X ) e. ( F " S ) ) $= - ( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtr4d + ( wfn wss wcel w3a wfun cdm wa cima fnfun 3ad2ant1 simp2 wceq fndm sseqtrrd cfv jca simp3 funfvima2 sylc ) CAEZBAFZDBGZHZCIZBCJZFZKUFDCSCBLGUGUHUJUDUEU HUFACMNUGBAUIUDUEUFOUDUEUIAPUFACQNRTUDUEUFUABDCUBUC $. @@ -76904,7 +76904,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and cpred cab cvv wfun wfrfun funfvop mpan cwrecs df-wrecs eqtri eleq2i eluni cuni bitri sylib simprr vex wfrlem3a 3simpa simprlr elssuni syl6sseqr syl eqid predeq3 sseq1d simprrr adantl wbr simprll sylibr fvex breldmg mp3an2 - df-br syldan simprrl fndm eleqtrd rspcdva sseqtr4d mp3an2i resex syl6eqel + df-br syldan simprrl fndm eleqtrd rspcdva sseqtrrd mp3an2i resex syl6eqel fun2ssres expr syl5 exlimdv mpd exlimddv ) ECUAZOZEECPZUBZIQZOZXBJQZKQZUC XEARABLQZUHZXERLXEUDSXFXDPXDXGUEDPTLXEUDUFKUGJUIZOZSZCABEUHZUEZUJOZIWSXAC OZXJIUGZCUKZWSXNABCDFGHULZECUMUNXNXAXHUTZOXOCXRXACABDUOXRHKLABJDUPUQZURIX @@ -79300,7 +79300,7 @@ currently used conventions for such cases (see ~ cbvmpox , ~ ovmpox and ( vx vy con0 wcel coa co wi wa csuc wss oveq2 sseq2d imbi2d ancoms ex cvv wceq onelon adantll word eloni ordsucss syl ad2antlr sucelon cv ssid 2a1i sssucid sstr2 mpi oasuc syl5ibr ad2antrr a2d wlim wral ciun sylbir limsuc - sucssel biimpd sylan9r imp ssiun2s adantr vex mpanr1 adantlr sseqtr4d a1d + sucssel biimpd sylan9r imp ssiun2s adantr vex mpanr1 adantlr sseqtrrd a1d oalim exp31 syl5bi com4r imp31 sseq1d ovex ax-mp syl6bi 3syld an32s mpdan tfindsg ) CFGZBFGZABGZCAHIZCBHIZGZJWHWIKZWJWMWNWJKAFGZWMWIWJWOWHBAUAUBWNW OWJWMWNWOKZWJWMWPWJALZBMZCWQHIZWLMZWMWIWJWRJZWHWOWIBUCXABUDABUEUFUGWHWIWO @@ -79640,7 +79640,7 @@ equinumerous even if they are not equal (which sometimes occurs, e.g., sylan adantr onelon ex cv csuc noel pm2.21i a1i coa elsuci omcl simpl jca wo oaword1 sseld imim2d imp adantrl oaord1 eleq1d syl5ibrcom adantrr jaod syl5 wb omsuc sylibrd exp43 com12 adantld impd wlim wral id ad2ant2r ciun - wss limsuc ssiun2s syl adantll cvv omlim mpanr1 sseqtr4d anabss1 eleqtrrd + wss limsuc ssiun2s syl adantll cvv omlim mpanr1 sseqtrrd anabss1 eleqtrrd sseldd exp53 com13 imp4c a1dd tfinds3 com23 exp4a mpdd com34 com24 imp31 vex ) BFGZCFGZHCGZABGZCAIJZCBIJZGZKZXAXDXCXBXGXAXDXBXCXGXAXDAFGZXBXCXGKZK XAXDXIBAUAUBXAXDXIXBXJXAXDXIXBLZXCXGXAXKXCLZXDXGADUCZGZXECXMIJZGZKZAHGZXE @@ -79800,7 +79800,7 @@ A C_ ( B .o A ) ) $= eleqtrid sylib adantlr onelon onnbtwn imnan sylibr com12 adantl mpd simpl jca anim2i anassrs coa ordsucelsuc oa1suc eleq2d bitr4d wss ordgt0ge1 1on ad5ant24 oaword mp3an1 syldan bitrd biimpa omsuc sseld eleq1 biimprd syl9 - sseqtr4d sylbid com23 adantlrl sucelon omord syl6bir syl3an2b 3comr 3expb + sseqtrrd sylbid com23 adantlrl sucelon omord syl6bir syl3an2b 3comr 3expb w3a syl6d an32s imp mtod rexlimdva2 pm2.01da nrexdv sylanbrc dflim3 ) AFG ZBCGZBUAZHZHZIAGZHZABJKZLZYFIMZYFDUBZUCZMZDFUDZUENZYFUAYCYGYDYBXSBFGZYGBC UFZXSYNHZYFFGYGABUGYFUNOUHPYEYHNZYLNYMYEYQAIMZNZBIMZNZHZYBYDUUBXSYAYDUUBX @@ -80075,7 +80075,7 @@ A C_ ( B .o A ) ) $= wb csuc oe0 eleqtrrid comu oecl omordi om0 eleq1d ad2antlr sylibd syldanl 0lt1o oesuc sylibrd exp31 com12 com34 impd wlim wral ciun wrex 0ellim syl eqimss2 sseq2d rspcev syl2an ssiun adantrr vex oelim mpanlr1 anasss an12s - sseqtr4d word limelon mpan ancoms sylan eloni ordgt0ge1 mpbird ex tfinds3 + sseqtrrd word limelon mpan ancoms sylan eloni ordgt0ge1 mpbird ex tfinds3 3syl a1dd expd imp31 ) AEFZBEFZGAFZGABHIZFZWLWKWMWOJWLWKWMWOGACKZHIZFZGAG HIZFZGADKZHIZFZGAXAUAZHIZFZWOWKWMLZCDBWPGMWQWSGWPGAHNOWPXAMWQXBGWPXAAHNOW PXDMWQXEGWPXDAHNOWPBMWQWNGWPBAHNOWKWTWMWKGPWSULAUBZUCQXAEFZWKWMXCXFJXIWKX @@ -85063,7 +85063,7 @@ the first case of his notation (simple exponentiation) and subscript it wn ssun2 reldom brrelex2i adantl ssexg wex brdomi cen cres wf1o vex resex cv simprr difss f1ores sylancl f1oen3g wf ccnv df-f1 imadif simplbiim cfv wfun snex simprl unexg difexg syl f1f fimass ssdifd f1fn snid elun1 ax-mp - fnsnfv difeq2d sseqtr4d sylc ffvelrn mp1i difsnen syl3anc domentr syl2anc + fnsnfv difeq2d sseqtrrd sylc ffvelrn mp1i difsnen syl3anc domentr syl2anc ssdomg eqbrtrd endomtr uncom difeq1i difun2 eqtri difsn ad2antrr ad2antlr wfn syl5eq 3brtr3d expr exlimdv syl5 impancom mpd en2sn mp2an endom simpr cin c0 incom disjsn biimpri undom syl21anc impbida ) ACHUBZBDHUBZIZAJZCKZ @@ -88067,7 +88067,7 @@ of Infinity (see comments in ~ fin2inf ). (Contributed by NM, ( b e. ( ~P A i^i Fin ) |-> ( F " b ) ) : ( ~P A i^i Fin ) -1-1-onto-> ( ~P B i^i Fin ) ) $= ( va wf1o cpw cfn cin cv cima wcel wa cres wfo wss syl adantl wceq adantr - ccnv cmpt eqid simpr elin2d cdm f1ofun elinel1 elpwi f1odm sseqtr4d fores + ccnv cmpt eqid simpr elin2d cdm f1ofun elinel1 elpwi f1odm sseqtrrd fores wfun syl2an2r fofi syl2anc imassrn f1ofo forn sseqtrid elpwd elind dff1o3 crn simprbi f1ocnv dfdm4 syl5eqr wb anim12i foimacnv syl2an eqcomd imaeq2 eqeq2d syl5ibrcom wf1 f1of1 f1imacnv impbid sylan2 f1o2d ) ABCFZDEAGZHIZB @@ -92643,7 +92643,7 @@ of all inductive sets (which is the smallest inductive set, since a1i eqimss biantrurd eleq2 bitr3d fveq2 sseq2d imbi12d imbi2d anbi12d csuc sseq1 noel pm2.21i adantl wo fvex elsuc sssucid sstr mpan simprr ad2antrl pm2.27 cantnfvalf ffvelrni ad2antlr ad3antrrr suppssdm fssdm - coa sucidg sseldd oif onelon oecl oaword2 cantnfsuc ad4ant13 sseqtr4d + coa sucidg sseldd oif onelon oecl oaword2 cantnfsuc ad4ant13 sseqtrrd simpr omcl expcom adantrr syld expr fveq2d f1ocnvfv2 ad2antrr eqtr3d oveq2d oveq12d oaword1 eqsstrrd a1dd jaod syl5bi expimpd com23 finds2 vtoclga mpcom mpd cantnfval om0 0ss syl6eqss pm2.61ne ) ACEUCUDZEHUEZ @@ -92702,7 +92702,7 @@ of all inductive sets (which is the smallest inductive set, since wbr sstrd syl31anc mpd fveq2d simprl simpl eleq1 oveq2d eleq12d imbi12d suceq sselda sylan2 ffvelrnd onelon syl2anc oecl cantnfsuc omcl 3eltr4d oesuc ad2antrr ad2antrl epeli peano2 omord2 peano1 a1i oveq2i oa0 eqtrd - syl5eq ex wtr ordtr trsuc mpan imim1i omwordi sseqtr4d sucex mpbir wiso + syl5eq ex wtr ordtr trsuc mpan imim1i omwordi sseqtrrd sucex mpbir wiso wwe ovexd cantnfcl oiiso isorel syl12anc mpbii fvex ad2antlr cantnfvalf sylib oaord omsuc exp32 a2d syl5 expcom finds2 syl3c eqeltrd rexlimdvaa wo simpl2im elnn nn0suc mpjaod ) AKUDUEZKJUFZCEUGUHZUIZKUBUJZUKZUEZUBUL @@ -93172,7 +93172,7 @@ of all inductive sets (which is the smallest inductive set, since simpld ffvelrnd omcl com cdm csupp wf1o cep wiso cvv wwe ovexd cantnfcl wa oiiso isof1o syl f1ocnv f1of 3syl cantnflem1a simprd elnn cantnfvalf ffvelrni oaword1 cantnfsuc mpdan f1ocnvfv2 oveq2d fveq2d oveq12d oveq1d - eqtrd sseqtr4d wo wb onss sselda adantr onsseleq orcom syl6bb mpteq2dva + eqtrd sseqtrrd wo wb onss sselda adantr onsseleq orcom syl6bb mpteq2dva ifbid ffvelrnda wne ne0d on0eln0 mpbird ifcld fmpttd 0ex a1i fsuppmptif mpbir2and cdif wn eldifn adantl iffalsed suppss2 ifeq1da syl3anc sseldd fveq2 eleq1w ifbieq1d eqid fvex fvmpt ifeq2d syl6reqr mpteq2ia cantnfp1 @@ -94510,7 +94510,7 @@ of the previous layer (and the union of previous layers when the ( vx vy cr1 cfv wcel c0 wceq con0 wa cpw wss ax-mp sylib wb limsuc sylibr wrex syl cv csuc cvv wlim w3o cdm word wfun r1funlim simpri limord elfvdm ordsson sseldi onzsl noel fveq2 syl6eq eleq2d biimpcd pm2.21d simpl simpr - r10 mtoi fveq2d adantr eqeltrrd r1sucg eqtrd eleqtrd elpwi sspwb sseqtr4d + r10 mtoi fveq2d adantr eqeltrrd r1sucg eqtrd eleqtrd elpwi sspwb sseqtrrd ex rexlimdvw wtr r1tr ciun r1limg sylan eliun simprl ad2antlr simprr trss mpbid mpsyl ad2antrr wi ordtr1 syl2anc fvex elpw2 eleqtrrd rspcev adantld rexlimddv 3jaod mpd ) ABEFZGZBHIZBCUAZUBZIZCJSZBUCGZBUDZKZUEZALZXAMZXBBJG @@ -95621,7 +95621,7 @@ C_ suc suc suc ( rank ` ( A u. B ) ) $= wlim limuni2 cun cv 0ellim n0i unieq uni0 syl6eq con3i 3syl rankon onsuci elexi sucid ontri1 mp2an con2bii mpbi rankxpu sstr mpan2 reeanv wi simprl mto simpr rankuni unieqi eqtri wne df-ne xpex rankeq0 notbii bitr2i sylib - unixp syl fveq2d syl5reqr eqimss eqsstrrd adantrr limuni sseqtr4d cvv vex + unixp syl fveq2d syl5reqr eqimss eqsstrrd adantrr limuni sseqtrrd cvv vex word onordi orduni ax-mp ordelsuc sylibr limsuc mpbid eqeltrd ordsucelsuc onsucuni2 mpan ad2antll eleqtrd onsucssi ex a1d rexlimdvv syl5bir mtoi wo ianor un00 animorl sylbir xpeq0 unex w3o ordzsl 3ori sylan orim12d syl5bi @@ -98140,7 +98140,7 @@ x finSupp (/) } |-> ( ( _I |` _om ) o. ( y o. `' ( Y o. `' X ) ) ) ) $. acndom2 $p |- ( X ~<_ Y -> ( Y e. AC_ A -> X e. AC_ A ) ) $= ( vf vx vh vg vk cv wex wcel wa cfv wral c0 wf wss wne syl cvv wbr wf1 wi cdom wacn brdomi cpw csn cdif cmap co cima simplr crn imassrn simplll f1f - frn 3syl sstrid cdm cin wceq elmapi adantl ffvelrnda elpwid f1dm sseqtr4d + frn 3syl sstrid cdm cin wceq elmapi adantl ffvelrnda elpwid f1dm sseqtrrd eldifad sseqin2 sylib eldifsni eqnetrd imadisj necon3bii sylibr ralrimiva jca acni2 syl2anc ccnv acnrcl ad3antlr wf1o simp-4l f1f1orn simprr sseldi f1ocnvfv2 eqeltrd wb f1ocnv f1of ffvelrnd ad2ant2r f1elima mpbid ralimdva @@ -98215,7 +98215,7 @@ x finSupp (/) } |-> ( ( _I |` _om ) o. ( y o. `' ( Y o. `' X ) ) ) ) $. fodomfi2 $p |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> B ~<_ A ) $= ( wcel cfn wfo w3a wceq cdom wbr wss 3ad2ant3 syl cdm adantl syl2anc sylc vx cv cima cpw cin wrex wfn fofn forn eqimss2 simp2 fipreima syl3anc ccrd - crn cres elinel2 finnum wfun simpl3 fofun elinel1 elpwid fof fdm sseqtr4d + crn cres elinel2 finnum wfun simpl3 fofun elinel1 elpwid fof fdm sseqtrrd wa 3syl fores fodomnum simpl1 ssdomg domtr breq1 syl5ibcom rexlimdva mpd wf ) ADEZBFEZABCGZHZCSTZUAZBIZSAUBZFUCZUDZBAJKZVTCAUEZBCUMZLZVRWFVSVQWHVR ABCUFMVSVQWJVRVSWIBIWJABCUGBWIUHNMVQVRVSUIBACSUJUKVTWCWGSWEVTWAWEEZVEZWBA @@ -101158,7 +101158,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality wlim ccrd elex csuc limsuc biimpd sseq1 rexbidv sucssel elv reximi eluni2 rspcv sylibr syl6com syl9 ralrimdv dfss3 syl6ibr adantr limuni sseq2d imp uniss syl5ibr jctird eqss imdistanda anim2d eximdv ss2abdv intss syl con0 - adantl limelon cfval sseqtr4d cfub eqimss anim2i eximi ss2abi ax-mp sstri + adantl limelon cfval sseqtrrd cfub eqimss anim2i eximi ss2abi ax-mp sstri jctil sylan ) CDHCIHZCUCZCUAJZAKBKZUDJLZWMCMZCWMNZLZOZOZBPZAQZRZLZCDUEWJW KOZWLXBMZXBWLMZOXCXDXFXEXDXBWNWOEKZFKZMZFWMSZECTZOZOZBPZAQZRZWLWKXBXPMZWJ WKXOXAMXQWKXNWTAWKXMWSBWKXLWRWNWKWOXKWQWKWOOZXKCWPMZWPCMZOWQXRXKXSXTWKXKX @@ -101400,7 +101400,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality cdm wofi syl2an2r wefr ssidd unieq syl6eq eqeq1 syl5ib nlim0 limeq mtbiri uni0 syl6 necon2ad cab fvex sseq2 bibi12d mpbiri rexlimivw wo fveq2 ordom impcom 3adant2 adantr syl22anc cardon ordtri1 mp2b syl3an2b 3expb sylan2b - fri ralrimiva ssiin cflim3 sseqtr4d dfiin2 cardlim ss2abi eleq1 syl6eqelr + fri ralrimiva ssiin cflim3 sseqtrrd dfiin2 cardlim ss2abi eleq1 syl6eqelr cint abssi intex onint sylancr sseldi eqeltrd elab csuc w3o ordzsl df-3or mpbid orcom df-or 3bitri cf0 1n0 df1o2 unieqi unisn eqtri neeqtrri limuni csn necon3ai ax-mp cfsuc sylan9eqr rexlimiva jaoi con4d fdmi eleq2i ndmfv @@ -103287,7 +103287,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality isf34lem5 $p |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) ) $= ( wcel cpw wss c0 wne wa cima cfv cint wceq adantr sylib ccnv crpss eqtrd - cuni crn imassrn wf isf34lem2 frnd sstrid cdm cin simprl sseqtr4d sseqin2 + cuni crn imassrn wf isf34lem2 frnd sstrid cdm cin simprl sseqtrrd sseqin2 fdmd simprr eqnetrd imadisj necon3bii sylibr jca isf34lem4 syldan adantrr isf34lem3 inteqd fveq2d compsscnv fveq1i wf1o compssiso isof1o sspwuni wb wiso syl elpw2g mpbird f1ocnvfv1 syl2an2r syl5eqr eqtr3d ) BDGZEBHZIZEJKZ @@ -103305,7 +103305,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality cpw csuc w3a cima cint cuni wfn isf34lem2 adantr ffnd imassrn frnd sstrid ccom simp1 fco sylan cdif sscon simpr peano2 syl2an simpll ffvelrn elpwid wa fvco3 isf34lem1 adantll sseq12d syl5ibr ralimdva 3impia fin33i syl3anc - rnco2 inteqi 3eltr3g fnfvima simpl cdm cin incom frn adantl fdmd sseqtr4d + rnco2 inteqi 3eltr3g fnfvima simpl cdm cin incom frn adantl fdmd sseqtrrd wb df-ss sylib syl5eq peano1 ne0i eqnetrd dm0rn0 necon3bii imadisj sylibr fdm mp1i isf34lem5 syl12anc isf34lem3 unieqd eleq12d mpbid ) CGHZICUBZEJZ BUAZEKZXKUCZEKZLZBIMZUDZDENZUEZUFZDKZDXSUEZHZXRUGZXRHZXQDXIUHXSXILZXTXSHY @@ -103332,7 +103332,7 @@ _Cardinal Arithmetic_ (1994), p. xxx (Roman numeral 30). The cofinality fveq1 ralbidv rneq rnco2 syl6eq unieqd eleq12d rspccv syl5 cdif ffvelrnda imbi12d sscon elpwid isf34lem1 syl2an2r peano2 ffvelrn syl2an fvco3 sylan syl5ibr sylibrd ralimdva wfn ffnd imassrn frnd fnfvima cdm cin incom fdmd - sstrid sseqtr4d df-ss syl5eq peano1 ne0i eqnetrd dm0rn0 necon3bii imadisj + sstrid sseqtrrd df-ss syl5eq peano1 ne0i eqnetrd dm0rn0 necon3bii imadisj sylib mp1i isf34lem4 syl12anc inteqd eqtrd sylibd imim12d sylcom ralrimiv isf34lem3 isfin3-3 impbid2 ) CFIZCUAIZBUEZDUEZJZYGUBZYHJZKZBLMZYHNZUCZYNI ZOZDCUFZLUDUGZMZYFYQDYSYHYSIYFLYRYHPZYQYHYRLUHYFUUAYMYPABCEYHGUIUJUKULYTY @@ -104584,7 +104584,7 @@ that every (nonempty) pruned tree has a branch. This axiom is redundant cdm crn wss crab ffvelrn eldifsni sylib syl ralrimiva rabid2 sylibr copab n0 dmeqi cab 19.42v abbii dmopab df-rab eqtri syl6reqr neeq1d biimparc wi 3eqtr4i eldifi elelpwi expcom expimpd exlimdv alrimiv rneqi rnopab sseq1i - wal 3syl abss bitri sseqtr4d adantl cun cuni cxp fvrn0 ax-mp sseli anim2i + wal 3syl abss bitri sseqtrrd adantl cun cuni cxp fvrn0 ax-mp sseli anim2i elssuni ssopab2i df-xp 3sstr4i pwex difexi ssex p0ex unexg sylancl uniexg frn xpexg sylancr ssexg vex eldm exbii bitr2i dmeq syl5bb sseq12d anbi12d rneq ralbidv exbidv imbi12d ax-dc vtoclg mp2and simpr fveq2 suceq breq12d @@ -105995,7 +105995,7 @@ proved by Ernst Zermelo (the "Z" in ZFC) in 1904. (Contributed by Mario ( va con0 wcel wss cfv wi wceq vy wa cv sseq2 fveq2 sseq2d imbi12d imbi2d wral r19.21v wo wb onsseleq ad4ant23 cuni c0 cima cif csn cun wfn cvv cdm crn cmpt tfr1 simplr onss syl simprr fnfvima mp3an2i elssuni iffalse 3syl - wn n0i sseqtr4d adantr simplrl csuc vuniex sucid word orduniorsuc orcanai + wn n0i sseqtrrd adantr simplrl csuc vuniex sucid word orduniorsuc orcanai eloni eleqtrrid mpd ssun1 syl6ss ifbothda ttukeylem3 ad4ant13 expr eqimss rspcdva a1i jaod sylbid ex expcom a2d syl5bi tfis3 expdcom 3imp2 ) AFOPZG OPZFGQZFIRZGIRZQZXIAXHXJXMSZAXHUBZFUAUCZQZXKXPIRZQZSZSZXOFNUCZQZXKYBIRZQZ @@ -110028,7 +110028,7 @@ prove that every set is contained in a weak universe in ZF (see fneq1i mpbir fnunirn ax-mp bitri csuc elssuni ad2antll ssun2 ssun1 syl6ss 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 + id uneq2d iuneq12d frsucmpt2 sylancl sseqtrrd fvssunirn rexlimdvaa syl5bi 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 @@ -136701,7 +136701,7 @@ subset of a (possibly extended) finite sequence of integers. (Contributed -> ( S u. { I } ) C_ ( M ... if ( I <_ N , N , I ) ) ) $= ( cfz co wss cz wcel cuz cfv w3a cle wbr cif 3ad2ant3 wceq cxr zre rexrd csn simp1 eluzel2 simp2 eluzelz ssfzunsnext syl13anc eluz2 3ad2ant2 xrmineq - cun 3ad2ant1 simp3 syl3anc eqcomd sylbi oveq1d sseqtr4d ) ACDEFGZDHIZBCJKIZ + cun 3ad2ant1 simp3 syl3anc eqcomd sylbi oveq1d sseqtrrd ) ACDEFGZDHIZBCJKIZ LZABUAUKZBCMNBCOZBDMNDBOZEFZCVEEFVBUSCHIZUTBHIZVCVFGUSUTVAUBVAUSVGUTCBUCPUS UTVAUDVAUSVHUTCBUEPABCDUFUGVBCVDVEEVAUSCVDQZUTVAVGVHCBMNZLZVICBUHVKVDCVKBRI ZCRIZVJVDCQVHVGVLVJVHBBSTUIVGVHVMVJVGCCSTULVGVHVJUMBCUJUNUOUPPUQUR $. @@ -149770,7 +149770,7 @@ computer programs (as last() or lastChar()), the terminology used for 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 + ccatval1 simpl3 cuz uzidd uzaddcl fzoss2 sseqtrrd 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 @@ -150420,7 +150420,7 @@ computer programs (as last() or lastChar()), the terminology used for ( x e. ( 0 ..^ ( L - F ) ) |-> ( S ` ( x + F ) ) ) ) $= ( wcel cc0 cfz co cfv cfzo wss c0 wceq elfzelz 3ad2ant2 3ad2ant3 cuz syl cz cword chash w3a cop csubstr cdm cmin cv caddc cmpt cif swrdval syl3anc - simp1 elfzuz fzoss1 elfzuz3 fzoss2 sstrd wrddm 3ad2ant1 sseqtr4d iftrued + simp1 elfzuz fzoss1 elfzuz3 fzoss2 sstrd wrddm 3ad2ant1 sseqtrrd iftrued eqtrd ) CBUAZFZDGEHIFZEGCUBJZHIFZUCZCDEUDUEIZDEKIZCUFZLZAGEDUGIKIAUHDUIIC JUJZMUKZVOVJVFDTFZETFZVKVPNVFVGVIUNVGVFVQVIDGEOPVIVFVRVGEGVHOQACDEVEULUMV JVNVOMVJVLGVHKIZVMVJVLGEKIZVSVJDGRJFZVLVTLVGVFWAVIDGEUOPDGEUPSVJVHERJFZVT @@ -152478,7 +152478,7 @@ Splicing words (substring replacement) 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 + wss eleq2s oveq1d lencl 3eqtrd sseqtrrd nn0zd fzoaddel eqeltrd ccatval1 cz sseldd syl3anc ccatval3 ) AFGMNZDFECUAUBNZOGDFUFNZUCOZMNZWMCUDNZDEDU COZUGUENZUDNZOZWOWPOZGCOZAWKWOWLWSADBUHZPZFQEUINZPZEQWQUINZPZCXCPZWLWSR HIJKCDEFXCXEXGXCUJUKAWKGFMNZWOAFGAFAXFFSPZIFEULTZUMZAGAGQCUCOZUNNPZGSPL @@ -152623,7 +152623,7 @@ Splicing words (substring replacement) 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 + fzoss1 eleq2s sseqtrrd ccatval2 jaodan syldan eqfnfvd ) BAUBZDZCYADZUJZUA ECFGZBFGZHIZJIZBCUCIZUDGZCUDGZBUDGZUCIZYDYJEYJFGZJIZUEZYJYHUEYDYIYADZYJYA DYPABCUFZAYIUGAYJUHUKYDYOYHYJYDYNYGEJYDYNYIFGZYFYEHIZYGYDYQYNYSKYRAYIUIRA ABCULZYBYFLDZYELDZYTYGKYCYBYFABUMZUNZYCYEACUMZUNZYFYEUOUPZUQMURUSYDYMEYMF @@ -153806,7 +153806,7 @@ the symbol at any position is repeated at multiples of L (modulo the ex jca caddc 3ad2antl3 wb eqcoms eleq1d mpbird modfzo0difsn eqcomd eqeq2d modsumfzodifsn rexbidv fveq2 3ad2ant3 simpl1 elfzoelz eleq2d sseli syl6bi cz fzo0ss1 cshwidxmod syl3anc 3adant3 eqtrd eqeq1d rexxfrd2 abbidv anim2i - imp 3adant2 cshwfn syl6eqss fndm sseqtr4d mpdan eqtr4d ) BAUAHZDBUBIZJZCK + imp 3adant2 cshwfn syl6eqss fndm sseqtrrd mpdan eqtr4d ) BAUAHZDBUBIZJZCK DLMZHZUCZBXICUDZUEZUFZEUGZBIZFUGZJZEXMNZFUHZBCUIMZUPDLMZUFZXKBUJZXMBUKZUL ZTXNXTJXKYDYFXFXHYDXJXFBKXGLMZUMYDABUNYGBUOOUQXFXHXJYFXFYEYGJZXHXJYFURZUR ABUSYHXHYIYHXHTZYFXJYJYGXLUEZYGXMYEYJYGXLUTXHXMYKJYHXHXIYGXLDXGKLPVAVBYHX @@ -153834,7 +153834,7 @@ the symbol at any position is repeated at multiples of L (modulo the zcnd npcan1 eqcomd rspcedvd fveq2 cmo 1cnd add32r syl3anc fvoveq1d simpl1 cc peano2zd fzossrbm1 sseld oveq2 3ad2ant2 mpbid cshwidxmod fzo0ss1 sylan eleq2d sseldi 3eqtr4rd 3adant3 eqeq1d rexxfrd2 abbidv wfun cdm wfn anim2i - eqtrd 3adant2 cshwfn fnfun sseqtrid sseqtr4d mpdan dfimafn eqcoms 3eqtr4d + eqtrd 3adant2 cshwfn fnfun sseqtrid sseqtrrd mpdan dfimafn eqcoms 3eqtr4d fndm jca ) BAUBGZDBUCHZIZCJDKLZGZUDZBXLCUEUFUGBCUHLZMDKLZUGZBCMNLZUHLZJDM UILZKLZUGZABCDUJXNUAUKZXOHZEUKZIZUAXPUNZEULZFUKZXSHZYEIZFYAUNZEULZXQYBXNY GYLEXNYFYKUAFYIMNLZXPYAXNYIYAGZYNXPGZXMXIYOYPUOZXKXMDUPGZYQCJDUMZYRYOYPYI @@ -156321,7 +156321,7 @@ the symbol at any position is repeated at multiples of L (modulo the rtrclreclem1 $p |- ( ph -> ( _I |` U. U. R ) C_ ( t*rec ` R ) ) $= ( vr vn cuni crtrcl cfv wss wi cvv cn0 cv crelexp co ciun cc0 wcel wceq cres cmpt wrex 0nn0 ssid relexp0d sseqtrrid oveq2 sseq2d rspcev sylancr - cid ssiun syl nn0ex ovex iunex oveq1 iuneq2d fvmptg sylancl sseqtr4d wb + cid ssiun syl nn0ex ovex iunex oveq1 iuneq2d fvmptg sylancl sseqtrrd wb eqid df-rtrclrec fveq1 imbi2d ax-mp mpbir ) AULBGGUAZBHIZJZKZAVJBELFMEN ZFNZOPZQZUBZIZJZKZAVJFMBVOOPZQZVSAVJWBJZFMUCZVJWCJARMSVJBROPZJZWEUDAVJV JWFVJUEABCDUFUGWDWGFRMVORTWBWFVJVORBOUHUIUJUKFMWBVJUMUNABLSWCLSVSWCTDFM @@ -156338,7 +156338,7 @@ the symbol at any position is repeated at multiples of L (modulo the $( The reflexive, transitive closure is indeed a closure. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) $) rtrclreclem2 $p |- ( ph -> R C_ ( t*rec ` R ) ) $= - ( vr vn crtrcl cfv wss wi cvv cn0 cv crelexp co ciun wcel sseqtr4d wceq + ( vr vn crtrcl cfv wss wi cvv cn0 cv crelexp co ciun wcel sseqtrrd wceq c1 sseq2d cmpt wrex ssidd relexp1d oveq2 rspcev sylancr ssiun syl eqidd 1nn0 oveq1 iuneq2d adantl nn0ex ovex iunex a1i fvmptd df-rtrclrec fveq1 wb imbi2d ax-mp mpbir ) ABBFGZHZIZABBDJEKDLZELZMNZOZUAZGZHZIZABEKBVJMNZ @@ -164831,7 +164831,7 @@ seq m ( + , ( n e. ZZ |-> if ( n e. A , [_ n / k ]_ B , 0 ) ) ) ~~> x ) \/ ( cuz cfv cc cc0 wcel co wceq adantl adantr cz vn wss caddc addid2 0cnd wa cv wf cif iftrue eqeltrd ex wn iffalse 0cn syl6eqel pm2.61d1 eluzelz fmptd syl ffvelrnd c1 cmin cfz cdif elfzelz simplr zcnd ad2antrr ax-1cn - npcan sylancl fveq2d sseqtr4d fznuz ssneldd eldifd eldifi eldifn fvmpt2 + npcan sylancl fveq2d sseqtrrd fznuz ssneldd eldifd eldifi eldifn fvmpt2 fveqeq2 syl2anc eqtrd vtoclga seqid ) ABGKLZUBZUFZUAUCMEFGNUAUGZMONWIUC PWIQWHWIUDRWHUEAGFKLOZWGJSWHTMGEATMEUHWGADTDUGZBOZCNUIZMEAWMMOZWKTOZAWL WNAWLWNAWLUFWMCMWLWMCQAWLCNUJRIUKULWLUMZWMNMWLCNUNZUOUPUQSHUSSAGTOZWGAW @@ -170154,7 +170154,7 @@ seq m ( x. , ( k e. ZZ |-> if ( k e. A , B , 1 ) ) ) ~~> x ) \/ ( cuz cfv wa cc c1 wcel co wceq adantl cz vn cmul cv mulid2 1cnd adantr wss iftrue adantlr eqeltrd ex wn iffalse ax-1cn syl6eqel pm2.61d1 fmptd cif uzssz sseldi ffvelrnd cmin cfz cdif caddc simplr zcnd npcand fveq2d - elfzelz sseqtr4d fznuz ssneldd eldifd fveqeq2 eldifi eldifn syl syl2anc + elfzelz sseqtrrd fznuz ssneldd eldifd fveqeq2 eldifi eldifn syl syl2anc fvmpt2 eqtrd vtoclga seqid ) ABGKLZUGZMZUAUBNEFGOUAUCZNPOWGUBQWGRWFWGUD SWFUEAGFKLZPWEJUFAGELNPWEATNGEADTDUCZBPZCOURZNEAWITPZMZWJWKNPZWMWJWNWMW JMWKCNWJWKCRWMWJCOUHSAWJCNPWLIUIUJUKWJULZWKONWJCOUMZUNUOUPHUQAWHTGFUSJU @@ -184088,7 +184088,7 @@ reduced fraction representation (no common factors, denominator sylib cn cuz wo cv elnn1uz2 csn phi1 0z hashsng ax-mp rabid2 elsni oveq1d syl6eq mprgbir fveq2i 3eqtr2i fveq2 oveq2 eqeq1d rabeqbidv fveq2d 3eqtr4a gcd1 fzo01 cfz eluz2nn phival fzossfz a1i sseqin2 fzo0ss1 syl6eqr rabeqdv - mpbi inrab2 3eqtr4g cmin phibndlem eluzelz fzoval sseqtr4d df-ss wi wa wn + mpbi inrab2 3eqtr4g cmin phibndlem eluzelz fzoval sseqtrrd df-ss wi wa wn wral cabs gcd0id eluzelre eluzge2nn0 nn0ge0d absidd eqtrd eluz2b3 simprbi eqnetrd adantr eleq2s neeq1d syl5ibrcom necon2bd 1z fzospliti sylancl ord simpr syld ralrimiva rabss sylibr 3eqtr3d jaoi sylbi ) BUACZBDEZBFUBGCZUC @@ -189163,7 +189163,7 @@ with complex numbers (gaussian integers) instead, so that we only have syl r19.21bi cnvimass fssdm adantr sstrd cmul cc0 nnm1nn0 nn0uz ffvelrnda nncnd mul02d addid1d eqtr2d oveq1 rspceeqv syl2anc vdwapval mpbird sseldd cmin wb ralrimiva rspcdva elfzle2 letrd nnzd elfz5 eqidd wfn ffn fniniseg - cz wf mpbir2and snssd oveq12d fveq2d sneqd imaeq2d sseq12d sseqtr4d unssd + cz wf mpbir2and snssd oveq12d fveq2d sneqd imaeq2d sseq12d sseqtrrd unssd 3syl eqsstrd sseq1d oveq2 rspc2ev fvex sneq 2rexbidv spcev ovex peano2nn0 sseq2d vdwmc ) AHUEUFUGZFUHUIUAUJZUBUJZYTUKULZUGZFUMZUCUJZUNZUOZUPZUBUQUS UAUQUSZUCURZAUUDUUEBFULZUNZUOZUPZUBUQUSUAUQUSZUUKABUQUTZGCULZUQUTZBUURUUC @@ -194023,7 +194023,7 @@ could not be used in an extensible structure (slots must be positive basprssdmsets $p |- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) ) $= ( cnx cbs cdm wcel wo elun sylibr syl cvv cfv cpr csn cun csts orcd snidg - cop co olcd prssd wceq cstr wbr structex setsdm syl2anc sseqtr4d ) ALMUAZ + cop co olcd prssd wceq cstr wbr structex setsdm syl2anc sseqtrrd ) ALMUAZ EUBBNZEUCZUDZBEDUHUEUINZAUSEVBAUSUTOZUSVAOZPUSVBOAVDVEKUFUSUTVAQRAEUTOZEV AOZPEVBOAVGVFAECOVGIECUGSUJEUTVAQRUKABTOZDFOVCVBULABGUMUNVHHBGUOSJDBETFUP UQUR $. @@ -196922,7 +196922,7 @@ topology is based on the order and not the extended metric (which would ( vx vw vz wceq wral wcel syl vy wfun cdm cxp wfn cv wbr wmo cfv cop co wrel csn ciun opex relsnop rgenw reliun mpbir releqd mpbiri crn wss cvv fvex wa wf wfo fof ffvelrn anim12dan sylan opelxpi sylancl snssd iunssd - anassrs eqsstrd dmss c0 wne vn0 dmxp ax-mp syl6sseq sqxpeqd sseqtr4d wi + anassrs eqsstrd dmss c0 wne vn0 dmxp ax-mp syl6sseq sqxpeqd sseqtrrd wi forn wal wrex wb eleq2d adantr df-br rexbii bitr2i 3bitr4g w3a elsn vex eliun opth syl5bi eqeq2 biimprd syl6 3expa rexlimdvva sylbid ralrimivva impd alrimiv mo2icl fofn opeq2 breq1d mobidv ralrn ralbidv mpbird opeq1 @@ -196962,7 +196962,7 @@ topology is based on the order and not the extended metric (which would ( ( F ` X ) .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) ) $= ( cfv co cop wceq wss wcel w3a df-ov wfun cxp wfn imasaddfnlem 3ad2ant1 fnfun syl csn cv ciun fveq2 fvoveq1 opeq12d sneqd ssiun2s 3ad2ant2 wral - opeq1d opeq2d oveq2 fveq2d ralrimivw ss2iun 3ad2ant3 sseqtr4d opex snss + opeq1d opeq2d oveq2 fveq2d ralrimivw ss2iun 3ad2ant3 sseqtrrd opex snss sstrd sylibr funopfv sylc syl5eq ) AGFUAZHFUAZUBZGEPZHEPZCQVSVTRZCPZGHD QEPZVSVTCUCVRCUDZWAWCRZCUAZWBWCSAVPWDVQACBBUEZUFWDABCDEFIJKLMNOUGWGCUIU JUHVRWEUKZCTWFVRWHJFIFJULZEPZIULZEPZRZWIWKDQZEPZRZUKZUMZUMZCVRWHJFWJVTR @@ -197055,7 +197055,7 @@ topology is based on the order and not the extended metric (which would csn eqid fnmpoi fnrel ax-mp rgenw reliun mpbir imasvsca releqd mpbiri crn fvex wss cvv wcel wa dffn2 mpbi fssxp wfo fof ffvelrnda snssd xpss2 xpss1 syl 3syl sstrid ralrimiva iunss sylibr eqsstrd dmss wne vn0 dmxp syl6sseq - wf c0 forn xpeq2d sseqtr4d cop wi wal df-br wb eleq2d adantr eliun df-3an + wf c0 forn xpeq2d sseqtrrd cop wi wal df-br wb eleq2d adantr eliun df-3an wrex w3a mpofun funopfv df-ov opex vex opeldm dmmpo syl6eleq opelxp sylib fvoveq1 eqidd cbvmpov ovmpo syl5eqr syl5bi ralrimivva opeq2 ralrn ralbidv weq mobidv mpbird ralxp sylanbrc eqtr3d adantl elsni simpl2im ex sylan2br @@ -197097,7 +197097,7 @@ topology is based on the order and not the extended metric (which would ( X .xb ( F ` Y ) ) = ( F ` ( X .x. Y ) ) ) $= ( vx wcel w3a cfv co csn cv cmpo cop wfun wss cdm wceq cxp wfn imasvscafn fnfun syl 3ad2ant1 ciun eqidd fveq2 sneqd oveq2 fveq2d mpoeq123dv ssiun2s - 3ad2ant3 imasvsca sseqtr4d simp2 fvex snid opelxpi sylancl eqid syl6eleqr + 3ad2ant3 imasvsca sseqtrrd simp2 fvex snid opelxpi sylancl eqid syl6eleqr dmmpo funssfv syl3anc df-ov 3eqtr4g fvoveq1 ovmpo eqtrd ) AKIUGZLJUGZUHZK LGUIZDUJZKWNOUFIWNUKZOULZLEUJZGUIZUMZUJZKLEUJZGUIZWMKWNUNZDUIZXDWTUIZWOXA WMDUOZWTDUPXDWTUQZUGXEXFURAWKXGWLADIBUSZUTXGABCDEFGHIJMNOPQRSTUAUBUCUDUEV @@ -198687,7 +198687,7 @@ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) -> csn wral wss animorrl mreexexlem3d wne n0 biimpi adantl wn mreexexlem2d wex simpr w3a 3anass cvv ad2antrr elfvexd simpr2 difsnb ssdifssd ssdifd sylib difun1 syl6sseqr simpr1 uncom uneq2i unass difsnid uneq1d syl5eqr - eqsstrrd syl5eq syl fveq2d sseqtr4d simpr3 csuc wo com wi simplr dif1en + eqsstrrd syl5eq syl fveq2d sseqtrrd simpr3 csuc wo com wi simplr dif1en 3anan12 sylbir expcom syl2anc orim12d mpd mreexexlemd ad3antrrr difss2d wal ssexd simprl simplr1 snssd unssd sselpwd ad3antlr cin simprrl en2sn elpwid el2v a1i incom disjdif ssdifin0 unen syl22anc eqbrtrrd rexlimddv @@ -198823,7 +198823,7 @@ f C_ ( N ` ( g u. h ) ) /\ ( f u. h ) e. I ) -> special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.) $) mreexfidimd $p |- ( ph -> S ~~ T ) $= - ( cdom wbr cen cfv mrcssidd sseqtrd cfn wcel orcd mreexdomd sseqtr4d olcd + ( cdom wbr cen cfv mrcssidd sseqtrd cfn wcel orcd mreexdomd sseqtrrd olcd mrissd sbth syl2anc ) AEFSTFESTEFUATABCDEFGHIJKLMNAEEHUBZFHUBZADEHIKLADEG IMKOUKZUCRUDADFGIMKPUKZAEUEUFZFUEUFZQUGOUHABCDFEGHIJKLMNAFUOUNADFHIKLUQUC RUIUPAURUSQUJPUHEFULUM $. @@ -201398,7 +201398,7 @@ which when given operations from the base category (using ~ df-resc ) ( vt vx vs cv cssc wbr cab crn cuni cpw cpm cxp cixp wcel wa cvv wss vex cdm co ovex wfn cfv wrex wex brssc wf simpl xpex fnex sylancl rnexg pwexg uniexg 4syl wceq fndm adantr syl6eqel ss2ixp fvssunirn mpbi simprr sseldi - sspwb mprg elixpconst sylib elpwi ad2antrl xpss12 syl2anc sseqtr4d elpm2r + sspwb mprg elixpconst sylib elpwi ad2antrl xpss12 syl2anc sseqtrrd elpm2r a1i syl22anc rexlimdvaa imp exlimiv sylbi abssi ssexi ) AFZBGHZAIBJZKZLZB UAZMUBZWIWJMUCWFAWKWFBCFZWLNZUDZWEDEFZWONZDFZBUEZLZOZPZEWLLZUFZQZCUGWEWKP ZDCWEBEUHXDXECWNXCXEWNXAXEEXBWNWOXBPZXAQZQZWIRPZWJRPWPWIWEUIZWPWJSXEXHBRP @@ -213863,7 +213863,7 @@ net proof size (existence part)? $) acsmap2d $p |- ( ph -> E. f ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) ) $= ( cuni cfv wss wa wex adantr cpw cfn cin crn wceq acsmred mrissd mrcssidd - sseqtr4d acsmapd simprl cmre wcel simprr mrcssvd mrcssd frn unissd unifpw + sseqtrrd acsmapd simprl cmre wcel simprr mrcssvd mrcssd frn unissd unifpw cv wf syl6sseq ad2antrl sstrd mrcidmd sseqtrd eqsstrd mrissmrcd ex eximdv jca mpd ) ADCUAUBUCZEUTZVAZDVNUDZOZGPZQZRZESVOCVQUEZRZESABCDEGHIJABCFHKAB HIUFZLUGADDGPZCGPZABDGHWCJMUHNUIUJAVTWBEAVTWBAVTRZVOWAAVOVSUKWFBCVQFGHABH @@ -215499,7 +215499,7 @@ arbitrary magmas (then it should be called "iterated sum"). If the magma is ( vy vm vn vz vf crn cv wceq wral crab wss c0g cfv cfz wcel cseq cuz wrex co wa wex cio ccnv cvv csn cdif cima chash wf1o ccom cif cplusg cbs oveq1 cgsu eqeq1d ovanraleqv sseldd ralrimiva elrabd snssd mgmidsssn0 syl elsni - eqid sneqd sseqtr4d eqssd sselda syldan ressbas2 syl6eqel ressplusg oveqd + eqid sneqd sseqtrrd eqssd sselda syldan ressbas2 syl6eqel ressplusg oveqd c1 fvex anbi12d raleqbidv rabeqbidv eleqtrd cress ovexi a1i eqtr3d sseq2d seqeq2d fveq1d eqeq2d anbi2d rexbidv exbidv iotabidv ifeq12d difeq2d fssd ifbieq12d imaeq2d gsumval wf feq3d mpbid 3eqtr4d ) AGUGZUBUHZBUHZEUTZYFUI @@ -216807,7 +216807,7 @@ everywhere defined internal operation (see ~ mndcl ), whose operation is subsubm $p |- ( S e. ( SubMnd ` G ) -> ( A e. ( SubMnd ` H ) <-> ( A e. ( SubMnd ` G ) /\ A C_ S ) ) ) $= ( csubmnd cfv wcel wss cbs c0g cress cmnd eqid adantl wceq adantr submmnd - wa co submss submbas sseqtr4d sstrd subm0cl eqeltrd oveq1i ressabs syl5eq + wa co submss submbas sseqtrrd sstrd subm0cl eqeltrd oveq1i ressabs syl5eq subm0 syldan eqeltrrd w3a wb submrcl issubm2 syl mpbir3and simprr sseqtrd jca ad2antrl adantrl impbida ) BCFGZHZADFGHZAVEHZABIZSZVFVGSZVHVIVKVHACJG ZIZCKGZAHZCALTZMHZVKABVLVKADJGZBVGAVRIZVFVRADVRNZUAOVFBVRPZVGBDCEUBZQUCZV @@ -217618,7 +217618,7 @@ proposition to be be proved (the first four hypotheses tell its values frmdsssubm $p |- ( ( I e. V /\ J C_ I ) -> Word J e. ( SubMnd ` M ) ) $= ( vx vy wcel wss wa cword cfv c0 cv co wral adantl wceq eqid adantr sswrd - csubmnd cbs cplusg frmdbas sseqtr4d wrd0 cconcat sselda anim12dan frmdadd + csubmnd cbs cplusg frmdbas sseqtrrd wrd0 cconcat sselda anim12dan frmdadd a1i syl ccatcl eqeltrd ralrimivva cmnd w3a frmdmnd frmd0 issubm mpbir3and wb ) ADHZBAIZJZBKZCUBLHZVGCUCLZIZMVGHZFNZGNZCUDLZOZVGHZGVGPFVGPZVFVGAKZVI VEVGVRIVDBAUAQVDVIVRRVEVIACDEVISZUETUFZVKVFBUGULVFVPFGVGVGVFVLVGHZVMVGHZJ @@ -217661,7 +217661,7 @@ proposition to be be proved (the first four hypotheses tell its values chash wrdf ad2antll frnd cores wfn vrmdf 3ad2ant1 fnssres syl2an2r df-ima ffnd simprl eqsstrrid df-f sylanbrc wrdco eqeltrrd gsumwsubmcl expr ssrdv ex wi wral cs1 simp2 sselda vrmdval simpr s1cld eqeltrd ralrimiva wfun wb - cdm ffund fdmd sseqtr4d funimass4 mpbird sstr2 impbid ) CFJZDCKZAEUALJZUB + cdm ffund fdmd sseqtrrd funimass4 mpbird sstr2 impbid ) CFJZDCKZAEUALJZUB ZBDUDZAKZDMZAKZXBXDXFXBXDNIXEAXBXDIUEZXEJZXGAJXBXDXHNZNZEBXGOZUFPZXGAXJWS XGCMZJXLXGQWSWTXAXIUCXJXEXMXGXJWTXEXMKWSWTXAXIUGZDCUHRXBXDXHUIZUJBCEFXGGH UKSXJXAXKAMZJXLAJWSWTXAXIULXJBDUMZXGOZXKXPXJXGUNDKXRXKQXJUOXGUQLUPPZDXGXH @@ -221010,7 +221010,7 @@ by a normal subgroup (resp. two-sided ideal). (Contributed by Mario subsubg $p |- ( S e. ( SubGrp ` G ) -> ( A e. ( SubGrp ` H ) <-> ( A e. ( SubGrp ` G ) /\ A C_ S ) ) ) $= ( csubg cfv wcel wss wa cgrp cbs cress co adantr eqid subgss wceq subggrp - adantl subgrcl subgbas sseqtr4d sstrd oveq1i ressabs syl5eq syldan issubg + adantl subgrcl subgbas sseqtrrd sstrd oveq1i ressabs syl5eq syldan issubg eqeltrrd syl3anbrc jca simprr sseqtrd adantrl ad2antrl eqeltrd impbida ) BCFGZHZADFGHZAUSHZABIZJZUTVAJZVBVCVECKHZACLGZICAMNZKHZVBUTVFVABCUAOVEABVG VEADLGZBVAAVJIZUTVJADVJPZQTUTBVJRZVABCDEUBZOUCZUTBVGIVAVGBCVGPZQOUDVEDAMN @@ -223452,7 +223452,7 @@ permutation associated with the composition of these two elements (in ( F " S ) C_ ( Y ` ( F " T ) ) ) $= ( vx cmhm co wcel cfv wss wa cima wral cbs eqid cv ralrimiva ssralv mpan9 cntzmhm wfun cdm wb mhmf adantr ffund simpr cntzssv syl6ss fdmd funimass4 - wf sseqtr4d syl2anc mpbird ) CDEKLMZABGNZOZPZCAQCBQFNZOZJUAZCNVEMZJARZVAV + wf sseqtrrd syl2anc mpbird ) CDEKLMZABGNZOZPZCAQCBQFNZOZJUAZCNVEMZJARZVAV HJVBRVCVIVAVHJVBVGBCDEFGHIUEUBVHJAVBUCUDVDCUFACUGZOVFVIUHVDDSNZESNZCVAVKV LCUQVCVKVLDECVKTZVLTUIUJZUKVDAVKVJVDAVBVKVAVCULVKBDGVMHUMUNVDVKVLCVNUOURJ AVECUPUSUT $. @@ -232114,7 +232114,7 @@ an extension of the previous (inserting an element and its inverse at $( Discharge the centralizer assumption in a commutative monoid. (Contributed by Mario Carneiro, 24-Apr-2016.) $) cntzcmnf $p |- ( ph -> ran F C_ ( Z ` ran F ) ) $= - ( crn cfv frnd ccmn wcel wss wceq cntzcmn syl2anc sseqtr4d ) ADKZCUAFLZAB + ( crn cfv frnd ccmn wcel wss wceq cntzcmn syl2anc sseqtrrd ) ADKZCUAFLZAB CDJMZAENOUACPUBCQIUCCUAEFGHRST $. $} @@ -232436,7 +232436,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements $( All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.) $) ablcntzd $p |- ( ph -> T C_ ( Z ` U ) ) $= - ( cbs cfv csubg wcel wss eqid subgss syl ccmn wceq cabl cntzcmn sseqtr4d + ( cbs cfv csubg wcel wss eqid subgss syl ccmn wceq cabl cntzcmn sseqtrrd ablcmn syl2anc ) ABDJKZCEKZABDLKZMBUENHUEBDUEOZPQADRMZCUENZUFUESADTMUIGDU CQACUGMUJIUECDUHPQUECDEUHFUAUDUB $. $} @@ -233553,7 +233553,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements 24-Apr-2016.) (Revised by AV, 3-Jun-2019.) $) gsumzsubmcl $p |- ( ph -> ( G gsum F ) e. S ) $= ( co cfv eqid syl cress cgsu cbs c0g ccntz csubmnd wcel cmnd submmnd wceq - wf submbas feq3d mpbid crn cin ssind wss resscntz syl2anc sseqtr4d cfsupp + wf submbas feq3d mpbid crn cin ssind wss resscntz syl2anc sseqtrrd cfsupp frnd subm0 breqtrd gsumzcl gsumsubm 3eltr4d ) AECUAQZDUBQVIUCRZEDUBQCABVJ DVIFVIUDRZVIUERZVJSVKSVLSZACEUFRZUGZVIUHUGMCVIEVISZUITLABCDUKBVJDUKNACVJD BAVOCVJUJMCVIEVPULTZUMUNADUOZVRHRZCUPZVRVLRZAVRVSCOABCDNVCZUQAVOVRCURWAVT @@ -233962,7 +233962,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements cseq nnuz simpr adantr fvmpt2 syl2anc wf ad2antll ffvelrnda feqmptd eqidd fmptco fveq1d elfznn fvconst2g syl2an 3eqtr4d seqfveq csupp simpl1 simpl2 f1of ccntz fmpttd wss crn wb elcntzsn mpbir2and snssd snidg frnd cntzidss - wf1 f1of1 cdm suppssdm dmmptss sstrid f1ofo forn sseqtr4d gsumval3 mulgnn + wf1 f1of1 cdm suppssdm dmmptss sstrid f1ofo forn sseqtrrd gsumval3 mulgnn a1i wfo expr exlimdv expimpd wo fz1f1o 3ad2ant2 mpjaod ) EUAKZAUBKZFBKZUC ZALMZEDAFUDZUGNZAUEOZFCNZMZYMPKZQYMUFNZAIUHZUIZIUJZRZYIYJYOYIYJRZUKFCNZEU TOZYNYLUUBYHUUCUUDMYFYGYHYJULBCEFUUDGUUDSZHUMTUUBYMUKFCUUBYMLUEOZUKYJYMUU @@ -234044,7 +234044,7 @@ elements of arbitrarily large orders (so ` E ` is zero) but no elements wf syl6eqss cbs feqmptd fveq2 fmptco mpteq2dv eqtrd oveq2d fveq2d 3eqtr4d mhmf cplusg cseq mndcl 3expb sylan wf1 f1of1 ad2antll cnvimass fssdm f1ss ex wss f1f fco syl2an2r ffvelrnda cuz simprl syl6eleq mhmlin coass fveq1i - nnuz fvco3 syl5req seqhomo crn wfo f1ofo sseqtr4d gsumval3 ccntz cntzmhm2 + nnuz fvco3 syl5req seqhomo crn wfo f1ofo sseqtrrd gsumval3 ccntz cntzmhm2 forn rnco2 fveq2i 3sstr4g eldifi syl2an suppssr 3eqtrd suppss cfn exlimdv 3eqtr4rd expr expimpd wo fsuppimpd eqeltrrd fz1f1o mpjaod ) ADUEUFIUGUHZU IZUJUKZFGDULZUMUNZEDUMUNZGUOZUKZUUKUPUOZUQUSZURUURUTUNZUUKUAVAZVBZUAVCZVD @@ -236162,7 +236162,7 @@ mapping the (infinite, but finitely supported) cartesian product of ( H dom DProd S <-> ( G dom DProd S /\ ran S C_ ~P A ) ) ) $= ( vx vy cfv wcel cvv wss wa adantr wb csn wral cin wceq eqid ad2antrr cdm csubg cdprd wbr crn cpw wi reldmdprd brrelex2i wf cv ccntz cdif cima cuni - a1i c0g cbs ffvelrn ad2ant2lr subgss syl subgbas sseqtr4d biantrud simpll + a1i c0g cbs ffvelrn ad2ant2lr subgss syl subgbas sseqtrrd biantrud simpll cmrc simplr eldifi ad2antll ffvelrnd resscntz syl2anc sseq2d ssin syl6bbr bitr4d anassrs cmre cgrp cacs subgrcl subgacs acsmre 3syl subggrp imassrn ralbidva frn ad2antlr sstrid mresspw sstrd sspwuni sylib mrcssidd subsubg @@ -236310,7 +236310,7 @@ mapping the (infinite, but finitely supported) cartesian product of dprdcntz2 $p |- ( ph -> ( G DProd ( S |` C ) ) C_ ( Z ` ( G DProd ( S |` D ) ) ) ) $= ( cdprd cfv cdm wss wceq wcel adantr vx vy cres co dprdres simpld dmres - wbr cin sseqtr4d df-ss sylib syl5eq cgrp cbs csubg dprdgrp eqid dprdssv + wbr cin sseqtrrd df-ss sylib syl5eq cgrp cbs csubg dprdgrp eqid dprdssv syl cntzsubg cv wa fvres adantl dprdsubg sselda dprdf2 ffvelrnda syldan sylancl subgss syl2anc ad2antrr wn simpr wi c0 noel elin eleq2d syl5bbr wne mtbiri imnan sylibr imp nelne2 dprdcntz eqsstrd dprdlub cntzrecd ) @@ -236432,7 +236432,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $. c2nd sneq oveq1 mpteq12dv breq2d adantr 1stdm sylan rspcdva 3ad2antr1 a1i sseldd cop 1st2nd simpr df-br sylibr elrelimasn mpbird sneqd eqeq12d fvex wb opth necon3bid mpbid dprdcntz df-ov oveq2 fvmpt3i fveq2d 3eqtr4a eqtrd - oveq1d oveq2d sseqtr4d ad2antrr eqsstrrd eqsstrd sstrd cdif cin cun snssd + oveq1d oveq2d sseqtrrd ad2antrr eqsstrrd eqsstrd sstrd cdif cin cun snssd cuni sylib syl6eq eleq1d simplbi crn imassrn frnd sstrid sspwuni mrcssidd mrccl difss ax-mp elrnmpt1s unissd mrcssd syl6eqr syl3anc sseqtrd subg0cl c0 dprdspan ccntz cgrp dprdgrp resiun2 iunid reseq2i eqtr3i wrel relssres @@ -236628,7 +236628,7 @@ G dom DProd ( j e. ( A " { i } ) |-> ( i S j ) ) ) $. sstrd syl difundir difeq1d c0 simpr snssd sslin incom syl5eqr sseq0 3syl syl2anc disj3 sylib uneq2d 3eqtr4a imaeq2d syl6eq unieqd uniun imaundi difss imass2 uniss imassrn sstrid sspwuni mrcssidd dprdspan - crn df-ima unieqi syl6eqr sseqtr4d dprdsubg syl3anc mrcsscl sseqtrd + crn df-ima unieqi syl6eqr sseqtrrd dprdsubg syl3anc mrcsscl sseqtrd fveq2i adantl subg0cl mp2b cpw mresspw unss12 mrccl lsmunss eqsstrd frnd mrcssd lsmsubg sselda ffvelrnda syldan wb lsmlub ssrind lsmub1 mpbi2and sseldd elind eqssd resima2 mp1i ineq12d dprddisj eqtr3d cv @@ -237357,7 +237357,7 @@ factorization into prime power factors (even if the exponents are ssfi mpd 3eqtrd cvv hashen mpbid fisseneq disjdif dprdsplit simprd eqtr3d cin cun syldan adantlr cbs fvexi rabex dmmpti odsubdvds cuz ablgrp grpbn0 cgrp cabl pccld eqbrtrrd pcdvdsb eluz2 syl3anbrc dvdsexp eqbrtrd nnexpcld - odcl hashcl ssrabdv oveq12d breq2d rabbidv fvmpt3i sseqtr4d nnnn0d pcdvds + odcl hashcl ssrabdv oveq12d breq2d rabbidv fvmpt3i sseqtrrd nnnn0d pcdvds oveq1 ne0d sneq difeq2d breq12d notbid wral cmpt ssrab3 ssidd clsm fssres difss eldif iddvdsexp sylan eqeltrrd breq1 elrab2 sylanbrc ex con3d elnn0 impr wo ord nncnd exp0d fvex hashsng ax-mp syl6reqr snfi sylancr eqsstrrd @@ -237841,7 +237841,7 @@ factorization into prime power factors (even if the exponents are 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 + a1i dmdprdsplit2 clsm dprdsplit oveq12d cabl lsmcom subgss sseqtrrd subglsm breq2 eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHVEVFZVGZK HVEVHZIVIZKRVJZUXJVGZKUXNVEVHZIVIZVQZRUXHVKADGHPVLZMVMZVBUSAUXTKVNV MZVDZKUXTVOVHZVPVRVSZVTZVDZUXTDVDZAUYBUXTIWAZAUXTLVNVMZVDZUYBUYHVQZ @@ -237887,7 +237887,7 @@ factorization into prime power factors (even if the exponents are ( va cv cdprd cdm wbr co wceq wa cword wrex wpss wss wne csubg cfv wb wcel subsubg syl mpbid simprd chash cfn subgss ssfid hashcl nn0red c1 cn0 cmul csn clt cvv c0g fvexi hashsng ax-mp csdm cgrp cacs cmre eqid - subgrcl subgacs 4syl mrcssvd subgbas sseqtr4d eleqtrd mrcsncl syl2anc + subgrcl subgacs 4syl mrcssvd subgbas sseqtrrd eleqtrd mrcsncl syl2anc cbs acsmre subg0cl snssd mrcssidd snssg mpbird eqnetrd od1 3syl elsni fveqeq2d syl5ibrcom necon3ad mpd ssnelpssd php3 snfi hashsdom sylancr wn eqbrtrrid cr cc0 1red cn c0 ne0i hashnncl nngt0d cabl wi adantr @@ -243213,7 +243213,7 @@ nonzero elements form a group under multiplication (from which it eleqtrd simprd subrg0 neeqtrd drngunit mpbir2and syl2anc subrginv 3eltr4d wne ringinvcl ralrimiva subrguss isdrng simprbi ad2antrr unitss sseqtrrid cin sseqtrd ssind subrgss difin2 simprl sseldd simprr subrgunit mpbir3and - sseqtr4d w3a expr ralimdva imp dfss3 sylibr eqssd sneqd difeq12d sylanbrc + sseqtrrd w3a expr ralimdva imp dfss3 sylibr eqssd sneqd difeq12d sylanbrc eqtrd impbida ) CJKZBCUALKZMZDJKZAUBZELZBKZABFUEZNZUCZWTXAMZXDAXFXHXBXFKZ MZXBDUDLZLZDUFLZXCBXJDUGKZXBDUHLZKZXLXMKXJWSXNWRWSXAXIUIZBCDGUJZOXJXPXBXM KZXBDUKLZVFZXJXBBXMXJXBBKZXBFVFZXJXIYBYCMZXHXIULXBBFUMZUNZUOXJWSBXMPZXQBC @@ -243237,7 +243237,7 @@ nonzero elements form a group under multiplication (from which it subsubrg $p |- ( A e. ( SubRing ` R ) -> ( B e. ( SubRing ` S ) <-> ( B e. ( SubRing ` R ) /\ B C_ A ) ) ) $= ( csubrg cfv wcel wss crg cress cbs adantr wceq eqid adantl subrgring jca - wa co cur subrgrcl subrgss subrgbas sseqtr4d oveq1i ressabs syl5eq syldan + wa co cur subrgrcl subrgss subrgbas sseqtrrd oveq1i ressabs syl5eq syldan eqeltrrd sstrd subrg1 subrg1cl eqeltrd issubrg syl21anbrc ad2antrl simprr adantrl sseqtrd impbida ) ACFGZHZBDFGHZBVBHZBAIZSZVCVDSZVEVFVHCJHZCBKTZJH ZBCLGZIZCUAGZBHZSVEVCVIVDACUBMVHDBKTZVJJVCVDVFVPVJNZVHBDLGZAVDBVRIZVCBVRD @@ -243598,7 +243598,7 @@ nonzero elements form a group under multiplication (from which it subdrgint $p |- ( ph -> L e. DivRing ) $= ( wcel cfv cdif cress co cgrp wss syl wceq eqid crg cmgp cbs c0g csn cint cdr csubrg c0 wne subrgint syl2anc subrgring fveq2i oveq1i mgpress oveq1d - difssd subrgss ressbas2 3syl sseqtr4d ressabs eqtr3d ciin intiin syl5reqr + difssd subrgss ressbas2 3syl sseqtrrd ressabs eqtr3d ciin intiin syl5reqr cv difeq1d oveq2d cmpt crn vex difexi dfiin3 iindif1 syl5eqr difss mgpbas cvv ax-mp fvexi ciun iinssiun csubg subrgsubg ssriv syl6ss subgint adantr subg0 sneqd difeq2d sselda ssdifd eqsstrrd iunssd sstrd eqsstrrid sylancr @@ -247457,7 +247457,7 @@ Absolute value (abstract algebra) ( F " ( K ` U ) ) = ( L ` ( F " U ) ) ) $= ( wcel wss cfv cima eqid adantr clmod clss lspcl syl2anc clmhm co wa wfun ccnv cbs wf lmhmf ffund lmhmlmod1 lmhmlmod2 crn imassrn sstrid lmhmpreima - frnd syldan cdm incom wceq simpr fdmd sseqtr4d df-ss sylib syl5req dminss + frnd syldan cdm incom wceq simpr fdmd sseqtrrd df-ss sylib syl5req dminss cin syl6eqss lspssid imass2 sstrd lspssp syl3anc funimass2 lmhmima eqssd syl ) DABUAUBKZCGLZUCZDCEMZNZDCNZFMZWADUDWBDUEZWENZLZWCWELWAGBUFMZDVSGWID UGVTGWIABDHWIOZUHPZUIWAAQKZWGARMZKZCWGLWHVSWLVTABDUJPZVSVTWEBRMZKZWNWABQK @@ -248285,7 +248285,7 @@ Absolute value (abstract algebra) lspsntri $p |- ( ( W e. LMod /\ X e. V /\ Y e. V ) -> ( N ` { ( X .+ Y ) } ) C_ ( ( N ` { X } ) .(+) ( N ` { Y } ) ) ) $= ( clmod wcel w3a co csn cfv cun wss snssd cpr lspvadd df-pr syl6sseq wceq - fveq2i simp1 simp2 simp3 lsmsp2 syl3anc sseqtr4d ) ELMZFDMZGDMZNZFGAOPCQZ + fveq2i simp1 simp2 simp3 lsmsp2 syl3anc sseqtrrd ) ELMZFDMZGDMZNZFGAOPCQZ FPZGPZRZCQZURCQUSCQBOZUPUQFGUAZCQVAACDEFGHIJUBVCUTCFGUCUFUDUPUMURDSUSDSVB VAUEUMUNUOUGUPFDUMUNUOUHTUPGDUMUNUOUITBURUSCDEHJKUJUKUL $. $} @@ -249079,7 +249079,7 @@ division ring is an Abelian group (vectors) together with a division ring lspindpi $p |- ( ph -> ( ( N ` { X } ) =/= ( N ` { Y } ) /\ ( N ` { X } ) =/= ( N ` { Z } ) ) ) $= ( csn cfv wne wcel wss syl2anc cpr wn wceq clsm co csubg clss clmod clvec - lveclmod syl eqid lsssssubg sseldd lsmub1 lsmpr sseqtr4d sseq1 syl5ibrcom + lveclmod syl eqid lsssssubg sseldd lsmub1 lsmpr sseqtrrd sseq1 syl5ibrcom lspsncl lspprcl lspsnel5 sylibrd necon3bd mpd lsmub2 jca ) AEOBPZFOBPZQZV HGOBPZQZAEFGUABPZRZUBZVJNAVNVHVIAVHVIUCZVHVMSZVNAVQVPVIVMSAVIVIVKDUDPZUEZ VMAVIDUFPZRZVKVTRZVIVSSADUGPZVTVIADUHRZWCVTSADUIRWDJDUJUKZWCDWCULZUMUKZAW @@ -250961,7 +250961,7 @@ left ideal which is also a right ideal (or a left ideal over the opposite ( vy wcel wceq cv wbr wral wa wss crg w3a csn simp1 simp3 rspssid syl2anc cfv snssd snssg 3ad2ant3 mpbird cab rspsn 3adant2 eleq2d vex breq2 syl6bb wb elab biimpd ralrimiv jca eleq2 raleq anbi12d syl5ibrcom wi df-ral ssab - sylbb2 ad2antll adantr sseqtr4d simpl1 simpl2 snssi adantl rspssp syl3anc + sylbb2 ad2antll adantr sseqtrrd simpl1 simpl2 snssi adantl rspssp syl3anc wal adantrr eqssd ex impbid ) DUANZGENZFBNZUBZGFUCZHUHZOZFGNZFAPZCQZAGRZS ZWJWRWMFWLNZWPAWLRZSWJWSWTWJWSWKWLTZWJWGWKBTXAWGWHWIUDWJFBWGWHWIUEUIBDWKH KIUFUGWIWGWSXAUTWHFWLBUJUKULWJWPAWLWJWOWLNZWPWJXBWOFMPZCQZMUMZNWPWJWLXEWO @@ -255290,7 +255290,7 @@ series in the subring which are also polynomials (in the parent ring). un0 a1i cop df-br vex ideq bitr3i brin simprbi brxp sonr syl2im pm2.01d wn ex breq2 syl6bb notbid syl5ibcom syl5bi con2d opex syl5eq syl6eqssr eldif mpbiran syl6ibr relssdv disj2 disj3 3eqtr4a soeq1 opsrbas reseq2d - sylibr mpbid ssun2 sseqtr4d sylanbrc ) AQULUMZQUNUMZUOZUPUUQUQZPURZQUSU + sylibr mpbid ssun2 sseqtrrd sylanbrc ) AQULUMZQUNUMZUOZUPUUQUQZPURZQUSU TZAGUURUOZUUSAUVCGBCDVAZGGVBZVCZUOZAGUVDUOZUVGAUVHJULUMZIVDVHZUVDUOZIVE UTAIHVFUVILUOZUVKNVIVJVKVLVMUTNWBOVDVHIUHWBOVDVNVOAHIMNORVEUGUHTAMOOVBV EAOORRTTVPUBVQUCVRAUVLUPUVIUQJVSUMZURZAJUSUTZUVLUVNWCZUAUVOUVPUVILJUVMU @@ -261237,7 +261237,7 @@ univariate polynomial evaluation map function for a (sub)ring is a ring c1 cfz wf1o wex mpteq1 mpt0 syl6eq oveq2d cnfld0 gsum0 sum0 eqtr4i sumeq1 wi eqtr4d a1i caddc ccom cseq cfn csupp ccntz cnfldbas cnfldadd eqid cmnd crg cnring ringmnd mp1i adantr wf fmpttd ccmn ringcmn cntzcmnf simprl wf1 - simprr f1of1 syl crn suppssdm fssdm wfo f1ofo forn 3syl sseqtr4d gsumval3 + simprr f1of1 syl crn suppssdm fssdm wfo f1ofo forn 3syl sseqtrrd gsumval3 sumfc fveq2 ffvelrnda f1of fvco3 fsum syl5eqr expr exlimdv expimpd fz1f1o sylan wo mpjaod ) ABIJZKDBCUBZLMZBCDNZJZBUCOZUDPZUEXNUFMZBGQZUGZGUHZRZXIX MURAXIXKICDNZXLXIXKKILMZYAXIXJIKLXIXJDICUBIDBICUIDCUJUKULYBSYAKSUMUNCDUOU @@ -264616,7 +264616,7 @@ S C_ ( ._|_ ` ( ._|_ ` S ) ) ) $= mrccss $p |- ( ( W e. PreHil /\ S C_ V ) -> ( F ` S ) = ( ._|_ ` ( ._|_ ` S ) ) ) $= ( cphl wcel wss wa cfv cmre cssmre adantr sylan ocv2ss ocvocv a1i mrcsscl - ocvss ocvcss sylan2 syl3anc mrcssid 3syl wceq mrccl cssi sseqtr4d eqssd + ocvss ocvcss sylan2 syl3anc mrcssid 3syl wceq mrccl cssi sseqtrrd eqssd syl ) FKLZBEMZNZBCOZBDOZDOZURAEPOLZBVAMVAALZUSVAMUPVBUQAEFGIQZRBDEFGHUAUQ UPUTEMZVCVEUQBDEFGHUDUBAUTDEFGIHUEUFABCVAEJUCUGURVAUSDOZDOZUSURBUSMZVFUTM VAVGMUPVBUQVHVDABCEJUHSUSBDFHTUTVFDFHTUIURUSALZUSVGUJUPVBUQVIVDABCEJUKSAU @@ -266517,7 +266517,7 @@ is used for the (special) unit vectors forming the standard basis of free frlmsslss2 cv wa csupp co uvcff simp3 sselda ffvelrnd cbs simpl2 frlmbasf wral wf syl2anc cdif simpll1 simpll2 eldifi adantl wne cin c0 wceq disjne disjdif mp3an1 adantll uvcvv0 suppss oveq1 sseq1d sylanbrc ralrimiva wfun - elrab2 cdm ffund fdmd sseqtr4d funimass4 mpbird lspssp syl3anc cvsca cgsu + elrab2 cdm ffund fdmd sseqtrrd funimass4 mpbird lspssp syl3anc cvsca cgsu wb cof simpl1 ssrab3 a1i uvcresum c0g cabl lmodabl syl csubg imassrn frnd crn sstrid lspcl lsssubg wfn 3ad2antl2 inidm syldan adantrr simprr fnfvof ffnd offn syl22anc csca sylan2 adantrl frlmsca fveq2d eleqtrd cvv suppssr @@ -270340,7 +270340,7 @@ matrix over (the ring) ` R ` . (Contributed by AV, 18-Dec-2019.) $) matrices. (Contributed by AV, 21-Aug-2019.) $) scmatsgrp1 $p |- ( ( N e. Fin /\ R e. Ring ) -> S e. ( SubGrp ` C ) ) $= ( vx vy wcel cfv cfn crg wa csubg cbs wss c0 wne cv csg co wral scmatdmat - ssrdv wceq dmatsgrp ancoms subgbas eqcomd syl sseqtr4d cur scmatid adantr + ssrdv wceq dmatsgrp ancoms subgbas eqcomd syl sseqtrrd cur scmatid adantr ne0d wi com12 impcom a1d imp32 eqid subgsub syl3anc scmatsubcl ralrimivva w3a eqeltrd cgrp csubrg dmatsrng subrgring ringgrp issubg4 3syl mpbir3and wb ) HUASZEUBSZUCZFCUDTSZFCUETZUFZFUGUHZQUIZRUIZCUJTZUKZFSZRFULQFULZWIFDW @@ -281325,7 +281325,7 @@ _Introduction to General Topology_ (1983), p. 114) and it is convenient $( A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) $) toponss $p |- ( ( J e. ( TopOn ` X ) /\ A e. J ) -> A C_ X ) $= - ( ctopon cfv wcel wa cuni wss elssuni adantl wceq toponuni adantr sseqtr4d + ( ctopon cfv wcel wa cuni wss elssuni adantl wceq toponuni adantr sseqtrrd ) BCDEFZABFZGABHZCQARIPABJKPCRLQCBMNO $. $( If ` K ` is a topology on the base set of topology ` J ` , then ` J ` is a @@ -282724,7 +282724,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ssntr $p |- ( ( ( J e. Top /\ S C_ X ) /\ ( O e. J /\ O C_ S ) ) -> O C_ ( ( int ` J ) ` S ) ) $= ( ctop wcel wss wa cpw cin cuni cnt cfv elin elpwg pm5.32i bitr2i elssuni - sylbi adantl wceq ntrval adantr sseqtr4d ) BFGADHIZCBGZCAHZIZICBAJZKZLZAB + sylbi adantl wceq ntrval adantr sseqtrrd ) BFGADHIZCBGZCAHZIZICBAJZKZLZAB MNNZUICULHZUFUICUKGZUNUOUGCUJGZIUICBUJOUGUPUHCABPQRCUKSTUAUFUMULUBUIABDEU CUDUE $. @@ -282948,7 +282948,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on elcls3 $p |- ( ph -> ( P e. ( ( cls ` J ) ` S ) <-> A. x e. B ( P e. x -> ( x i^i S ) =/= (/) ) ) ) $= ( vy vz cfv wcel c0 wi wceq wa ccl cv cin wne wral ctop cuni wss ctg tgcl - wb ctb syl eqeltrd sseqtrd eleqtrd eqid elcls bastg sseqtr4d sseld imim1d + wb ctb syl eqeltrd sseqtrd eleqtrd eqid elcls bastg sseqtrrd sseld imim1d syl3anc ralimdv2 eleq2w ineq1 neeq1d imbi12d cbvralv syl6ib wrex ad2antrr simprl simprr tg2 syl2anc rspccva imp ssdisj necon3d exp31 imp4a rexlimdv ex syl5com ad2antlr mpd exp43 ralrimdv impbid bitrd ) ADEFUAOOPZDMUBZPZWM @@ -283126,7 +283126,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on c0 wi simpl adantrl intss1 sstrd uniopn syl2anc ralrimiv vuniex elint2 ex expr alrimiv simpll simplrl sseldd simplrr inopn ralrimiva vex ralrimivva syl3anc inex1 wb intex biimpi istopg mpbir2and 3adant1 n0 ad2antlr ancoms - cvv wex elssuni toponuni sseqtr4d exlimdv unissb eqid topopn 3syl eqeltrd + cvv wex elssuni toponuni sseqtrrd exlimdv unissb eqid topopn 3syl eqeltrd mpd elintg 3ad2ant1 mpbird unissel eqcomd istopon sylanbrc ismred ) ABGZA UAUBZAUCZCXFXGUCHXEAUDUEABUFXECRZXFHZXHULUGZUHZXHUIZIGZAXLJZKXLXFGXIXJXMX EXIXJSZXMDRZXLHZXPJZXLGZUMZDUJZXPERZUKZXLGZEXLLDXLLZXOXTDXOXQXSXOXQSZXRYB @@ -283601,7 +283601,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on 7-Jan-2018.) $) neiptoptop $p |- ( ph -> J e. Top ) $= ( wcel wss wral wa cvv ve vf vc ctop cv cuni wi wal cin uniss adantl wceq - cfv neiptopuni adantr sseqtr4d w3a simp-4l ad3antrrr simpllr jca ad2antlr + cfv neiptopuni adantr sseqtrrd w3a simp-4l ad3antrrr simpllr jca ad2antlr sseldd elssuni simpr sselda neipeltop simprbi syl r19.21bi adantllr sseq1 3jca 3anbi2d eleq1 anbi12d imbi1d cpw crab ssidd ralrimiva sylanbrc pwexg imbi2d rabexg eqeltrid ssexd uniexg 3anbi23d anbi1d imbi12d vtoclg chvarv @@ -283640,7 +283640,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on mpan2 rexeqbidv sselda ancrd ralimdva reximdva ralbii rexbii sylibr dfss3 a1d mpd biimpri reximi syl21anc r19.29a sylan2b ralrimiva sylanbrc anim1i neipeltop nfv nfrab1 nfcv rabid elequ1 elequ2 ex syl5bi ssrd eleq2 rspcev - syl12anc nfan jca nfre1 sseqtr4d syl31anc simprbi r19.21bi anasss reximi2 + syl12anc nfan jca nfre1 sseqtrrd syl31anc simprbi r19.21bi anasss reximi2 ad2antll r19.29af impbida eleqtrd eqid isneip syl2an2r bitr4d eqrdv eqtrd mpteq2dva ) ACFDFULZCUCZUDFDUUTUEBUFUCUCZUDAFDDUGZUGZCJUHAFDUVAUVBAUUTDRZ SZPUVAUVBUVFPULZUVARZUVGBUIZUJZFQUMZQULZUVGUJZSZQBUNZSZUVGUVBRZUVFUVHUVPU @@ -284249,7 +284249,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ( ( nei ` ( J |`t A ) ) ` B ) = ( ( ( nei ` J ) ` B ) |`t A ) ) $= ( vc vd va vb ve ctop wcel wss cv wa wrex cin wceq syl2anc cvv crest cnei w3a cfv cuni wex nfv nfre1 nfan simpl anim2i cdif cun simp-5r simp1 simp2 - co restuni ad5antr sseqtr4d sstrd eltopss ssdifssd unssd simpr1l 3anassrs + co restuni ad5antr sseqtrrd sstrd eltopss ssdifssd unssd simpr1l 3anassrs simplr simpr sseqtrd inss1 inundif simpr1r eqsstrrd unss1 eqsstrrid sseq2 syl6ss sseq1 anbi12d rspcev syl12anc c0 indir incom disjdif eqtr3i uneq2i syl un0 3eqtri df-ss biimpi syl5req vex difexi unex anbi2d rexbidv eqeq2d @@ -284532,7 +284532,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on ( vm vn ctsr wcel cun cvv wss vz csn cfi cfv cv cin wceq wrex w3o wb snex ssun2 cuni ordtuni cdm dmexg eqeltrid eqeltrrd uniexb sylibr ssexg elfiun sylancr wi ssun1 eqsstri sseli a1i ordtbas2 syl6eqss sseld wa cpw fipwuni - fisn elpwid ad2antll unissi sseqtrrid adantr simprl syl6eleq syl sseqtr4d + fisn elpwid ad2antll unissi sseqtrrid adantr simprl syl6eleq syl sseqtrrd sstrd elsni sseqin2 sylib sselda eleq1 syl5ibrcom rexlimdvva 3jaod sylbid adantrl eqeltrd ssrdv ssfii unssad fiss sylancl unssd eqssd unass syl6eqr eqsstrrd ) FPQZGUBZCDRZRZUCUDZXHXIERZRZXJERXGXKXMXGUAXKXMXGUAUEZXKQZXNXHU @@ -284566,7 +284566,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on { x e. X | -. x R P } e. ( ordTop ` R ) ) $= ( vy wcel cv wbr wn crab cmpt crn cun cfv cvv wceq eqid adantr wa csn cfi cordt ctg wss ordtuni cdm dmexg eqeltrid eqeltrrd uniexb sylibr ssfii syl - cuni ctb fibas bastg ax-mp syl6ss ordtval sseqtr4d ssun2 ssun1 wrex simpr + cuni ctb fibas bastg ax-mp syl6ss ordtval sseqtrrd ssun2 ssun1 wrex simpr eqidd breq2 notbid rabbidv rspceeqv syl2anc wb rabexg elrnmpt 3syl mpbird sseldi sseldd ) CDHZBEHZUAZEUBZGEAIZGIZCJZKZAELZMZNZGEWFWECJKAELMNZOZOZCU DPZWEBCJZKZAELZWCWNWNUCPZUEPZWOWCWNWSWTWCWNQHZWNWSUFWCWNUPZQHXAWCEXBQWAEX @@ -284581,7 +284581,7 @@ we show (in ~ tgcl ) that ` ( topGen `` B ) ` is indeed a topology (on { x e. X | -. P R x } e. ( ordTop ` R ) ) $= ( vy wcel cv wbr wn crab cmpt crn cun cfv cvv wceq eqid adantr wa csn cfi cordt ctg wss ordtuni cdm dmexg eqeltrid eqeltrrd uniexb sylibr ssfii syl - cuni ctb fibas bastg ax-mp syl6ss ordtval sseqtr4d ssun2 wrex simpr eqidd + cuni ctb fibas bastg ax-mp syl6ss ordtval sseqtrrd ssun2 wrex simpr eqidd breq1 notbid rabbidv rspceeqv syl2anc rabexg elrnmpt mpbird sseldi sseldd wb 3syl ) CDHZBEHZUAZEUBZGEAIZGIZCJKAELMNZGEWEWDCJZKZAELZMZNZOZOZCUDPZBWD CJZKZAELZWBWMWMUCPZUEPZWNWBWMWRWSWBWMQHZWMWRUFWBWMUPZQHWTWBEXAQVTEXARWAGA @@ -285265,7 +285265,7 @@ converges to zero (in the standard topology on the reals) with this ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ x C_ ( `' F " y ) ) ) ) ) ) $= ( ctopon cfv wcel cv cima wss wa wrex wi wral wb ad2antlr ccnp co wf ccnv w3a iscnp wfun cdm ffun toponss adantlr wceq fdm funimass3 syl2anc anbi2d - sseqtr4d rexbidva imbi2d ralbidv pm5.32da 3ad2ant1 bitrd ) EGIJKZFHIJKZCG + sseqtrrd rexbidva imbi2d ralbidv pm5.32da 3ad2ant1 bitrd ) EGIJKZFHIJKZCG KZUEDCEFUAUBJKGHDUCZCDJBLZKZCALZKZDVJMVHNZOZAEPZQZBFRZOZVGVIVKVJDUDVHMNZO ZAEPZQZBFRZOZABCDEFGHUFVDVEVQWCSVFVDVGVPWBVDVGOZVOWABFWDVNVTVIWDVMVSAEWDV JEKZOZVLVRVKWFDUGZVJDUHZNVLVRSVGWGVDWEGHDUITWFVJGWHVDWEVJGNVGVJEGUJUKVGWH @@ -285337,7 +285337,7 @@ converges to zero (in the standard topology on the reals) with this tgcn $p |- ( ph -> ( F e. ( J Cn K ) <-> ( F : X --> Y /\ A. y e. B ( `' F " y ) e. J ) ) ) $= ( vx vz wcel cv wral wa wi ctop syl ccn co wf ccnv cima ctopon wb syl2anc - cfv iscn wss ctg ctb topontop eqeltrrd tgclb sylibr bastg sseqtr4d ssralv + cfv iscn wss ctg ctb topontop eqeltrrd tgclb sylibr bastg sseqtrrd ssralv cuni wceq wex eleq2d eltg3 bitrd ciun iunopn sylan9r imaeq2 imauni syl6eq eleq1d imbi2d syl5ibrcom expimpd exlimdv sylbid imp cbvralv syl6ib impbid ex ralrimdva anbi2d ) ADEFUAUBNZGHDUCZDUDZBOZUEZENZBFPZQZWGWKBCPZQAEGUFUI @@ -285356,7 +285356,7 @@ converges to zero (in the standard topology on the reals) with this ( F e. ( ( J CnP K ) ` P ) <-> ( F : X --> Y /\ A. y e. B ( ( F ` P ) e. y -> E. x e. J ( P e. x /\ ( F " x ) C_ y ) ) ) ) ) $= ( cfv wcel wss wa wi syl vz ccnp co wf cv cima wrex wral wb iscnp syl3anc - ctopon ctg ctb ctop topontop eqeltrrd tgclb sylibr sseqtr4d ssralv anim2d + ctopon ctg ctb ctop topontop eqeltrrd tgclb sylibr sseqtrrd ssralv anim2d bastg sylbid eleq2d biimpa tg2 r19.29 sstr expcom reximdv com12 rexlimivw imim2i imp32 ex com23 ralrimdva sylibrd impbid ) AFEGHUBUCOPZIJFUDZEFOZCU EZPZEBUEZPZFWFUFZWDQZRZBGUGZSZCDUHZRZAWAWBWLCHUHZRZWNAGIULOPZHJULOPZEIPZW @@ -285406,7 +285406,7 @@ converges to zero (in the standard topology on the reals) with this ( ( _I |` X ) e. ( J Cn K ) <-> K C_ J ) ) $= ( vx ctopon cfv wcel wa cid cres ccn co ccnv cv cima wral wf syl6bbr wceq wss iscn wf1o f1oi f1of biantrur cnvresid imaeq1i elssuni adantl toponuni - ax-mp cuni ad2antlr sseqtr4d resiima syl5eq eleq1d ralbidva dfss3 bitrd + ax-mp cuni ad2antlr sseqtrrd resiima syl5eq eleq1d ralbidva dfss3 bitrd syl ) ACEFZGZBVBGZHZICJZABKLGZVFMZDNZOZAGZDBPZBATZVEVGCCVFQZVLHVLDVFABCCU AVNVLCCVFUBVNCUCCCVFUDUKUERVEVLVIAGZDBPVMVEVKVODBVEVIBGZHZVJVIAVQVJVFVIOZ VIVHVFVICUFUGVQVICTVRVISVQVIBULZCVPVIVSTVEVIBUHUIVDCVSSVCVPCBUJUMUNCVIUOV @@ -285719,7 +285719,7 @@ converges to zero (in the standard topology on the reals) with this 25-Aug-2015.) $) cnclsi $p |- ( ( F e. ( J Cn K ) /\ S C_ X ) -> ( F " ( ( cls ` J ) ` S ) ) C_ ( ( cls ` K ) ` ( F " S ) ) ) $= - ( ccn co wcel wss wa ccl cfv cima ccnv ctop cntop1 adantr cuni sseqtr4d + ( ccn co wcel wss wa ccl cfv cima ccnv ctop cntop1 adantr cuni sseqtrrd cnvimass wf eqid cnf fssdm cdm cin wceq simpr fdmd sylib dminss syl6eqssr sseqin2 clsss syl3anc imassrn frnd sstrid cncls2i syldan sstrd wfun ffund crn wb clsss3 sylan funimass3 syl2anc mpbird ) BCDGHIZAEJZKZBACLMZMZNBANZ @@ -285761,7 +285761,7 @@ converges to zero (in the standard topology on the reals) with this wi eqid ralrimdva jcad ccnv toponmax cdm cnvimass ad2antlr sseqtrid fveq2 fdm sselpwd imaeq2d imaeq2 fveq2d sseq12d rspcv topontop ad3antlr crn cin ctop wfun ffun funimacnv inss1 syl6eqss clsss syl3anc sstr2 syl5com eqtrd - wb clsss3 syl2anc sseqtr4d funimass3 sylibd syld imdistanda cncls2 impbid + wb clsss3 syl2anc sseqtrrd funimass3 sylibd syld imdistanda cncls2 impbid sylibrd ) CEHIJZDFHIJZKZBCDUCUAJZEFBUBZBALZCMIZIZNZBXCNZDMIZIZOZAEPZQZKZW TXAXBXLWRWSXAXBBCDEFUDUEWTXAXJAXKWTXCXKJZKZXCCUFZOZXAXJUNXOXCEXPXNXCEOWTX CEUGUHWREXPRZWSXNECUIZUJUKXAXQXJXCBCDXPXPUOZULUMSUPUQWTXMXBBURZGLZNZXDIZY @@ -285867,7 +285867,7 @@ converges to zero (in the standard topology on the reals) with this ccn wf iscn simprbda cuni eqid cncnpi adantl wceq toponuni raleqdv mpbird ad2antrr jca simprl wrex cdm cnvimass fdm sseqtrid ssralv simpllr wfn ffn wi ad2antlr elpreima simplbda syl2anc cnpimaex syl3anc wfun ffund simp-4l - simprr toponss sylan sseqtr4d funimass3 rexbidva mpbid expr ralimdva syld + simprr toponss sylan sseqtrrd funimass3 rexbidva mpbid expr ralimdva syld anbi2d impr an32s ctop topontop ad3antrrr eltop2 adantr mpbir2and impbida ) CEIJKZDFIJKZLZBCDUCMKZEFBUDZBANZCDUAMJKZAEOZLZWSWTLZXAXDWSWTXABUBGNZPZC KZGDOZGBCDEFUEZUFXFXDXCACUGZOZWTXMWSWTXCAXLXBBCDXLXLUHUIQUJXFXCAEXLWQEXLU @@ -286328,7 +286328,7 @@ F C_ ( CC X. X ) ) $= space converges to a point in the set. (Contributed by Mario Carneiro, 30-Dec-2013.) $) lmcld $p |- ( ph -> P e. S ) $= - ( ccl cfv wcel syl wceq cuni ccld wss eqid cldss ctopon toponuni sseqtr4d + ( ccl cfv wcel syl wceq cuni ccld wss eqid cldss ctopon toponuni sseqtrrd lmcls cldcls eleqtrd ) ABCFPQQZCABCDEFGHIJKLMNACFUAZHACFUBQRZCUMUCOCFUMUM UDUESAFHUFQRHUMTKHFUGSUHUIAUNULCTOCFUJSUK $. $} @@ -287615,7 +287615,7 @@ require the space to be Hausdorff (which would make it the same as T_3), sseqtrrid wb elpw2g ad2antrr mpbiri unieq eqeq2d pweq ineq1d rexeqdv mpid rspcv wex elfpw ad2antrl simplbi ssrab sseq2 ac6sfi crn frn wfo wfn dffn4 wf ffn fofi syl2an sylanbrc simplrr ciun uniiun eqsstrid ad2antll fniunfv - ss2iun sseqtrd sseqtr4d rspceeqv exlimddv rexlimdvaa syld com23 ralrimdva + ss2iun sseqtrd sseqtrrd rspceeqv exlimddv rexlimdvaa syld com23 ralrimdva unissd sylan2 tgcl baib bitrd sylibrd impbid ) CUDIZDCJZKZLZCUEUFZUAIZDAM ZJZKZDBMZJZKZBUUFNZOUGZPZQZACNZRZUUEUUDJZUUGKZUURUUJKZBUUMPZQZAUUDNZRZUUC UUQUUEUUDUBIZUVDABUUDUURUURUPZUHUIUUCUVDUUOAUVCRZUUQUUCUVBUUOAUVCUUCUUSUU @@ -288156,7 +288156,7 @@ require the space to be Hausdorff (which would make it the same as T_3), K e. Conn ) $= ( vx cconn wcel ccld cfv cin c0 wss wa wceq wne cima syl2anc syl wfo ctop ccn co w3a cpr cntop2 3ad2ant3 cv wo wn df-ne ccnv cuni cdm simpl1 simpl3 - eqid simprl elin1d cnima crn elssuni syl6sseqr simpl2 forn sseqtr4d df-rn + eqid simprl elin1d cnima crn elssuni syl6sseqr simpl2 forn sseqtrrd df-rn syl6sseq sseqin2 sylib simprr eqnetrd imadisj necon3bii sylibr cnclima wf elin2d connclo cnf fdm fof 3eqtr2d imaeq2d foimacnv foima 3eqtr3d syl5bir 3syl expr orrd vex elpr ex ssrdv isconn2 sylanbrc ) BHIZDEAUAZABCUCUDIZUE @@ -288223,7 +288223,7 @@ require the space to be Hausdorff (which would make it the same as T_3), Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) $) connima $p |- ( ph -> ( K |`t ( F " A ) ) e. Conn ) $= ( crest co cconn wcel cuni wfo ccn wss syl syl2anc cima cres wfun wf eqid - cdm cnf ffund fdmd sseqtr4d fores wceq wb ctop cntop2 imassrn frnd sstrid + cdm cnf ffund fdmd sseqtrrd fores wceq wb ctop cntop2 imassrn frnd sstrid crn restuni foeq3 mpbid cnrest ctopon toptopon2 sylib df-ima eqimss2 mp1i cfv cnrest2 syl3anc cnconn ) ADBKLZMNBECBUAZKLZOZCBUBZPZVRVNVPQLNZVPMNJAB VOVRPZVSACUCBCUFZRWAAFEOZCACDEQLNZFWCCUDHCDEFWCGWCUEZUGSZUHABFWBIAFWCCWFU @@ -288342,7 +288342,7 @@ require the space to be Hausdorff (which would make it the same as T_3), wceq eqid clsndisj syl32anc exlimddv simprl2 simprl3 sscls syl2anc sscond n0 sstrd ssdif ax-mp syl6ss disj2 sylibr simprr nconnsubb expr ralrimivva ssv mt2d ex wb simp1 sseq2d biimpa clsss3 syl2an2r 3adant3 connsub mpbird - sseqtr4d ) BCUAGHZACIZBAJUBUCHZUDZBABUEGGZJUBUCHZDKZXCLZMNZEKZXCLZMNZXEXH + sseqtrrd ) BCUAGHZACIZBAJUBUCHZUDZBABUEGGZJUBUCHZDKZXCLZMNZEKZXCLZMNZXEXH LZCXCOZIZUDZXCXEXHUFZIZUGZUHZEBUIDBUIZXBXRDEBBXBXEBHZXHBHZTZTZXNXQYCXNTXP XAWSWTXAYBXNUJYCXNXPXAUGYCXNXPTZTZAXEBXHCWSWTXAYBYDUKZWSWTXAYBYDULZXBXTYA YDUMZXBXTYAYDUNZYEFKZXFHZXEALMNZFYEXGYKFUOXGXJXMXPYCUPFXFVOUQYEYKTZBURHZA @@ -288399,7 +288399,7 @@ require the space to be Hausdorff (which would make it the same as T_3), ( ( J e. ( TopOn ` X ) /\ A e. X ) -> S e. ( Clsd ` J ) ) $= ( ctopon cfv wcel wa ccld ccl wss crest cconn cuni ctop syl2an2r syl3anc co topontop cpw crab ssrab2 sspwuni mpbi eqsstri toponuni adantr sseqtrid - cv wceq eqid clsss3 sseqtr4d sscls conncompid sseldd conncompconn clsconn + cv wceq eqid clsss3 sseqtrrd sscls conncompid sseldd conncompconn clsconn simpl a1i conncompss wb iscld4 mpbird ) DEGHIZBEIZJZCDKHIZCDLHHZCMZVIVKEM BVKIDVKNTOIZVLVIVKDPZEVGDQIZVHCVNMZVKVNMEDUAZVIECVNCBAUKZIDVRNTOIJZAEUBZU CZPZEFWAVTMWBEMVSAVTUDWAEUEUFUGZVGEVNULVHEDUHUIZUJZCDVNVNUMZUNRWDUOVICVKB @@ -289052,7 +289052,7 @@ require the space to be Hausdorff (which would make it the same as T_3), sylbird mpand sylbid rexlimdva syl5 embantd impcomd exlimdv exp4b com23 mpd expr impr imp mtoi cuni toponuni syl sseqtrid eqid 1stcelcls mtbird c1stc syl2anc ctop topontop eleqtrd elcls syl3anc mtbid wfun ffund fdmd - cdm toponss sseqtr4d funimass3 df-ss sylib sseq1d bitr4d bitr4i syl6bbr + cdm toponss sseqtrrd funimass3 df-ss sylib sseq1d bitr4d bitr4i syl6bbr nne inssdif0 anbi2d rexbidva rexanali syl6bb mpbird ralrimiva mpbir2and iscnp adantr impbida ) ADBEFUCUDOPZGHDUEZQGCUFZUEZUUSBEUGOUHZRZDUUSUIZB DOZFUGOUHZUJZCUKZRZAUUQRZUURUVGAEGULOPZFHULOPZRUUQUURAUVJUVKJKUMUVJUVKU @@ -290194,7 +290194,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually crab kgenval ssrab2 syl6eqss sspwuni sylib sylan9ssr iunin2 uniiun ineq2i ciun incom 3eqtr2i cmptop ad2antll simplr sselda simplrr eqeltrrid iunopn kgeni alrimiv inss1 elssuni ad2antrl ssidd elpwi sseqin2 resttopon sylan2 - toponmax syl eqeltrd eqssd sseqtr4d sstrid inindir simplrl simprr syl3anc + toponmax syl eqeltrd eqssd sseqtrrd sstrid inindir simplrl simprr syl3anc ex inopn eqeltrid ralrimivva cvv fvex istopg ax-mp sylanbrc istopon ) ABU AFZGZAUBFZUCGZBXDUDZUEZXDXBGXCCHZXDIZXHUDZXDGZJZCUFZXHDHZKZXDGZDXDLCXDLZX EXCXLCXCXIXKXCXIMZXKXJBIZAEHZUGUHZUIGZXJXTKZYAGZJZEBUJZLZXIXCXJXFBXHXDUKX @@ -290337,7 +290337,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually 21-Mar-2015.) $) llycmpkgen2 $p |- ( ph -> J e. ran kGen ) $= ( vz wcel cfv wss cv wa wceq syl cdif cin syl2anc wb ctop ckgen wrex wral - vu crn crest co ccmp csn cnei cuni elssuni adantl kgenuni adantr sseqtr4d + vu crn crest co ccmp csn cnei cuni elssuni adantl kgenuni adantr sseqtrrd sselda adantlr syldan ad3antrrr difss ntropn sylancl simprl neii1 syl3anc cnt inopn cun simplr ntrss2 snssd neiint mpbid snss sylibr sseldd simpllr vex elind simprr kgeni cvv resttop inss2 restuni sseqtrid eqid isopn3 a1i @@ -290446,7 +290446,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually ssun1 lmss mpbid ffvelrnda simprl eldifbd eldifd cin difin wceq frn snssd difss2d unssd restuni difeq1d syl5eqr incom ccmp simplr 1stckgenlem kgeni eqeltrid opncld eqeltrd lmcld exlimdv sylbid iscld4 mpbird elssuni adantl - ex ssrdv kgenuni syl sseqtr4d isopn2 iskgen2 sylanbrc ) AUBEZAUCEZAUDFZAG + ex ssrdv kgenuni syl sseqtrrd isopn2 iskgen2 sylanbrc ) AUBEZAUCEZAUDFZAG AUDUEEAUFZXTBYBAXTBHZYBEZYDAEZXTYEIZYFAJZYDKZAUGFEZYGYJYIAUHFFZYIGZYGCYKY IYGCHZYKEZLYIDHZMZYOYMAUIFULZIZDUMZYMYIEZXTYNYSNZYEXTYIYHGZUUAYHYDUJZYMYI DAYHYHSZUKUNOYGYRYTDYGYRYTYGYRIZYMYHYDUUEAYHUOFEZYQYMYHEUUEYAUUFYGYAYRXTY @@ -290512,7 +290512,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually A. z e. Comp A. g e. ( z Cn J ) ( F o. g ) e. ( z Cn K ) ) ) ) $= ( vk cfv wcel wa ccn co cv ccmp cres wi wral wss wceq ctopon ckgen wf cpw crest ccom kgencn crn rncmp adantl cuni simprr eqid cnf frn 3syl toponuni - ad3antrrr sseqtr4d rnex sylibr oveq2 eleq1d reseq2 oveq1d eleq12d imbi12d + ad3antrrr sseqtrrd rnex sylibr oveq2 eleq1d reseq2 oveq1d eleq12d imbi12d vex elpw rspcv syl mpid wb simplll ssidd cnrest2 syl3anc mpbid cnco cores ssid ax-mp eleq1i syl6ib syld ralrimdvva oveq1 eleq2d raleqbidv cid elpwi ex resabs1d sseqtrd cnrest syl2anc eqeltrrd coeq2 coires1 syl9r ralrimdva @@ -290999,7 +290999,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually Carneiro, 3-Feb-2015.) $) ptopn $p |- ( ph -> X_ k e. A S e. ( Xt_ ` F ) ) $= ( vg vy vz vx cv cfv wcel wceq wfn wral cuni cdif cfn wrex w3a wa wex cab - cixp cpt ctg ctb wss ctop wf eqid ptbas syl2anc bastg ffnd ptval sseqtr4d + cixp cpt ctg ctb wss ctop wf eqid ptbas syl2anc bastg ffnd ptval sseqtrrd syl elptr2 sseldd ) AMQZBUANQZVHRZVIERZSNBUBVJVKUCTNBOQUDUBOUEUFUGPQNBVJU KTUHMUIPUJZEULRZDBCUKAVLVLUMRZVMAVLUNSZVLVNUOABFSZBUPEUQVOHIPNOBVLMEFVLUR ZUSUTVLUNVAVEAVPEBUAVMVNTHABUPEIVBPNOBVLMEFVQVCUTVDAPNOBVLCMDEFGVQHJKLVFV @@ -291299,7 +291299,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually simprrl sylib simpld clsndisj syl32anc n0 simplrr simprlr simprd exdistrv opelxpi inxp syl6eleqr elin1d sselda sylan2 elin2d adantl inelcm exlimdvv simprrr ex syl5bir mp2and expr rexlimdvva syl5 expd sylbid txtopon clsss3 - ralrimiv syl sseqtr4d adantrr adantrl opelxpd eleqtrd elcls mpbird syl5bi + ralrimiv syl sseqtrrd adantrr adantrl opelxpd eleqtrd elcls mpbird syl5bi syl3anc relssdv eqssd ) CEUALZMZDFUALZMZNZAEOZBFOZNZNZABPZCDUEUFZUGLLZACU GLLZBDUGLLZPZYNYTYPUBLMZYOYTOZYQYTOYNYRCUBLMZYSDUBLMZUUAYNCUHMZACUIZOZUUC YGUUEYIYMECUJQZYNAEUUFYJYKYLUKYGEUUFRYIYMECULQZUMZACUUFUUFUNZUPSYNDUHMZBD @@ -291481,7 +291481,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually ptuni 3adant3 syl6reqr mpteq1d ccnv cima cpt eqeltrid ffvelrn 3adant1 wfn wral pttop vex elixp simprbi fveq2 unieqd eleq12d rspcva sylan2 3ad2antl3 fmpttd feq2d mpbird cdif cfn wrex wex cab wss ctg ctb ptbas bastg syl ffn - eqid ptval syl5eq sseqtr4d adantr ptpjpre2 sseldd expr ralrimiv 3impa jca + eqid ptval syl5eq sseqtrrd adantr ptpjpre2 sseldd expr ralrimiv 3impa jca iscn2 syl21anbrc eqeltrd ) BFOZBPCUBZDBOZUCZAGDAQZRZUDAJBJQZCRZUEZUFZXEUD ZEDCRZUGUHZXCAGXIXEXCXIEUEZGWTXAXIXMSXBJBCEFIUIUJHUKZULXCEPOXKPOZGXKUEZXJ UBZXJUMUAQZUNZEOZUAXKUTZTXJXLOXCECUORZPIWTXAYBPOXBBCFVAUJUPXAXBXOWTBPDCUQ @@ -291605,7 +291605,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually ineq12d rexbidv acni3 ixpin ineq1i eqtr4i neeq1i 3imtr3i sylan2br sylan ne0i exlimiv expr syldan eleq2 ineq1 imbi12d syl5ibrcom expimpd exlimdv 3adantr3 syl5bi ralrimiv ctg fmpttd ffnd ptval syl5eq pttopon ctb ptbas - clsss3 sseqtr4d sselda elcls3 mpbird eqelssd ) AUAEBDUFZFUGPPZEBDCUGPPZ + clsss3 sseqtrrd sselda elcls3 mpbird eqelssd ) AUAEBDUFZFUGPPZEBDCUGPPZ UFZAUWCFUHPZQUVTUWCRZUWAUWCRAUWCEBCULZUIPZUHPUWDABUWBECGJAEUJZBQZSZCHUK PZQZCUMQZKHCUNUOZUWJUWMDCUPZRZUWBCUHPQUWNUWJDHUWOLUWJUWLHUWOUQKHCURUOZU SZDCUWOUWOUTZVCVAVBFUWGUHIVDVEADUWBRZEBTUWEAUWTEBUWJUWMUWPUWTUWNUWRDCUW @@ -291761,7 +291761,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually fmpttd xkobval abeq2i ad5ant15 simplr imaeq2d 0ss eqsstri syl6eqss imaeq1 wrex ima0 sseq1d sylanbrc ralrimiva rabid2 sylibr simpllr toponmax adantr elrab syl eqeltrrd wne cin cif ifnefalse ad2antlr eleq2d snss syl6bb cres - vex df-ima simplrl ad2antrr elpwid toponuni ad5antr sseqtr4d rneqd syl5eq + vex df-ima simplrl ad2antrr elpwid toponuni ad5antr sseqtrrd rneqd syl5eq xpssres eqtrd biantrurd 3bitr2d syl6bbr rabbi2dva simplrr toponss syl2anc rnxp bitr3d sseqin2 eqtr3d eqeltrd pm2.61dane imaeq2 mptpreima syl5ibrcom sylib syl6eq eleq1d expimpd rexlimdvva ralrimiv cvv simpr ovex pwex xkotf @@ -291810,7 +291810,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually simpr nfop nfeq eqeq12d rspc sylc eleq1d adantr ad2antrr simplrl cnpimaex fveq2 simprl simplrr simprr jca opelxp reeanv 3imtr4g sylbid cin an4 elin ex biimpri a1i simpl toponss syl2an ssinss1 adantl sselda elin1d wfun cdm - ffund sseqtr4d sseldd funfvima mpd elin2d eqeltrd funimass4 mpbird syldan + ffund sseqtrrd sseldd funfvima mpd elin2d eqeltrd funimass4 mpbird syldan fdmd adantlr xpss12 sstr2 syl2im anim12d syl5bi ctop topontop inopn 3expb syl sylan eleq2 vex jctild expimpd imaeq2 sseq1d anbi12d rspcev syl6 expd rexlimdvv syld ralrimivva xpex rgen2w sseq2 anbi2d rexbidv ralrnmpo ax-mp @@ -291887,7 +291887,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually sylc cint ad2antrr toponuni ineq1d topontop ad2antrl simpl sylanbrc cdm mpbird wfun funmpt dmmptg funimass4 sylancr nfel1 eleq1d cbvral expimpd ssralv frn wfo wfn dffn4 fofi rintopn ralimi ad2antll ralrn elrint cdif - ffn simp-4l simpllr simplr ffvelrnd toponss elin1d sseqtr4d inss1 mpsyl + ffn simp-4l simpllr simplr ffvelrnd toponss elin1d sseqtrrd inss1 mpsyl syl6bb fnfvelrn intss1 sstrid r19.26 sylan biimpd ralimia sylbir syl6an wi sylbid ralimdaa impr eldifi syl2an ralbidv cun inundif raleqi ralunb bitr3i ralcom mptexg syl2anr ralbidva syl5bb ralrimivw sseqtrrid rspcev @@ -292617,7 +292617,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually ffn frnd vex fvex iunex elpw sylibr simplr sseldi fss sylancl ffvelrn syl inss2 iunfi elind simprl uniiun syl6eq xpeq2d xpiundi simprrr xpeq2 fveq2 unieqd sseq12d cbvralv ss2iun ffvelrnda elpwi uniss ad3antrrr iunss eqssd - sseqtr4d iuncom4 rspceeqv expr exlimdv expimpd rexlimdva mpd ) AGUAUEZOZP + sseqtrrd iuncom4 rspceeqv expr exlimdv expimpd rexlimdva mpd ) AGUAUEZOZP ZXREUFZQUGZUBUEZUHZFUCUEZUIZYEYCUJZOZRZUCXRUKZSZUBULZSZUADUFZQUGZUMZFGUIZ BUEZOZPBYBUMZADVDTZUDUEZYETYFYSRZBYBUMSUCDUMZUDGUKYPKAUUDUDGAUUBGTZSBUCUU BCDEFGHIACVDTUUEJUNAUUAUUEKUNAECDUOUPRUUELUNAYQEOZPZUUEMUNAUUEUQURUSUUCYI @@ -293108,7 +293108,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually snssd eqtrd sneqd mpteq1d cnveqd imaeq1d ralrimivw jca ralrimiva mpoeq123 fssd sylancr rneqd uneq12d fveq2d syl5eq oveq1d firest fveq2i fvex tgrest ovex mp2an eqtri syl6eqr wss xkotop cin mpoexga sylan rnexg unexg restval - snex syl sylancl wo elun wb elmapg syl2anr syl5ibr ssrdv sseqtr4d sseqin2 + snex syl sylancl wo elun wb elmapg syl2anr syl5ibr ssrdv sseqtrrd sseqin2 cnf elsni sylib xkouni eqeltrd eleq1d bitrd ccmp sylan2b eqsstrd elpreima rnmpo abeq2i crab cres cnvresima resmptd syl5eqr rgenw fnmpt fveq1 sselda mp1i fvmpt snss elmapi 3syl simplrl fnsnfv sseq1d syl5bb pm5.32da abbi2dv @@ -293164,7 +293164,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually eqidd ctopon toptopon2 cndis ancoms sylanb rabeqdv mpoeq123dv rneqd rnmpo rabbidva syl6eq cif cixp elmapi wral wi eleq2 imbi2d bibi1d simprl elin1d elpwid sselda 2thd imbi1d wn ffvelrn pm2.21 ifbothda ralbidv2 wfn ffn vex - elixp baib syl wfun cdm ffun sseqtr4d funimass4 3bitr4d rabbi2dva elssuni + elixp baib syl wfun cdm ffun sseqtrrd funimass4 3bitr4d rabbi2dva elssuni fdm ad2antll ssid sseq1 ifboth sylancl ralrimivw ss2ixp cvv simplr uniexg ex ad2antrr ixpconstg eqsstrd eqtrd sseqtrd sseqin2 eqtr3d elin2d simplrr sylib topopn ad3antrrr fvconst2g ad4ant14 eleqtrrd eldifn iffalsed eldifi @@ -293271,7 +293271,7 @@ arbitrary neighborhoods (such as "locally compact", which is actually ( vx vk vv vy vh co cv wcel cima wa syl2anc eqid ccn ccom cmpt cxko crest wf ccnv ccmp cuni cpw crab wss cmpo crn wral adantr cnco fmpttd wceq wrex simpr xkobval abeq2i ad3antrrr imaeq1 imaco syl6eq sseq1d elrab3 syl wfun - wb cdm cnf ffund imassrn frnd fdmd sseqtr4d funimass3 bitrd rabbidva ctop + wb cdm cnf ffund imassrn frnd fdmd sseqtrrd funimass3 bitrd rabbidva ctop sstrid ad2antrr cntop1 simplrl elpwid simplrr cnima xkoopn eqeltrd imaeq2 mptpreima eleq1d syl5ibrcom expimpd rexlimdvva syl5bi ralrimiv cvv ctopon cfv xkotopon ovex pwex cxp xkotf frn ax-mp a1i cfi cntop2 xkoval subbascn @@ -293317,7 +293317,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cnt wss ccmp wcel wa wral wex cfn cin wrex cop ccn crab cxko ctx imacmp wel syl2anc w3a cnlly adantr cnima ccom imaco eqsstrrid wfun cdm wb cnf eqid ffun 3syl crn imassrn frn sstrid fdm funimass3 mpbid sselda nlly2i - sseqtr4d syl3anc cvv nllytop syl ad3antrrr imaexg simprl sstrd syl22anc + sseqtrrd syl3anc cvv nllytop syl ad3antrrr imaexg simprl sstrd syl22anc ctop eleq2 anbi12d rspcev syl12anc mpd ciun cxp ad2antrr xkotop simprrl sylib simprrr ralimi xkoopn eqsstrd iunss ex ralimdv sylc imaeq1 sseq1d elrabd ss2iun elrab vex coex ssrab2 mp2an elrestr simprr1 simpllr elind @@ -293989,7 +293989,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= ( vz vk vv vw wcel wa co cv wss wceq syl cvv vf vt vr ctopon cfv ctx cxko ccn wf ccnv cima ccmp cuni cpw crab cmpo crn wral cop cmpt simplr cnmptid crest simpll simpr cnmptc cnmpt1t fmptd wrex xkobval abeq2i csn cxp sylan - eqid wb imaeq1 sseq1d elrab3 wfun funmpt simplrl elpwid toponuni sseqtr4d + eqid wb imaeq1 sseq1d elrab3 wfun funmpt simplrl elpwid toponuni sseqtrrd cdm simprd adantr dmmptg opex a1i mprg syl6sseqr funimass4 sylancr sselda wel opeq1 fvmpt eleq1d vex opeq2 ralsn syl6bbr ralbidva dfss3 eleq1 ralxp weq bitri 3bitrd rabbidva sneq xpeq2d elrab ctop ad2antrr topontop adantl @@ -294109,7 +294109,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= A e. X ) ) -> ( R " { A } ) e. K ) $= ( vy ctop wcel wa cima cuni wss adantr eqid syl2anc wceq ad2antll ctopon co ctx csn cv cop cmpt ccnv crab nfv nfcv nfrab1 cxp txtop simprl eltopss - txuni sseqtr4d imass1 syl xpimasn sseqtrd sseld pm4.71rd wb elimasng elvd + txuni sseqtrrd imass1 syl xpimasn sseqtrd sseld pm4.71rd wb elimasng elvd cvv anbi2d bitrd rabid syl6bbr eqrd mptpreima syl6eqr ccn toptopon biimpi cfv ad2antlr ad2antrr simprr cnmptc cnmptid cnmpt1t cnima eqeltrd ) CHIZD HIZJZBCDUATZIZAEIZJZJZBAUBZKZGDLZAGUCZUDZUEZUFBKZDWMWOWRBIZGWPUGZWTWMGWOX @@ -294126,7 +294126,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= imasncld $p |- ( ( ( J e. Top /\ K e. Top ) /\ ( R e. ( Clsd ` ( J tX K ) ) /\ A e. X ) ) -> ( R " { A } ) e. ( Clsd ` K ) ) $= ( vy ctop wcel wa co ccld cfv cima cuni wss eqid syl wceq ad2antll ctx cv - csn cop cmpt ccnv crab nfv nfcv nfrab1 simprl cldss txuni adantr sseqtr4d + csn cop cmpt ccnv crab nfv nfcv nfrab1 simprl cldss txuni adantr sseqtrrd cxp imass1 xpimasn sseqtrd sseld pm4.71rd cvv elimasng anbi2d bitrd rabid elvd syl6bbr eqrd mptpreima syl6eqr ccn ctopon toptopon ad2antlr ad2antrr wb biimpi simprr cnmptc cnmptid cnmpt1t cnclima syl2anc eqeltrd ) CHIZDHI @@ -294150,7 +294150,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= cop ccnv ctx csn ccn cuni toptopon biimpi ad2antlr ad2antrr simprr cnmptc cnmptid cnmpt1t simprl wceq txuni adantr eqid cncls2i syl2anc crab nfrab1 nfv nfcv imass1 syl xpimasn sseld pm4.71rd cvv elimasng elvd anbi2d bitrd - rabid syl6bbr eqrd mptpreima syl6eqr fveq2d txtop clsss3 sseqtr4d 3sstr4d + rabid syl6bbr eqrd mptpreima syl6eqr fveq2d txtop clsss3 sseqtrrd 3sstr4d wb ) CJKZDJKZLZBEFUAZMZAEKZLZLZIFAIUBZUDZUCZUEZBNZDOPZPZXABCDUFQZOPPZNZBA UGZNZXCPXFXHNZWQWTDXEUHQKBXEUIZMZXDXGMWQIAWRDCDFWKDFRPKZWJWPWKXMDFHUJUKUL ZWQIADCFEXNWJCERPKZWKWPWJXOCEGUJUKUMWLWNWOUNUOWQIDFXNUPUQWQBWMXKWLWNWOURZ @@ -294366,7 +294366,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= (Contributed by Mario Carneiro, 9-Apr-2015.) $) qtopkgen $p |- ( ( J e. ran kGen /\ F Fn X ) -> ( J qTop F ) e. ran kGen ) $= - ( vx ckgen crn wcel wa co ctop cfv wss cuni wceq syl sseqtr4d syl2anc ccn + ( vx ckgen crn wcel wa co ctop cfv wss cuni wceq syl sseqtrrd syl2anc ccn sylib cqtop kgentop qtoptop sylan cv ccnv cima elssuni adantl adantr eqid wfn kgenuni wfo simpll simplr dffn4 qtopuni ctopon qtopid kgencn3 eleqtrd toptopon cnima sylancom wb elqtop2 mpbir2and ex ssrdv iskgen2 sylanbrc ) @@ -294387,7 +294387,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= wfn simpr simpl3l elin2d elind basis2 syl22anc adantr inss1 simp2l sstrid sselda f1ocnvfv2 simprrr syl6ss cdm cnvimass f1odm sseqtrid sstrd simprrl eqeltrrd crn imassrn forn wf1 f1of1 f1imacnv eqeltrd mpbir2and wfun fnfun - simprl inpreima sseqtr4d funimass3 mpbird inex1 elpw2 sylibr rexlimddv ex + simprl inpreima sseqtrrd funimass3 mpbird inex1 elpw2 sylibr rexlimddv ex vex elunii ssrdv 3expib sylbid ralrimivv cvv ovex isbasisg ax-mp ) BIJZCD AUCZKZFLZGLZMZBAUDUEZYDUBZMZUFZNZGYEUGFYEUGZYEIJZYAYIFGYEYEYAYBYEJZYCYEJZ KZYBDNZAUHZYBOZBJZKZYCDNZYPYCOZBJZKZKZYIXTXSCDAUIZYNUUDPCDAUJZXSUUEKYLYSY @@ -294414,7 +294414,7 @@ z C_ ( `' F " { h e. ( R Cn T ) | ( h " K ) C_ V } ) ) ) $= ( ( topGen ` J ) qTop F ) = ( topGen ` ( J qTop F ) ) ) $= ( vx vy vz vw ctb wcel wa cfv cv wss cima cuni wb syl syl2anc wf1o ctg co cqtop ccnv cpw cin wral wfun cdm f1ocnv ad2antlr simpr crn df-rn wfo wceq - f1ofun f1ofo forn syl5eqr sseqtr4d funimass4 simprl elin1d elqtop2 sylan2 + f1ofun f1ofo forn syl5eqr sseqtrrd funimass4 simprl elin1d elqtop2 sylan2 dfss3 ad3antrrr mpbid simprd elin2d elpwid imass2 elpwd elind wfn simp-4r wrex f1ofn sstrd simprr fnfvima syl3anc eleq2 rspcev rexlimdvaa funimass2 ad2antrr wf1 f1of1 elssuni syl6sseqr f1imacnv eqeltrd mpbir2and vex elpw2 @@ -294562,7 +294562,7 @@ K C_ ( J qTop F ) ) $= ( wcel cfv wceq wss syl syl2anc wa ad2antrr vx cqtop co crest cres ctopon ccn crn cuni wfn wfo fofn qtopid ccnv cima cnvimass fndm sseqtrid eqsstrd toponuni sseqtrd eqid cnrest wb qtoptopon df-ima imaeq2d foimacnv syl5eqr - cdm eqtrd eqimss cnrest2 syl3anc mpbid resttopon qtopss cv fnfun sseqtr4d + cdm eqtrd eqimss cnrest2 syl3anc mpbid resttopon qtopss cv fnfun sseqtrrd fores foeq3 elqtop3 cin cnvresima imass2 adantl adantr df-ss sylib syl5eq wfun eleq1d ccld simplrl ctop cvv topontop toponmax fornex ssexd restopn2 sylc sstrd sylan simprbda adantrl mpbir2and elrestr eqeltrrd cdif difeq1d @@ -294769,7 +294769,7 @@ indistinguishability map (in the terminology of ~ qtoprest ). kqsat $p |- ( ( J e. ( TopOn ` X ) /\ U e. J ) -> ( `' F " ( F " U ) ) = U ) $= ( vz ctopon cfv wcel wa ccnv cima cv wb wfn syl adantr wceq kqffn kqfvima - elpreima 3expa biimprd expimpd sylbid ssrdv cdm cin toponss fndm sseqtr4d + elpreima 3expa biimprd expimpd sylbid ssrdv cdm cin toponss fndm sseqtrrd wss sseqin2 sylib dminss syl6eqssr eqssd ) EFIJZKZCEKZLZDMDCNZNZCVCHVECVC HOZVEKZVFFKZVFDJVDKZLZVFCKZVAVGVJPZVBVADFQZVLABDEUTFGUAZFVFVDDUCRSVCVHVIV KVCVHLVKVIVAVBVHVKVIPABVFCDEFGUBUDUEUFUGUHVCCDUIZCUJZVEVCCVOUNVPCTVCCFVOC @@ -294799,7 +294799,7 @@ indistinguishability map (in the terminology of ~ qtoprest ). ( cfv wcel wa cima wb adantr cdif wn c0 cin wss wceq adantl vz ctopon wfn ccld ccnv cv kqffn elpreima syl wi noel elin incom cdm cuni eqid toponuni cldss fndm eqtrd sseqtrd dfss4 sylib imaeq2d ineq2d simpll difeq1d cldopn - sseqtr4d eqeltrd kqdisj syl2anc eqtr3d syl5eq eleq2d syl5bbr mtbiri imnan + sseqtrrd eqeltrd kqdisj syl2anc eqtr3d syl5eq eleq2d syl5bbr mtbiri imnan sylibr eldif baibr simpr kqfvima syl3anc bitrd sylibd sylbid ssrdv dminss con1bid expimpd sseqin2 syl6eqssr eqssd ) EFUBHZIZCEUDHIZJZDUEDCKZKZCWRUA WTCWRUAUFZWTIZXAFIZXADHZWSIZJZXACIZWPXBXFLZWQWPDFUCZXHABDEWOFGUGZFXAWSDUH @@ -295023,7 +295023,7 @@ topologically distinguishable points are separated (there is an open set ctopon ccl wrex ccld cpw cin wral crn kqtopon adantr topontop ccnv simplr ccn kqid ad2antrr simprl cnima simprr elin1d cnclima elin2d elpwi nrmsep3 co imass2 syl13anc simplll kqopn simprrl wfun kqffn fnfun cuni eqid cldss - wi wfn toponuni sseqtr4d funimass1 mpd ad2antrl clscld kqcld sscls clsss2 + wi wfn toponuni sseqtrrd funimass1 mpd ad2antrl clscld kqcld sscls clsss2 elssuni simprrr cdm clsss3 fndm eqtrd funimass3 mpbird sstrd sseq2 sseq1d wb fveq2 anbi12d rspcev syl12anc rexlimddv ralrimivva isnrm sylanbrc ) DE UDKZLZDUALZMZDUBKZUCLZGNZHNZOZXRXOUEKZKZINZOZMZHXOUFZGXOUGKZYBUHZUIZUJIXO @@ -295050,7 +295050,7 @@ topologically distinguishable points are separated (there is an open set ctopon ccl wrex ccld cpw wral topontop adantr simplr simpll simprl simprr cin kqopn elin1d kqcld elin2d imass2 nrmsep3 syl13anc ccnv ccn co simplll elpwi kqid cnima simprrl wfun cdm wb wfn kqffn fnfun cuni eqid cldss fndm - toponuni eqtrd sseqtr4d funimass3 kqtopon elssuni ad2antrl clscld cnclima + toponuni eqtrd sseqtrrd funimass3 kqtopon elssuni ad2antrl clscld cnclima mpbid sscls clsss2 simprrr kqsat sseqtrd sstrd sseq2 fveq2 sseq1d anbi12d crn rspcev syl12anc rexlimddv ralrimivva isnrm sylanbrc ) DEUDKZLZDUAKZUB LZMZDUCLZGNZHNZOZXPDUEKZKZINZOZMZHDUFZGDUGKZXTUHZUPZUIIDUIDUBLXJXNXLEDUJU @@ -295290,7 +295290,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario ( wcel wss cima cnt cfv ccnv ccn adantr wceq cnntri syl2anc syl imacnvcnv co chmeo wa cuni hmeocn crn imassrn wf1o wfo eqid hmeof1o f1ofo forn 3syl sseqtrid wf1 f1of1 f1imacnv sylancom fveq2d sseqtrd wfun wi f1ofun cntop2 - ctop ntrss3 sseqtr4d funimass1 mpd hmeocnvcn sylan fveq2i 3sstr3g eqssd ) + ctop ntrss3 sseqtrrd funimass1 mpd hmeocnvcn sylan fveq2i 3sstr3g eqssd ) BCDUATGZAEHZUBZBAIZDJKZKZBACJKZKZIZVQBLZVTIZWBHZVTWCHZVQWEWDVRIZWAKZWBVQB CDMTGZVRDUCZHZWEWIHVOWJVPBCDUDNZVQBUEZVRWKBAUFVQEWKBUGZEWKBUHWNWKOVOWOVPB CDEWKFWKUIZUJNZEWKBUKEWKBULUMZUNZVRBCDWKWPPQVQWHAWAVOVPEWKBUOZWHAOVQWOWTW @@ -295508,7 +295508,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario cntop2 ccnv simpll simprl cnima cuni wfn wf1o eqid hmeof1o ad2antlr f1ofn f1ocnv 3syl elssuni ad2antrl simprr fnfvima syl3anc regsep simpllr sseldd hmeoima simprrl elpreima mpbir2and imacnvcnv syl6eleq hmeocls simprrr cdm - wb wfun f1ofun regtop clsss3 f1odm sseqtr4d funimass3 eqsstrd eleq2 fveq2 + wb wfun f1ofun regtop clsss3 f1odm sseqtrrd funimass3 eqsstrd eleq2 fveq2 mpbird sseq1d anbi12d rspcev syl12anc rexlimddv ralrimivva isreg sylanbrc expcom exlimiv sylbi ) ABUAUBABUCHZUDUEZAIJZBIJZUFZABUGXKCKZXJJZCUHXNCXJU IXPXNCXLXPXMXLXPLZBMJZDKZEKZJZXTBUJNZNZFKZOZLZEBUKZDYDULFBULXMXQXOABUMHJZ @@ -295532,7 +295532,7 @@ Kolmogorov quotient is regular Hausdorff (T_3). (Contributed by Mario chmph wbr chmeo wne hmph wex ccl wrex ccld cpw cin wral ccn hmeocn adantl c0 n0 cntop2 ccnv simpll adantr simprl cnima simprr elin1d cnclima elin2d elpwid imass2 nrmsep3 syl13anc simpllr hmeoima simprrl wfun crn cuni wf1o - eqid hmeof1o f1ofun cldss wfo f1ofo forn 3syl sseqtr4d funimass1 ad2antrl + eqid hmeof1o f1ofun cldss wfo f1ofo forn 3syl sseqtrrd funimass1 ad2antrl elssuni hmeocls simprrr wb nrmtop ad3antrrr clsss3 f1odm funimass3 mpbird mpd cdm eqsstrd sseq2 sseq1d anbi12d rspcev syl12anc rexlimddv ralrimivva fveq2 isnrm sylanbrc expcom exlimiv sylbi ) ABUAUBABUCHZUPUDZAIJZBIJZKZAB @@ -298438,7 +298438,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the elfiun fmfnfmlem1 fmfnfmlem3 eleq2d fmfnfmlem2 syl5bi sylbid syl6bb sylan jcad elv fbssfi ad3antrrr cfm co w3a cfg filtop 3jca ssfg imaelfm adantrr sselda jca filin 3expa simprr elin fvelima simplrr simprl syl2an ad2antrr - wel ssel2 fbelss fdmd sseqtr4d fvimacnv biimpd impr ad2ant2rl elind inss2 + wel ssel2 fbelss fdmd sseqtrrd fvimacnv biimpd impr ad2ant2rl elind inss2 sstri funfvima2 sylc anassrs expr eleq1 rexlimdva imbi1d syl5ibrcom ssrdv imbi12d syl5ibcom syld impd adantrl sstrd filss syl13anc exp32 rexlimdvaa ineq2 imaeq2d imp syldan rexlimdvva 3jaod rexlimdv impcomd impbid ) ACULZ @@ -298485,7 +298485,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the wf foima 3syl filtop fgcl eqid imaelfm syl31anc eqeltrrd sseldd rnelfmlem ffn unssd ssun1 fbasne0 ssn0 sylancr cin wral wb cvv elrnmpt elv ad2antrr 0nelfil adantr 3jca ssfg sselda syl2anc filin 3expa sylan eleq1 syl5ibcom - w3a jca mtod wex neq0 elin wi wfun ffun fvelima ad3antrrr fbelss sseqtr4d + w3a jca mtod wex neq0 elin wi wfun ffun fvelima ad3antrrr fbelss sseqtrrd fdmd fvimacnv inelcm adantl sylbid imbi1d rexlimdva syld impd exlimdv mpd syl5bi ineq2 neeq1d syl5ibrcom expimpd ralrimivv fbunfip mpbird mpbir3and ex fsubbas unexg ssfii unssad sstrd syl3anc fmfnfmlem4 bitr4d eqrdv eqtrd @@ -298730,7 +298730,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the wb ctopon cfil cflim csn cnei elflim dfss3 ctop topontop ad2antrr opnneip co 3expb expr com23 cnt simpr cuni wceq toponuni eleqtrd snssd eqid neii1 simplr neiint syl3anc mpbid snssg ad2antlr mpbird syl2anc imbi12d simpllr - ntropn eleq2 mpid ntrss2 sseqtr4d filss 3exp2 com24 syl3c impbid pm5.32da + ntropn eleq2 mpid ntrss2 sseqtrrd filss 3exp2 com24 syl3c impbid pm5.32da syld syl5bb bitrd ) DEUAGHZCEUBGHZIZBDCUCULHBEHZBUDZDUEGGZCJZIWLBAKZHZWPC HZLZADMZIBCDEUFWKWLWOWTWOFKZCHZFWNMZWKWLIZWTFWNCUGXDXCWTXDXCWSADXDWPDHZIW QXCWRXDXEWQXCWRLZXDXEWQIZIWPWNHZXFXDDUHHZXGXHWIXIWJWLEDUIZUJZXIXEWQXHBDWP @@ -298916,7 +298916,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( F e. ( Fil ` X ) /\ S e. F /\ A e. ( J fLim F ) ) ) $= ( vx vy cfv wcel wss w3a c0 wne 3ad2ant1 syl wb cvv wa syl2anc ctopon ccl cfil cflim csn cnei cun cfi cfg cfbas cpw cuni ctop topontop eqid neisspw - co wn toponuni pweqd sseqtr4d toponmax elpw2g biimpar 3adant3 snssd unssd + co wn toponuni pweqd sseqtrrd toponmax elpw2g biimpar 3adant3 snssd unssd wceq ssun2 simp2 ssexd snnzg ssn0 sylancr cv cin wral wi sseqtrd neindisj simp3 expr syl21anc imp elsni ineq2d neeq1d syl5ibrcom ralrimiv ralrimiva clsss3 sseldd eleqtrrd 3ad2ant3 neifil syl3anc filfbas ne0i cls0 neeqtrrd @@ -299083,7 +299083,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( ctopon cfv wcel cfil co wss wa cv wral wb 3ad2ant1 syl3anc w3a cflf cfm wf cflim csn cnei cima wrex flfval eleq2d simp1 toponmax filfbas 3ad2ant2 cfbas simp3 fmfil elflim syl2anc dfss3 cuni ctop topontop eqid neii1 wceq - sylan toponuni adantr sseqtr4d baibd syldan ralbidva syl5bb anbi2d 3bitrd + sylan toponuni adantr sseqtrrd baibd syldan ralbidva syl5bb anbi2d 3bitrd elfm ) DFIJKZEGLJKZGFCUDZUAZACDEUBMJZKADEFCUCMJZUEMZKZAFKZAUFZDUGJJZWDNZO ZWGCHPUHBPZNHEUIZBWIQZOWBWCWEACDEFGUJUKWBVSWDFLJKZWFWKRVSVTWAULWBFDKZEGUP JKZWAWOVSVTWPWAFDUMSZVTVSWQWAEGUNUOZVSVTWAUQZDECFGURTAWDDFUSUTWBWJWNWGWJW @@ -299329,7 +299329,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( vx vk vj vy cfv wcel cz cv cuz wa wss ctopon w3a wral wrex cima clm wbr wf wi cflf co wfn wb cpw uzf ffn ax-mp uzssz eqsstri imaeq2 sseq1d rexima wceq mp2an wfun cdm simpl3 ffund uzss eleq2s adantl fdmd syl6eq funimass4 - sseqtr4d syl2anc rexbidva syl5rbb imbi2d ralbidv anbi2d simp1 simp2 simp3 + sseqtrrd syl2anc rexbidva syl5rbb imbi2d ralbidv anbi2d simp1 simp2 simp3 eqidd lmbrf cfbas uzfbas flffbas syl3an2 3bitr4d ) CFUANOZEPOZGFBUHZUBZAF OZAJQZOZKQZBNZWQOKLQZRNZUCZLGUDZUIZJCUCZSWPWRBMQZUEZWQTZMRGUEZUDZUIZJCUCZ SZBACUFNUGABCDUJUKNOZWOXFXMWPWOXEXLJCWOXDXKWRXKBXBUEZWQTZLGUDZWOXDRPULZGP @@ -299361,7 +299361,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the wb reeanv cin filin 3expb sylan inss1 ssralv inss2 anim12i anbi12d rspcev cfil syl2an ex rexlimdvva syl5bir impbid2 cmpt cres df-ima filelss resmpt reseq1i syl5eq syl rneqd sseq1d ralbii wf fmpt wfn fnmpti syl6bb rexbidva - opex wfun cdm adantr ffund fdmd sseqtr4d funimass4 syl2anc 3bitr4d ralrab + opex wfun cdm adantr ffund fdmd sseqtrrd funimass4 syl2anc 3bitr4d ralrab ralbidv c0 wne ctopon toponmax rabn0 sylibr bitrd ffvelrnda syl3anc isflf df-f mpbiran bitri r19.26 3bitr3i impexp r19.21v bitr3i syl6bbr anim12dan r19.28zv r19.27zv sylan9bbr syl5bb pm5.32da anbi1i an4 3bitr4g ctg oveq1d @@ -299517,7 +299517,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the A. o e. J ( A e. o -> A. s e. F ( o i^i s ) =/= (/) ) ) ) ) $= ( ctopon cfv wcel cfil wa cfcls cv wral c0 wne wi wss ad2antrr ad3antrrr co ccl cin isfcls2 wrex filn0 adantl r19.2z ex cuni ctop topontop filelss - syl adantll wceq toponuni sseqtrd clsss3 syl2anc sseqtr4d sseld rexlimdva + syl adantll wceq toponuni sseqtrd clsss3 syl2anc sseqtrrd sseld rexlimdva eqid syld pm4.71rd wb adantlr simplr eleqtrd elcls syl3anc ralcom r19.21v ralbidva ralbii bitri syl6bb pm5.32da 3bitrd ) DEGHIZCEJHIZKZADCLUAIAFMZD UBHHZIZFCNZAEIZWGKWHABMZIZWIWDUCOPZFCNQZBDNZKACDEFUDWCWGWHWCWGWFFCUEZWHWC @@ -299736,7 +299736,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the E. g e. ( Fil ` X ) ( F C_ g /\ A e. ( J fLim g ) ) ) ) $= ( vx vy cfv wcel co cv wss cflim wa c0 wne adantr syl wb cvv cfil csn cun cfcls wrex cnei cfi cfg cfbas wn filsspw cuni ctop fclstop adantl neisspw - cpw eqid filunibas wceq fclsfil sylan9req pweqd sseqtr4d unssd ssun1 ssn0 + cpw eqid filunibas wceq fclsfil sylan9req pweqd sseqtrrd unssd ssun1 ssn0 filn0 sylancr cin wral incom fclsneii eqnetrrid 3com23 adantll ralrimivva w3a 3expb filfbas ctopon istopon sylanbrc fclselbas eleqtrrd snssd neifil snnzg syl3anc fbunfip syl2anc mpbird filtop fsubbas mpbir3and unexg mpan2 @@ -300119,7 +300119,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ad2antlr ad2antrl elfpw sselda anasss anim1i syl2anc ex mt3d expr ssrdv elunii eldifsni elinel2 elfir syl13anc filfi eleqtrd imbi12d syl5ibrcom eleq2 eleq1 rexlimdva sylbid imp32 adantrrr elssuni fibas tgtopon ax-mp - ctopon ctb syl6eqel fiuni eqtrd fveq2d eleqtrrd toponuni sseqtr4d filss + ctopon ctb syl6eqel fiuni eqtrd fveq2d eleqtrrd toponuni sseqtrrd filss simprrr rexlimddv ralrimiva imdistanda syl5bi flimopn sylibrd ralrimivw difexg unieq syl6eqr eqnetrd necon3bii sylib ciun sseq0 difss sseqtrrid ssdif0 unissi eqssd jctil sseq1 eqeq2d anbi12d anbi2d pweq ineq1d mpcom @@ -300590,7 +300590,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the cab fveq2 syl6eleq vex elixp simp-4r simplrr eleqtrd sylib fveq1 sylibr cixp ex ralrimiva cif fvex uniex ifex wi breq1d exlimddv rspcdva eluni2 wb mptpreima baib ad2antlr rexbidva mpbird eliun dfiun2g unieqi syl6eqr - ssrdv sseqtrd unissd ad3antrrr sseqtr4d rspceeqv rexlimddv syl5bir mtod + ssrdv sseqtrd unissd ad3antrrr sseqtrrd rspceeqv rexlimddv syl5bir mtod unieq neq0 rexv sylan2 eleq1 ac6num mptexd rgenw notbid ralrab ad2antll fnmpt iftrue adantrr csn adantl en1b elsni eqeltrrd adantlr eqeltrd a1d expr pm2.27 iffalse pm2.61d1 ralimdva syl5bi impr fneq1 ifbieq12d fvmpt @@ -300879,7 +300879,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the cfil ad3antrrr 3an1rs adantllr creg simprl regsep expcom ad2antll syl5bir reximi 3expib anim1d eltopss 3expa cfm cflim flfval cfbas uniexg eqeltrid elrestr cfg filfbas fgfil eleqtrrd eqid imaelfm flimclsi eqsstrd eqsstrrd - sylibr adantlr expl wfun cdm cnextf ffund fdmd sseqtr4d funimass4 biimprd + sylibr adantlr expl wfun cdm cnextf ffund fdmd sseqtrrd funimass4 biimprd reximdva cnnei mpbird ) AFGHUIUJRZGHUKUJSZUXQUAULZUMUBULZUNZUABULZUQZGUOR ZRZUPZUBUYBUXQRZUQZHUORRZURZBEURZAUYJBEAUYBESZTZUYFUBUYIUYMUXTUYISZTZAUCU LZUXQRZUXTSZUCUXSURZUAUYEUPZUYFAUYLUYNUSUYOUXSGSZFUXSCUTZUMZHVARZRZUXTUNZ @@ -301461,7 +301461,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the vf vt vv cgrp ctmd cminusg ctopn ctgp symggrp cmnd ctps cplusg ctx grpmnd syl cbs ctopon cpw csn cxp cpt crest symgtopn distopon pttoponconst mpdan cmap cv wf1o elsymgbas f1of elmapg anidms sylbid ssrdv resttopon eqeltrrd - istps sylibr cxko ccom cmpo symgplusg ctop distop xkotopon cndis sseqtr4d + istps sylibr cxko ccom cmpo symgplusg ctop distop xkotopon cndis sseqtrrd syl5ibr ccmp cnlly clly disllycmp llynlly xkococn syl3anc cnmpt2res xkopt eqeltrid mpancom oveq1d eqtrd oveq12d eleqtrd crn cplusf wfn coex plusfeq vex fnmpoi ax-mp eqcomi grpplusf frn cnrest2 mpbid oveq2d istmd syl3anbrc @@ -301562,7 +301562,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the cv csubmnd wa ctopn crest submmnd cress tmdtps resstps sylan eqeltrid cbs cmnd wceq submbas ressplusg mpoeq123dv plusffval syl6reqr ctopon tmdtopon oveqd adantr submss tmdcn eqeltrrid cnmpt2res eqeltrd crn wb cxp mndplusf - wf frn 3syl sseqtr4d cnrest2 syl3anc mpbid resstopn istmd syl3anbrc ) BGH + wf frn 3syl sseqtrrd cnrest2 syl3anc mpbid resstopn istmd syl3anbrc ) BGH ZABUAIZHZUBZCULHZCJHCKIZBUCIZAUDLZWIMLZWINLHZCGHWDWFWBACBDUEOZWECBAUFLZJD WBBJHWDWMJHBUGABWCUHUIUJWEWGWJWHNLZHZWKWEWGEFAAETZFTZBPIZLZQZWNWEWTEFCUKI ZXAWPWQCPIZLZQWGWEEFAAWSXAXAXCWDAXAUMWBACBDUNOZXDWEWRXBWPWQWDWRXBUMWBAWRB @@ -301578,7 +301578,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( vx ctgp wcel cfv cgrp ctmd cminusg ccn adantl cmpt cbs wceq eqid adantr co wf csubg wa ctopn crest subggrp csubmnd tgptmd subgsubm submtmd syl2an cv subgbas mpteq1d subginv adantll mpteq2dva grpinvf syl feqmptd 3eqtr4rd - ctopon tgptopon wss subgss tgpgrp eqeltrrd cnmpt1res eqeltrd crn sseqtr4d + ctopon tgptopon wss subgss tgpgrp eqeltrrd cnmpt1res eqeltrd crn sseqtrrd tgpinv wb frnd cnrest2 syl3anc mpbid resstopn istgp syl3anbrc ) BFGZABUAH GZUBZCIGZCJGZCKHZBUCHZAUDSZWGLSGZCFGWAWCVTABCDUEMZVTBJGABUFHGWDWABUGABUHA BCDUIUJWBWEWGWFLSZGZWHWBWEEAEUKZBKHZHZNZWJWBEAWLWEHZNECOHZWPNWOWEWBEAWQWP @@ -301652,7 +301652,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( ( cls ` J ) ` S ) e. ( SubGrp ` G ) ) $= ( vx vy vz wcel cfv wa wss cv co wral eqid adantr syl adantl wceq syl2anc ctgp csubg ccl cbs c0 wne csg cuni ctop ctopon tgptopon topontop toponuni - subgss sseqtrd clsss3 sseqtr4d sscls c0g subg0cl ne0d ssn0 cop df-ov ccnv + subgss sseqtrd clsss3 sseqtrrd sscls c0g subg0cl ne0d ssn0 cop df-ov ccnv cxp cima opelxpi ctx txcls syl22anc txtopon cnvimass cgrp wf tgpgrp fssdm grpsubf subgsubcl 3expb ralrimivva fveq2 syl6eqr eleq1d ralxp sylibr wfun cdm ffund xpss12 fdmd funimass5 mpbird clsss syl3anc ccn tgpsubcn cncls2i @@ -301681,7 +301681,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the ( vx vy wcel cfv wa cv co wral cmpt crn wss cima cres eqid ad2antrr syl ctgp cnsg ccl csubg cplusg csg nsgsubg clssubg sylan2 wf df-ima cuni ctop cbs ctopon tgptopon topontop ad2antlr subgss wceq toponuni sseqtrd clsss3 - syl2anc sseqtr4d resmptd rneqd syl5eq ccn tgptmd simpr cnmptc cnmpt1plusg + syl2anc sseqtrrd resmptd rneqd syl5eq ccn tgptmd simpr cnmptc cnmpt1plusg ctmd cnmptid tgpsubcn cnmpt12f cnclsi nsgconj ad4ant234 fmpttd frnd clsss ctx eqsstrd syl3anc sstrd eqsstrrd wfn ovex fnmpti df-f mpbiran ralrimiva sylibr fmpt isnsg3 sylanbrc ) BUAGZABUBHZGZIZACUCHZHZBUDHZGZEJZFJZBUEHZKZ @@ -301757,7 +301757,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the chmeo toponuni sseqtrid conncompconn connima conncompss syl3anc wral wi hmeocn elpwi ccnv mptpreima ssrab3 grprid simprrl oveq2 eleq1d sylanbrc elrab2 hmeocnvcn simprl sseqtrd simprrr wfun cdm wf1o grplactcnv simpld - mp3an2i grplactfval f1oeq1 mpbid f1ocnv f1ofun f1odm sseqtr4d funimass3 + mp3an2i grplactfval f1oeq1 mpbid f1ocnv f1ofun f1odm sseqtrrd funimass3 3syl imacnvcnv syl6eqr expr sylan2 ralrimiva eleq2w anbi12d sylibr ralrab unissb eqssd ) EUCOZBGOZUDZBCUEZBAUFZOZFUUHPQZUGOZUDZAGUJZUHZUIZ UUFUUGGRBUUGOZFUUGPQZUGOUUGUUORUUFUUGBNUFZCUKZNULZGUUEUUGUUTSUUDNBCGUMU @@ -301927,7 +301927,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the [ A ] .~ = ( ( cls ` J ) ` { A } ) ) $= ( vx vy wcel cfv co cima wceq eqid ctgp wa cec cv cplusg cmpt csn imaeq2i ccl cgrp wss tgpgrp adantr cuni ctop ctopon tgptopon topontop syl grpidcl - snssd toponuni sseqtrd clsss3 syl2anc sseqtr4d eqsstrid simpr tgplacthmeo + snssd toponuni sseqtrd clsss3 syl2anc sseqtrrd eqsstrid simpr tgplacthmeo eqglact syl3anc chmeo hmeocls 3eqtr4a crn cres df-ima resmptd syl5eq wrex rneqd cab fvexi oveq2 eqeq2d rexsn grprid sylan syl5bb abbidv rnmpt df-sn c0g 3eqtr4g eqtrd fveq2d ) DUAOZAFOZUBZABUCZMFAMUDZDUEPZQZUFZGUGZRZEUIPZP @@ -302534,7 +302534,7 @@ given an amorphous set (a.k.a. a Ia-finite I-infinite set) ` X ` , the tsmscls $p |- ( ph -> ( ( cls ` J ) ` { X } ) C_ ( G tsums F ) ) $= ( vx vy cfv co wcel csn ccl cpw cfn cin cv wss crab cmpt crn cfg cres cfm cgsu cflim ctsu cflf ctps eqid tsmsval ctopon cfil wceq istps sylib cfbas - tsmsfbas fgcl syl tsmslem1 fmpttd flfval syl3anc eqtrd flimsncls sseqtr4d + tsmsfbas fgcl syl tsmslem1 fmpttd flfval syl3anc eqtrd flimsncls sseqtrrd wf eleqtrd ) AHUAFUBRRZFBUCUDUEZPVTPUFQUFZUGQVTUHUIZUJZUKSZCQVTEDWAULUNSZ UIZUMSRZUOSZEDUPSZAHWHTVSWHUGAHWIWHOAWIWFFWDUQSRZWHAQPBCVTDEFWCURGIJVTUSZ WCUSZLMNUTAFCVARTZWDVTVBRTZVTCWFVQWJWHVCAEURTWMLCFEIJVDVEAWCVTVFRTWNAQPBV @@ -304013,7 +304013,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed ssequn2 syl6eqr rspceeqv elrest biimpar syl21anc syldanl ad2antrr r19.29a biimpa ex ralrimiva ustincl simprl simprr ineq12d inindir reeanv sylanbrc ineq1 r19.29vva simp-4l ustdiag cdm inss1 resss sstri iss wbr ssel2 equid - resieq mpbiri breq2 rspcev dminxp sylibr reseq2d syl5req eqsstrd sseqtr4d + resieq mpbiri breq2 rspcev dminxp sylibr reseq2d syl5req eqsstrd sseqtrrd mpbi ssrin ad3antrrr ustinvel cnveqd cnvin cnvxp coss1 coss2 ax-mp mpbird 3jca ineq2i eqtri syl6eq sstr mpan inss2 xpidtr ssind id sseq1d ustexhalf coeq12d adantlr ad4ant13 sseq2d rexbidv wb isust syl ) BCUBUCZHZACIZJZBAA @@ -304169,7 +304169,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed csn biimpa inss2 sseq1 mpbiri rexlimivw syl simp-5l ad2antrr xpexd simplr ad6antr elrestr syl3anc inss1 imass1 ax-mp sstr mpan imassrn sstri rnxpid rnin sseqtri a1i ssind adantl simpllr imaeq1 sseq1d rspcev eleqtrd elin1d - sseqtr4d elutop simplbda r19.21bi syl21anc r19.29a trust biimpar syl12anc + sseqtrrd elutop simplbda r19.21bi syl21anc r19.29a trust biimpar syl12anc ralrimiva ex ssrdv ) BCIJZKZACLZMZDBNJZAUBUCZBAAUDZUBUCZNJZXCDOZXEKZXIXHK ZXCXJMZXCXIALZEOZFOZUOZPZXILZEXGQZFXIUAZXKXCXJUEXLXIGOZARZUFZGXDQZXMXCXJY DXCXDSKASKZXJYDUGXCBNUHXCACSXACSKXBBCIUIUJXAXBUKULZGXIAXDSSUMUNUPZYCXMGXD @@ -304193,7 +304193,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed cust crest co cxp wral elutop simprbda restutop syldan trust adantr sstrd csn simp-9l simplr simp-4r ustincl syl3anc inimass adantl simpllr imaeq1d syl ssrin ad5antr simp-5r sseldd ad2antrr inimasn elv ineq2d syl5eq incom - xpimasn syl6eq eqtrd sseqtr4d sstrid imaeq1 sseq1d rspcev syl2anc simp-4l + xpimasn syl6eq eqtrd sseqtrrd sstrid imaeq1 sseq1d rspcev syl2anc simp-4l simplbda r19.21bi r19.29a sqxpexg elrest sylan2 ralrimiva mpbir2and df-ss biimpa sylib eqcomd ineq1 rspceeqv fvex mpan ad2antlr mpbird eqelssd ) BC UCJZKZABUBJZKZLZUAXGAUDUEZBAAUFZUDUEZUBJZXFXHACMZXJXMMXFXHXNDNZENZUOZOZAM @@ -304475,7 +304475,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed ( vr cfv wcel cxp wss wa wceq c1st c2nd syl2anc syl wbr csn cvv cust ccnv vz ctx co ccl ccom cv cop wrel relxp cuni ctop utoptop eqeltrid ad3antrrr cutop txtop simpllr ctopon utoptopon toponuni sqxpeqd txuni eqtrd sseqtrd - eqid clsss3 sseqtr4d simpr sseldd 1st2nd sylancr cima cin simp-4l simpr1l + eqid clsss3 sseqtrrd simpr sseldd 1st2nd sylancr cima cin simp-4l simpr1l wral ustrel elin sylib simpld xp1st elrelimasn biimpa simp-4r xpss syl6ss 3anassrs df-rel sylibr 1st2ndbr xp2nd wb simpr1r fvex brcnv breq syl5rbbr simprd mpbird wi w3a brcogw ex mp3an sylan syl21anc ralrimiva c0 wne cnei @@ -304545,7 +304545,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed ( vx vb va vv vw cfv wcel wa cv wss wrex wceq ad2antrr syl simplr syl2anc cust cha ctop wral creg cutop utoptop adantr eqeltrid cima ccnv ccom cnei ccl csn simp-4l simpr cuni ad3antrrr simpllr eqid eltopss utopbas syl6eqr - unieqi sseqtr4d sseldd utopsnnei syl3anc neii2 simprl snss sylibr simplll + unieqi sseqtrrd sseldd utopsnnei syl3anc neii2 simprl snss sylibr simplll vex ad6antr ustimasn sseqtrd simprr clsss ctx cxp ustssxp sqxpeqd simp-5r co imasncls syl22anc utop3cls sstrd imass1 jca reximdva simp-5l ustex3sym ex mpd r19.29a cmpt opnneip utopsnneip eleqtrd wb elrnmpt mpbid ralrimiva @@ -305205,7 +305205,7 @@ unit group (that is, the nonzero numbers) to the field. (Contributed ( vb vw va vv cust cfv wcel ccfilu wa wss cxp cv wrex syl2anc cin cvv w3a c0 crest co wn cfbas wral simp1 simp2l iscfilu biimpa simpld simp3 simp2r trfbas2 biimpar syl21anc wceq ad5antr adantr elfvexd ssexd ad4antr simplr - elrestr syl3anc simpr ssrind simpllr sseqtr4d eqsstrrid id sqxpeqd sseq1d + elrestr syl3anc simpr ssrind simpllr sseqtrrd eqsstrrid id sqxpeqd sseq1d inxp rspcev simprd r19.21bi ad4ant13 r19.29a xpexd elrest ralrimiva trust wb syl mpbir2and ) BDIJZKZCBLJZKZUBCAUCUDZKUEZMZADNZUAZWLBAAOZUCUDZLJKZWL AUFJKZEPZXAOZFPZNZEWLQZFWRUGZWPCDUFJKZWOWMWTWPXGGPZXHOZHPZNZGCQZHBUGZWPWI @@ -307442,7 +307442,7 @@ the base set to the (extended) reals and which is nonnegative, symmetric blssec $p |- ( ( D e. ( *Met ` X ) /\ P e. X /\ S e. RR* ) -> ( P ( ball ` D ) S ) C_ [ P ] .~ ) $= ( cxmet cfv wcel cxr w3a cbl co cpnf cec wss wa cle wbr pnfge adantl ssbl - wi pnfxr 3expia mpanr2 mpd 3impa wceq xmetec 3adant3 sseqtr4d ) AEGHIZBEI + wi pnfxr 3expia mpanr2 mpd 3impa wceq xmetec 3adant3 sseqtrrd ) AEGHIZBEI ZDJIZKBDALHZMZBNUPMZBCOZUMUNUOUQURPZUMUNQZUOQDNRSZUTUOVBVADTUAVAUONJIZVBU TUCUDVAUOVCQVBUTABDNEUBUEUFUGUHUMUNUSURUIUOABCEFUJUKUL $. $} @@ -308077,7 +308077,7 @@ the product of closed balls in each coordinate (where closed ball means rnmpt unfi sylancl ssun2 snss mpbir ne0i mp1i fmpttd frnd 0xr snssd unssd fisupcl syl13anc eqeltrd rspcv syl5bir mpan2d sylbird ovex elabrex adantl cle ssun1 syl6eleqr sseldi supxrub syl2an2r breqtrrd mpand impbid 3bitrrd - prdsxmet xrlelttr ralrimdva pm5.32da blssm ss2ixp sseqtr4d sseld pm4.71rd + prdsxmet xrlelttr ralrimdva pm5.32da blssm ss2ixp sseqtrrd sseld pm4.71rd elbl 3bitr4d eqrdv ) AUGFCEUJUKULZBJBUMZFUKZCIUJUKULZUNZAUGUMZDUOZFUVCEUL ZCUPUQZURZUVDUVCUVBUOZURUVCUURUOZUVHAUVDUVFUVHAUVDURZUVHUUSUVCUKZUVAUOZBJ USZUUTUVKIULZCUPUQZBJUSZUVFUVJUVCBJKUNZUOZUVCJUTZUVHUVMVCAUVDUVRADUVQUVCA @@ -308143,7 +308143,7 @@ the product of closed balls in each coordinate (where closed ball means $( The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.) $) blssopn $p |- ( D e. ( *Met ` X ) -> ran ( ball ` D ) C_ J ) $= - ( cxmet cfv wcel cbl crn ctg ctb wss blbas bastg syl mopnval sseqtr4d ) A + ( cxmet cfv wcel cbl crn ctg ctb wss blbas bastg syl mopnval sseqtrrd ) A CEFGZAHFIZSJFZBRSKGSTLACMSKNOABCDPQ $. $( The union of a collection of open sets of a metric space is open. @@ -309107,7 +309107,7 @@ S C_ ( P ( ball ` D ) T ) ) $= A. w e. X ( ( P C w ) < z -> ( ( F ` P ) D ( F ` w ) ) < y ) ) ) ) $= ( cfv wcel co crp wral wa wi cxmet w3a ccnp cbl cima wss wrex clt metcnp3 wf cv wbr wb wfun cdm ffun ad2antlr simpll1 simpll3 rpxr ad2antll syl3anc - cxr blssm wceq fdm sseqtr4d funimass4 syl2anc elbl imbi1d impexp ad2antrr + cxr blssm wceq fdm sseqtrrd funimass4 syl2anc elbl imbi1d impexp ad2antrr simpl2 simplrl rpxrd simpllr adantr ffvelrnd simplr elbl2 syl22anc imbi2d ffvelrnda pm5.74da syl5bb ralbidv2 anassrs rexbidva ralbidva pm5.32da bitrd ) DJUANOZEKUANOZFJOZUBZGFHIUCPNOJKGUJZGFBUKZDUDNPZUEFGNZAUKZEUDNPZU @@ -309507,7 +309507,7 @@ the half element (corresponding to half the distance) is also in this cle wbr 0xr a1i rpxr 0le0 rpre rehalfcld rphalflt ltled icossico syl22anc c2 imass2 sseq1 elfg syl12anc cfiluexsm syl3anc funimass2 ex reximdv sylc rspcev ralrimiva jca simprl sseq2d rexbidv simp-4r rspcdva nfv nfcv nfre1 - simprd nfral nfan ad4antr fbelss sylancom xpss12 simp-6r sseqtr4d ralrimi + simprd nfral nfan ad4antr fbelss sylancom xpss12 simp-6r sseqtrrd ralrimi r19.29r cin sseqin2 biimpi adantl dminss syl6eqssr adantr reximi syl sstr sstrd r19.41v sylbir simp-5r biimpa r19.29a wb iscfilu mpbir2and impbida ) FUBUCZDFUDJKZLZCFFUEZEUFMZUGJKZCFUHJKZDBNZUULUEZUIZOANZUJMZPZBCQZARUKZL @@ -309680,7 +309680,7 @@ the half element (corresponding to half the distance) is also in this ( unifTop ` ( metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) ) $= ( va vx vb vv vd ve cfv wcel wa cv wss wrex cima wceq syl cvv wb crp cuni wne cpsmet cmetu cutop cbl crn ctg wral cpw csn crab cust metuust utopval - c0 eleq2d syl6bb biimpa simpld elpwid unirnblps ad2antlr sseqtr4d simp-5r + c0 eleq2d syl6bb biimpa simpld elpwid unirnblps ad2antlr sseqtrrd simp-5r rabid simplr ad3antrrr simpllr sseldd metustbl syl3anc sstr expcom anim2d simpr reximdv sylc simprd r19.21bi r19.29a ralrimiva fvex rnex eltg2 mp1i jca mpbird sseqtrd elpwg adantl ccnv cico co cmpt sselda blssexps syl2anc @@ -313404,7 +313404,7 @@ Normed space homomorphisms (bounded linear operators) ( vx cr wcel wa co cfv wss clt wbr c0 wceq cxr cle w3a wn ad2antrr adantl crp cicc cioo crn ctg cnt cpr cun wb rexr icc0 syl2an biimpar fveq2d ctop retop ntr0 ax-mp eqsstri syl6eqss iccssre uniretop ntrss2 sylancr anim12i - adantr uncom prunioo syl5eq 3expa sylan sseqtr4d simpr simpl ltlecasei cv + adantr uncom prunioo syl5eq 3expa sylan sseqtrrd simpr simpl ltlecasei cv 0ss cin wral cabs cmin ccom cxp cres wrex ntropn cxmet rexmet cmopn tgioo cbl eqid mopni2 mp3an1 cdiv rphalfcl ltsubrpd rpred resubcld ltnled mpbid caddc rpre rphalflt ltsub2dd readdcld ltaddrp sylancom lttrd rexrd elioo2 @@ -313493,7 +313493,7 @@ Normed space homomorphisms (bounded linear operators) sylib elpwid simpll snssd unssd unex elpw sylibr snfi sylancl simplr2 unfi elind ssun1 syl6ss biimpa simp1d simp2d simplr cbl simplr3 lttrd cfv cioo simp3d min1 crp rphalflt ltadd2dd elioo2 syl2an bl2ioo elun2 - eleqtrrd wo lelttric mpjaod ssrdv uniun unisng uneq2d sseqtr4d rspcev + eleqtrrd wo lelttric mpjaod ssrdv uniun unisng uneq2d sseqtrrd rspcev ex 3exp2 expimpd mpd syl6bb cbvrabv eqtri sylanbrc iftrue breq1d mtod syl5ibcom iffalse eqeltrrd ) AHEIALFUNUOUPZUQUPZEURUSZUTZHEVAAUXBUXAL URUSZALUXAVBUSUXDUTALUWTALIVCVBVDZVCUGAUIUJIAIDEVEUPZVCDBVFZVEUPZCVFZ @@ -313590,7 +313590,7 @@ Normed space homomorphisms (bounded linear operators) eleq1 eldifi simplrr elicc2 syl2anc simp2d simp1d leloed mt3d mnfxr rexrd ord elioo2 sylancr mpbir3and inelcm syl5ibrcom simp3d pnfxr sylancl inss1 cxr ltpnfd jctil jctir leidd ioodisj syl21anc sseq0 ioojoin syl32anc un12 - csn unass ioomax 3eqtr3g sseqtr4d disjsn sylibr disjssun syl nconnsubb ex + csn unass ioomax 3eqtr3g sseqtrrd disjsn sylibr disjssun syl nconnsubb ex eqtri mt2d eq0rdv ssdif0 ) ADEZFUBUCUDZAUEGUFHZIZBAHZCAHZIZIZBCUGGZAUHZJK YBAEYAUAYCYAUAUIZYCHZXPXNXPXTUJYAYEXPUKYAYEIZALYDFGZXOYDMFGZDXODULUDHYFUM NXNXPXTYEUOZYGXOHYFLYDUNNYHXOHYFYDMUNNYFBYGHZXRYGAOJUPYFYJBDHZLBPQZBYDPQZ @@ -314349,7 +314349,7 @@ Normed space homomorphisms (bounded linear operators) ( 0 < ( F ` A ) /\ if ( 1 <_ ( F ` A ) , 1 , ( F ` A ) ) e. RR+ ) ) $= ( wcel cc0 wbr c1 wa cfv clt cle cif crp wceq cin c0 wn adantr wne inelcm wi expcom adantl necon2bd mpd ccl eqcom cxmet wss wb cuni ccld eqid cldss - mopnuni sseqtr4d simpr sseldd metdseq0 syl3anc syl5bb cldcls eleq2d bitrd + mopnuni sseqtrrd simpr sseldd metdseq0 syl3anc syl5bb cldcls eleq2d bitrd syl mtbird wo cpnf cicc co wf metdsf syl2anc ffvelrnd cxr elxrge0 simprbi 0xr eliccxr xrleloe sylancr mpbid ord mt3d cr ifcl 1red 0lt1 breq2 ifboth 1xr xrltle xrmin1 xrrege0 syl22anc elrpd jca ) ADGQZUAZRDHUBZUCSZTXMUDSZT @@ -314369,7 +314369,7 @@ Normed space homomorphisms (bounded linear operators) metnrmlem1 $p |- ( ( ph /\ ( A e. S /\ B e. T ) ) -> if ( 1 <_ ( F ` B ) , 1 , ( F ` B ) ) <_ ( A D B ) ) $= ( wcel c1 co wa cfv cle wbr cif cxr 1xr cc0 cpnf cicc cxmet wss wf adantr - cuni ccld eqid cldss wceq mopnuni sseqtr4d metdsf syl2anc simprr ffvelrnd + cuni ccld eqid cldss wceq mopnuni sseqtrrd metdsf syl2anc simprr ffvelrnd syl sseldd eliccxr sylancr simprl xmetcl syl3anc xrmin2 metdstri syl22anc ifcl cxad xmetsym metds0 oveq12d xaddid1d eqtrd breqtrd xrletrd ) ADGRZEH RZUAZUAZSEIUBZUCUDZSWIUEZWIDEFTZWHSUFRZWIUFRZWKUFRUGWHWIUHUIUJTZRWNWHKWOE @@ -314386,7 +314386,7 @@ Normed space homomorphisms (bounded linear operators) (Revised by Mario Carneiro, 5-Sep-2015.) $) metnrmlem2 $p |- ( ph -> ( U e. J /\ T C_ U ) ) $= ( wcel wss cv c1 cfv cle wbr cif c2 cdiv cbl ciun ctop wral cxmet mopntop - co syl wa cxr adantr cuni ccld eqid cldss mopnuni sseqtr4d sselda cc0 clt + co syl wa cxr adantr cuni ccld eqid cldss mopnuni sseqtrrd sselda cc0 clt metnrmlem1a simprd rphalfcld rpxrd blopn syl3anc ralrimiva iunopn syl2anc wceq crp eqeltrid csn blcntr snssd ss2iun iunid eqcomi 3sstr4g jca ) AHJS GHTAHDGDUAZUBWIIUCZUDUEUBWJUFZUGUHUOZEUIUCUOZUJZJRAJUKSZWMJSZDGULWNJSAEKU @@ -314405,7 +314405,7 @@ Normed space homomorphisms (bounded linear operators) ( S C_ z /\ T C_ w /\ ( z i^i w ) = (/) ) ) $= ( wcel wss cin c0 wceq w3a wrex incom syl5eq metnrmlem2 simpld simprd cfv cv c1 cle wbr cif c2 cdiv co cbl ciun ineq1i iunin1 eqtr4i wral wa ineq2i - iunin2 cxmet cxr cxad adantr cuni ccld eqid cldss mopnuni sseqtr4d sselda + iunin2 cxmet cxr cxad adantr cuni ccld eqid cldss mopnuni sseqtrrd sselda syl adantrr adantrl crp metnrmlem1a rphalfcld rpxrd caddc rpred rehalfcld cc0 clt rexaddd recnd 2cnd wne 2ne0 a1i divdird eqtr4d metnrmlem1 ancom2s cxmu xmetsym syl3anc breqtrd xmetcl xle2add syl22anc mp2and cmul readdcld @@ -316033,7 +316033,7 @@ extended reals extends the topology of the reals (by ~ xrtgioo ), this mpan cneg syl5eqr eqtrd ltp1d eqbrtrd wb mpbir2and elind absge0d ge0p1rpd elbl oveq2 ineq1d rspceeqv eleq2 eqeq1 rexbidv anbi12d syl12anc ralrimiva rspcev syldan eqeq2d cmpcovf syl2anc ad4antr ad2antrl simprr simprl rpred - simpllr ad2antrr rspcdva rexlimdvaa eluni2 elssuni sseqtr4d simp-6l sstrd + simpllr ad2antrr rspcdva rexlimdvaa eluni2 elssuni sseqtrrd simp-6l sstrd syl6bb sseldd simplrl ffvelrnd id fveq2 oveq2d eqeq12d elin1d rpxrd mpbid eleqtrd simprd eqbrtrrd breq1d simplrr ltletrd ralrimiv fimaxre3 reximddv ltled sylbid elin2d ffvelrn ex exlimdv expimpd rexlimdva mpd ci cmul cmpo @@ -316287,7 +316287,7 @@ topological space to the reals is bounded (above). (Boundedness below lebnumlem1 $p |- ( ph -> F : X --> RR+ ) $= ( wcel ad2antrr cc0 vm vw cv cdif co cmpt crn cxr clt csu crp wa adantr cinf cfn cr wral wf cmet cfv wss c0 wne difssd cuni sselda elssuni wceq - syl cxmet metxmet mopnuni sseqtr4d wi wn notbid syl5ibrcom necon2ad imp + syl cxmet metxmet mopnuni sseqtrrd wi wn notbid syl5ibrcom necon2ad imp eleq1 pssdifn0 syl2anc eqid metdsre syl3anc fmpt sylibr simplr rsp sylc fsumrecl wbr wrex eleq2d biimpa eluni2 0red metdsval simprl sseldd 3syl sylib mpd ffvelrnd eqeltrrd cle cpnf metdsf elxrge0 simprd ccl ad2antll @@ -316354,7 +316354,7 @@ topological space to the reals is bounded (above). (Boundedness below mopnuni neeq1d biimpa evth2 raleq rexeqbi1dv syl mpbird wfn ffn ralbidv wb breq1 rexrn ssrexv sylc chash cdiv cn ad2antrr simplr eqnetrrd unieq uni0 syl6eq necon3i cfn hashnncl nnrpd rpdivcld wn ralnex clt cdif cmpt - cxr cinf csu cr simprl metdsval difssd elssuni adantl sseqtr4d wi eleq1 + cxr cinf csu cr simprl metdsval difssd elssuni adantl sseqtrrd wi eleq1 notbid syl5ibrcom necon2ad syl2anc syl3anc ffvelrnd rpred ltnled rpcnd weq ad3antrrr imp pssdifn0 metdsre simprr sseq2 rspccva sylan cin rpxrd eqeltrrd metdsge syl31anc blssm difin0ss syl5com sylbid eqbrtrrd fsumlt @@ -316468,7 +316468,7 @@ topological space to the reals is bounded (above). (Boundedness below ad2antlr ltdiv23 syl122anc mpbid ltsub23d ltaddrpd iccssioo syl22anc 0red eqbrtrd nn0ge0d divge0 iccss ssind cxmet eqid oveq2 rexmet mpbi syl6eleqr sseqin2 rpxr xpss12 resabs1 ax-mp eqcomi blres mp3an2i bl2ioo eqtrd sstr2 - ineq1d sseqtr4d reximdv syld ralrimdva oveq12d raleqbidv rspcev rexlimdva + ineq1d sseqtrrd reximdv syld ralrimdva oveq12d raleqbidv rspcev rexlimdva syl6an mpd ) BUDEZFGUBHZBUEIZUFZUAJZUCJZUGKUHZUVGUVGUIZUJZUKLZHZAJZEZABMZ UAUVGULZUCUMMCJZGKHZDJZNHZUWAUWCNHZUBHZUVQEZABMZCGUWCUNHZULZDUOMZUVIUAAUV NBUDUVGUCUPUVNUVGUQLOZUVIUVLPUQLOUVGPEUWLUSUVGQPURUTVAUVLUVGPVBVCVDUDVEOU @@ -321360,7 +321360,7 @@ vector spaces which are also normed vector spaces (that is, normed groups ( O ` ( ( cls ` J ) ` S ) ) = ( O ` S ) ) $= ( vy wcel wss cfv syl adantr sylib wceq eqid syl2anc cin cc ccph ccl ctop vx wa cuni ctopon ctps cngp cphngp ngptps istps topontop toponuni sseqtrd - simpr sscls ocv2ss cv cip co csca c0g wral clsss3 sseqtr4d ocvss a1i crab + simpr sscls ocv2ss cv cip co csca c0g wral clsss3 sseqtrrd ocvss a1i crab sselda ccld cab df-ss ineq1d dfrab3 ineq2i inass eqtr4i 3eqtr4g cmpt ccnv clscld csn cima cvv fvex mptiniseg ax-mp ccnfld ccn simpll cnmptc cnmptid ctopn cnmpt1ip cha cnfldhaus cclm cphclm clm0 ad2antrr syl6eqelr unicntop @@ -321844,7 +321844,7 @@ is an accumulation point (limit point) of subset ` y ` ". @) A. z e. y A. w e. y ( z D w ) < x ) ) ) $= ( cfv wcel cv cxp cc0 co wss wrex crp wral wa wbr cxr cxmet cfil cima clt ccfil cico iscfil cdm wb wf xmetf ad3antrrr ffund filelss ad4ant24 xpss12 - wfun syl2anc sseqtr4d funimassov cle 0xr a1i simpllr rpxrd simp-4l sselda + wfun syl2anc sseqtrrd funimassov cle 0xr a1i simpllr rpxrd simp-4l sselda fdmd adantrr adantrl xmetcl syl3anc xmetge0 elico1 df-3an syl6bb syl22anc w3a baibd 2ralbidva bitrd rexbidva ralbidva pm5.32da ) EGUAHIZFEUEHIFGUBH IZEBJZWGKZUCLAJZUFMZNZBFOZAPQZRWFCJZDJZEMZWIUDSZDWGQCWGQZBFOZAPQZRABEFGUG @@ -322453,7 +322453,7 @@ is an accumulation point (limit point) of subset ` y ` ". @) cfili com znnen nnenom entri raleq raleqbi1dv axcc4 syl cid cfz cdom ciin cn cen ad2antrr uzenom endom 3syl crab cin dfin5 wss wceq cuz fzn0 eleq2s biimpri cfil cxmet metxmet simpl cfilfil elfzelz ffvelrn filelss syl2an2r - simprr r19.2z syl2anr iinss elfvdm fvi sseqtr4d sseqin2 sylib syl5eqr cfi + simprr r19.2z syl2anr iinss elfvdm fvi sseqtrrd sseqin2 sylib syl5eqr cfi fvex cvv wb syl2anc weq fveq2 oveq2 raleqbidv cbvralv ex exlimdv mpd feq3 cfn adantl fzfid iinfi syl13anc filfi eleqtrd fileln0 eqnetrd rabn0 eleq1 adantrrr eliin syl6bb axcc4dom wal df-ral 19.29 sylanb simprrl 4syl mpbid @@ -322989,7 +322989,7 @@ any of the balls (i.e. it is in the intersection of the closures). ( ( D |` ( Y X. Y ) ) e. ( CMet ` Y ) <-> Y e. ( Clsd ` J ) ) ) $= ( vf ccmet cfv wcel cmet sylan cflim co wss eqid wceq syl2anc syl syl3anc c0 cxp cres ccld cmetmet metsscmetcld wa cmopn wne ccfil wral adantr cuni - cv cldss adantl cxmet metxmet mopnuni 3syl sseqtr4d metres2 cfg crest cin + cv cldss adantl cxmet metxmet mopnuni 3syl sseqtrrd metres2 cfg crest cin ad2antrr metrest eqcomd cfil cfilfil elfvdm trfg oveq12d ctopon mopntopon cdm cfbas cpw filfbas filsspw sspwb sylib sstrd fbasweak fgcl ssfg filtop sseldd flimrest ccl flimclsi cldcls ad2antlr df-ss 3eqtrd simpll cfilresi @@ -323052,7 +323052,7 @@ any of the balls (i.e. it is in the intersection of the closures). ( vf cfv wcel co wa syl wss cfil wceq syl2anc cmet cv cflim c0 wne wral ccfil ccmet cbl cxmet crp metxmet adantr simpr cfil3i syl3anc ccl crest wrex cfcls cin ctopon ad2antrr mopntopon cfilfil sylan simprr cuni ctop - topontop simprl rpxrd blssm toponuni sseqtrd eqid clsss3 sseqtr4d sscls + topontop simprl rpxrd blssm toponuni sseqtrd eqid clsss3 sseqtrrd sscls cxr filss syl13anc fclsrest cdm cfilfcls ad2antlr sseqtrid eqsstrd ccmp inss1 ad2ant2r wn cfbas filfbas fbncp wb trfil3 mpbird resttopon fveq2d cdif eleqtrd fclscmpi ssn0 rexlimddv ralrimiva iscmet sylanbrc ) ACFUAL @@ -323321,7 +323321,7 @@ be in all these balls (see ~ bcthlem3 ) and hence misses each cbl cmetmet metxmet 3syl ctop mopntop syl cpw ccld cn frnd eqid sspwuni cldss2 syl6ss sylib ntropn syl2anc jca mopni2 3expa sylan wn w3a c0 wne cdif cxp wf cop wceq caddc wral cvv csn wex mopnuni topopn eqeltrd reex - c1 cr rpssre ssexi xpexg sylancl 3ad2ant1 ntrss3 sseqtr4d simp2 opelxpd + c1 cr rpssre ssexi xpexg sylancl 3ad2ant1 ntrss3 sseqtrrd simp2 opelxpd sseldd simp3 cdiv clt wbr ccl copab opabssxp elpw2g adantr mpbiri simpl wb rspa syl2an ssdif0 c1st 1st2nd2 ad2antll fveq2d df-ov syl3anc wi cxr c2nd syl21anc mpd syl31anc simpr3 rpred simpr1 simpr2 rpxr sstr2 eximdv @@ -332102,7 +332102,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by adddird cima cvol wceq itg1val cin ciun wss inss2 a1i i1fima syl2anc wn inmbl ssdifssd eldifsni ad2antlr necon3ai itg1addlem3 syl21anc eqeltrrd wne simpl itg1addlem1 iunin2 mblss iunid imaeq2i imaiun cnvimarndm fdmd - 3eqtr3i syl5eq sseqtr4d df-ss sylib syl5req fveq2d 3eqtr4d oveq2d dfin4 + 3eqtr3i syl5eq sseqtrrd df-ss sylib syl5req fveq2d 3eqtr4d oveq2d dfin4 fsummulc2 difssd eqsstrri sseli elsni oveq1d 0re syl6eqel mul02d cuz wo olcd sumz sylan2 fsumss 3eqtrd inss1 simpr incom iuneq2i eqtri cnvimass sseqtrid cc anasss fsumcom oveq12d ) AEUAZFUAZLUBZUUAMUBZGNZUCNZMOZYTUU @@ -337302,7 +337302,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wi cpw cxp ciun ccnfld ctopn df-dv dmmpossx simpl sseldi oveq2 opeliunxp2 sylib simprd cvv wb cnex simpld elpm2g mpbid adantr sseli sstrd ctop cuni elpwid ctopon cnfldtopon resttopon topontop wceq toponuni sseqtrd syl2anc - syl ntrss2 sselda dvlem fmpttd ssdifssd clp ntrss3 sseqtr4d restabs simpr + syl ntrss2 sselda dvlem fmpttd ssdifssd clp ntrss3 sseqtrrd restabs simpr mp3an2i ntropn perfopn eqeltrrd cnfldtop toponunii restperf lpss3 lpdifsn limcmo moanimv sylibr eldv mobidv mpbird alrimiv wrel reldv dffun6 funfnd ex mpbiran wex vex elrn dvcl exlimdv syl5bi ssrdv sylanbrc wn df-ov ndmfv @@ -337501,7 +337501,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( A C_ CC /\ S C_ dom ( CC _D F ) ) ) -> ( S _D ( F |` S ) ) = ( ( CC _D F ) |` S ) ) $= ( cr cc cpr wcel wf wa wss cdv co cres wrel wceq reldv ad2antrr ssidd dvbss - cdm recnprss simplr simprr simprl fssresd ssdmres sseqtr4d relssres sylancr + cdm recnprss simplr simprr simprl fssresd ssdmres sseqtrrd relssres sylancr sstrd sylib wfun dvfg ffund dvres2 syl22anc funssres syl2anc eqtr3d ) BDEFG ZAECHZIZAEJZBECKLZTZJZIZIZBCBMZKLZVDBMZTZMZVJVKVHVJNVJTZVLJVMVJOBVIPVHVNBVL VHBBVIUTBEJZVAVGBUAQZVHAEBCUTVAVGUBZVHBVEAVBVCVFUCZVHAECVHERZVQVBVCVFUDZSUJ @@ -337520,7 +337520,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( cr cc cpr wcel wf wa cdv co cdm wceq cres wss cin ad2antrr sylancr wrel reldv recnprss simplr inss2 fssres sylancl rescom resres wfn fnresdm 3syl eqtri ffn reseq1d syl5eqr feq1d mpbid inss1 a1i dvbss dmres simprr ineq2d - syl5eq sseqtr4d relssres wfun dvfg ffund ctopon cnfldtopon simprl toponss + syl5eq sseqtrrd relssres wfun dvfg ffund ctopon cnfldtopon simprl toponss ssidd cfv dvres2 syl22anc funssres syl2anc eqtr3d ) BFGHIZAGCJZKZADIZGCLM ZNZAOZKZKZBCBPZLMZWFBPZNZPZWLWMWJWLUAWLNZWNQWOWLOBWKUBWJWPBARZWNWJWQBWKWB BGQZWCWIBUCSZWJWQGCWQPZJZWQGWKJWJWCWQAQXAWBWCWIUDZBAUEAGWQCUFUGWJWQGWTWKW @@ -338161,7 +338161,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( A x. ( ( S _D F ) ` C ) ) ) $= ( cxp cmul co cdv cfv cc0 cc syl wceq csn cof caddc wcel fconst6g ssidd wf cdm cr cpr recnprss dvbss sseldd cres dvconst dmeqd c0ex fconst fdmi - wss syl6eq sseqtr4d dvres3 xpssres oveq2d reseq1d eqtrd 3eqtr3d fconst2 + wss syl6eq sseqtrrd dvres3 xpssres oveq2d reseq1d eqtrd 3eqtr3d fconst2 syl22anc sylibr fdmd eleqtrrd fveq1d fvconst2 oveq1d ffvelrnd fvconst2g dvmul mul02d syl2anc dvfg mulcomd oveq12d mulcld addid2d 3eqtrd ) ACDDB UAZLZEMUBNONPCDWIONZPZCEPZMNZCDEONZPZCWIPZMNZUCNQBWOMNZUCNWRACDWIEDFABR @@ -338183,7 +338183,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by ( vx cmul co cdv cc wf syl cc0 cfv wceq cmpt csn cxp cof caddc wcel snssd fconstg fssd c0ex fconst ccnfld ctopn crest cnt wss cr cpr recnprss ssidd cres cdm dvbsss a1i eqsstrrd eqid dvres resmptd fconstmpt reseq1i 3eqtr4g - syl22anc oveq2d dvconst dmeqd fdmi syl6eq sseqtr4d dvres3 xpssres reseq1d + syl22anc oveq2d dvconst dmeqd fdmi syl6eq sseqtrrd dvres3 xpssres reseq1d eqtrd 3eqtr3d ctop cnfldtopon resttopon sylancr topontop toponuni sseqtrd ctopon ntrss2 syl2anc dvbssntr eqssd reseq12d 3eqtr4d feq1d mpbiri dvmulf cuni fdmd cin cv sseqin2 sylib mpteq1d cvv ssexd fvconst2g sylan wa eqidd @@ -339871,7 +339871,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by simplr elicc2 simp1d eleqtrrd simp2d simp3d absdifled mpbir2and xrlelttrd cicc elbl3 mpbird ssrdv syl6sseqr resabs1d dvbss fssresd eqsstrid crn ctg blssm cnt ccnfld ctopn tgioo2 ctop retop cmopn tgioo eqeltrd isopn3i dvcn - blopn syl31anc rescncf sylc oveq2d iccntr 3eqtr3d sseqtr4d fvres sylan9eq + blopn syl31anc rescncf sylc oveq2d iccntr 3eqtr3d sseqtrrd fvres sylan9eq sstrid reseq1d eqtr4d sselda syl2an2r dvlip mp2and exp32 sylbid rexlimdva simp-4l impd sylan2 mpd wfn absf subf fco ffn fnresdm mp2b syl6eq anbi12d mp2an 3sstr3d breq1d 3imtr3d imp dvlipcn syldan an32s wo elpri mpjaodan ) @@ -340597,7 +340597,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by mpbir3and elind xrletrd neeq1d ffvelrnda syldan subeq0ad cicc xrltletrd w3a adantl iccssioo fss dvcn cncffvrn rescncf sylc ctg cnt syl6ss dvres tgioo2 iccntr reseq2d dmeqd sstrid ssdmres ubicc2 lbicc2 eqeq12d eqcomd - ioossicc sseqtr4d biimpar rolle fveq1d fvres sylan9eq dvf feq2d eqeltrd + ioossicc sseqtrrd biimpar rolle fveq1d fvres sylan9eq dvf feq2d eqeltrd mpbii rexlimdva ex sylbid ralrimiva rspcdva sstrd iccssre cmvth mulcomd iooss1 divmuleqd bitr4d rexbidva fvoveq1d fvoveq1 rspcev adantlr ssrexv syl5ibrcom syl2imc anassrs expimpd inss2 difss sstri sseli ovex rexbiia @@ -341073,7 +341073,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by cioo ltsubrpd ltaddrpd lttrd ltled ccncf rescncf sylc evthicc2 wf cncff wss syl ffvelrnd adantr wiso cxr cle rexrd lbicc2 syl3anc w3a wb elicc2 syl2anc mpbir3and isorel biimpd exp32 com4l syl3c fvresd breq12d sylibd - ccnv wfun fdmd sseqtr4d funfvima2 df-ima simprr syl5eq eleqtrd ad2antrl + ccnv wfun fdmd sseqtrrd funfvima2 df-ima simprr syl5eq eleqtrd ad2antrl ffund mpd mpbid simp2d simprll lelttr mpand syld ubicc2 brcnv syl5bb wo fvex ax-resscn a1i fss sylancl sstrd ccnfld ctopn tgioo2 dvres syl22anc cc eqid iccntr reseq2d eqtrd dmeqd cin mpjaod simp3d ltletr mpan2d rnss @@ -341137,7 +341137,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by restabs syl5eq rpred resubcld readdcld eqid icccmp eqeltrrd sstrid sylc rescncf cncffvrn mpbird cncfcnvcn mpbid cncfcn eleqtrd cle ltled ctopon wbr resttopon sylancr toponuni fveq1d eqeltrid ltsubrpd ltaddrpd elicc2 - w3a mpbir3and wi fdm sseqtr4d funfvima2 cnfldtopon cncnpi ssexg sylancl + w3a mpbir3and wi fdm sseqtrrd funfvima2 cnfldtopon cncnpi ssexg sylancl mpd oveq1d eleqtrrd topontop sseqtrd cdif cun cin difssd unssd ffvelrnd ssun1 elind restntr 3eqtr4g fveq2d eqtr3d feq2d feq3 cnprest syl22anc jca ) ABEUBZJDUCUBZUBZUDEUEZUVPHGUFUGUBUDZAEBCUHUGZBCUIUGZUJUGZUKZUVQUB @@ -341191,7 +341191,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by adantr cicc clt wbr cle w3a sseldd rpred resubcld readdcld elicc2 syl2anc biimpa simp1d simplrl rphalflt ltsub2dd simp2d ltletrd simp3d lelttrd cxr ltadd2dd rexrd elioo2 mpbir3and ex ssrdv rpre ctopon syl6ss resttopon ccn - bl2ioo sseqtr4d simprr dvcnvrelem2 rexlimddv simpld eqeltrrd eqelssd wral + bl2ioo sseqtrrd simprr dvcnvrelem2 rexlimddv simpld eqeltrrd eqelssd wral sstrd simprd fveq2d eleqtrd ralrimiva cnfldtopon cncnp mpbir2and eleqtrrd cncfcn dvcnv ) ABJCUCUGKZUDUEUFKZDEUVBLZUVBUVDUHZJJMUINAUJUKAEUVCNZEUVCUL KZKZEOZABUVHEAUVCUMNZEJPZUVHEPUNAECUEZJADECUOZDECUTUVLEOIDECUPDECUQURACDJ @@ -341241,7 +341241,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by wa sylancl ltmul2 syl112anc eqbrtrrd ltsubadd2d breqtrrdi wss ccncf leidd cle simpld ltsub2dd eqbrtrd ltaddsub2d ltled syl22anc rescncf sylc cdm cc iccss wf ax-resscn iccssre syl2anc dvres iccntr reseq2d dmeqd dmres rexrd - cin sseqtr4d df-ss sylib syl5eq mvth fveq1d fvres adantr sylan9eq syl3anc + cin sseqtrrd df-ss sylib syl5eq mvth fveq1d fvres adantr sylan9eq syl3anc cxr ubicc2 fvresd lbicc2 oveq12d ad2antrl ad2antll sselda sseldd ffvelrnd eqeq12d resubcld gt0ne0d redivcld mulcomd subdid breq12d 3bitr3d divdiv1d jca eqtr3d oveq2d 3eqtr4d mpbird eqbrtrid crn ctg cnt a1i cncff fss ctopn @@ -345428,7 +345428,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ N ) ) ) $= ( cn0 wcel cc wa cfv cc0 wceq wne cle wbr clt adantr syl2anc ad2antrr mpd wi wb vn wf c1 caddc co cuz cima csn wral simprr wfun cdm wss ffun adantl - cv wn peano2nn0 eluznn0 ex syl ssrdv fdm sseqtr4d funfvima2 nn0z peano2zd + cv wn peano2nn0 eluznn0 ex syl ssrdv fdm sseqtrrd funfvima2 nn0z peano2zd cz ad2antrl eluz simplr eleq2d fvex elsn syl6bb 3imtr3d necon3ad cr nn0re zred ltnled mpbird zleltp1 expr ralrimiva ccnv simpr syl2an nn0red eluzle ltp1d ad2antll ltletrd mpbid fveq2 neeq1d imbi12d simprl rspcdva necon1bd @@ -345953,7 +345953,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by max1 eluz2 fzss2 syl cdif csn c1 cima wn eldifn adantl cun wo eldifi cmin nn0uz peano2nn0 syl6eleq uzsplit syl5eq nn0cnd ax-1cn pncan oveq2d uneq1d sylancl ad2antrr eleqtrd elun sylib ord mpd wi wfun ffund ssun2 sseqtrrid - cdm fdmd sseqtr4d funfvima2 elsni mul02d fsumss max2 oveq12d 3eqtr4d + cdm fdmd sseqtrrd funfvima2 elsni mul02d fsumss max2 oveq12d 3eqtr4d mpteq2dva eqtr4d ) AGHUAUBZUCBUDUEIUFUCZFUGZCUHZBUGZUUEUIUCZUJUCZFUKZUEJU FUCZUUEDUHZUUHUJUCZFUKZUAUCZULBUDUEIJUMUNZJIUOZUFUCZUUECDUUCUCUHZUUHUJUCZ FUKZULABUDUUJUUNUAGHURURURUDURUPAUQUSUUJURUPAUUGUDUPZUTZUUDUUIFVAUSUUNURU @@ -345998,7 +345998,7 @@ or are almost disjoint (the interiors are disjoint). (Contributed by nn0zd syl2anc eqeltrd addid2d fveq2d eleqtrd fzss2 adantlr cdif csn c1 wn eldifn cun wo eldifi peano2nn0 uzsplit syl5eq ax-1cn pncan sylancl uneq1d cima eqtrd ad3antrrr elun sylib ord mpd wi wfun cdm ffund ssun2 sseqtrrid - fdmd sseqtr4d funfvima2 elsni oveq1d simplr syl2an mul02d fsumss sumeq2dv + fdmd sseqtrrd funfvima2 elsni oveq1d simplr syl2an mul02d fsumss sumeq2dv fzfid eqtr4d mul01d fsum2mul addcomd fsumcl olcd 3eqtr3d simpll fsummulc1 sumz mul4d expaddd ad2antlr pncan3d eqtr3d 3eqtr4rd cbvsumv oveq2i syl6eq cfn mpteq2dva ) AHIUCUDUEBUFUGJUHUEZFUIZCUJZBUIZUVBUKUEZUCUEZFULZUGKUHUEZ @@ -348946,7 +348946,7 @@ of all kernels (preimages of ` { 0 } ` ) of all polynomials in wrex caddc cicc 1re resubcl sylancl peano2re syl cpr ccpn cres reelprrecn crn cint wss cn0 wfn ssid fncpn ax-mp 1nn0 fnfvelrn intss1 cz cply plycpn mp2an sseldi cpnres sylancr cima df-ima wf zssre ax-resscn plyss plyreres - frnd eqsstrid cdm iccssre syl2anc syl6ss plyf fdmd sseqtr4d c1lip3 wa w3a + frnd eqsstrid cdm iccssre syl2anc syl6ss plyf fdmd sseqtrrd c1lip3 wa w3a simp2 recnd adantr 3ad2ant1 abssubd eqbrtrd 1red elicc4abs syl3anc mpbird simp3 wb subidd fveq2d abs0 0le1 eqbrtri eqbrtrdi wceq fveq2 oveq2d oveq2 breq12d fvoveq1d fvoveq1 rspc2v simp1l ffvelrnd eqtrd breq1d mpd cdiv wne @@ -350818,7 +350818,7 @@ evaluate the derivatives (generally ` RR ` or ` CC ` ), ` F ` is the sub4d divsubdird absdivd csn syl6eleq eluzelz ralrimiva climsub climabs fvexi remulcld eqimss2i climconst2 uztrn2 fvconst2 cof ulmscl ccom cres cbl ovresd cnmetdval cxr cnxmet xmetres2 sylancr rpxrd elbl3 crp blcntr - syl3anc jca offval2 fmpt3d dvmptsub dmeqd dmmpti syl6eq sseqtr4d sselda + syl3anc jca offval2 fmpt3d dvmptsub dmeqd dmmpti syl6eq sseqtrrd sselda wb fvmpt2 mpan2 sylan9eq dvmptcl abssubd ralbidv ltled eqbrtrrd 3brtr4d dvlip2 mpdan climle absrpcld ledivmul2d lttrd leltaddd 2halvesd breqtrd mpd lelttrd abs3lemd ) ABVBZRLVCZELVCZVDVEZREVDVEZVFVEZEMVCEGOKVCZVGVEZ @@ -397960,7 +397960,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to f1of wf1 f1of1 f1elima mpbird adantr simpl1 simpl2 3jca axcontlem3 sylan2 wb sselda adantrl jca breq1 anbi1d breq2 anbi2d rspc2va sylan an32s letrd simpld expr exlimdv mpd sylanbrc ex ccnv ssrab3 f1ocnvdm syl2an funfvima2 - wo sseqtr4d syl2anc sylc breq2d weq adantrr simplr cdm f1ofun fdm anim12d + wo sseqtrrd syl2anc sylc breq2d weq adantrr simplr cdm f1ofun fdm anim12d imp simprll rspc2v f1ocnvfv2 breq1d anbi12d axcontlem8 anassrs ralrimivva wfun opeq1 opeq2 cbvral2v 2ralbidv rspcev syld com23 rexlimdv ) JUFPZDJUG UHZUIZEUVFUIZAUJZKBUJZUKULQBERADRZUMZSZKUVFPZGDPZEUOUNZUMZKGUNZSZSZUAUJZU @@ -398314,7 +398314,7 @@ the Axiom of Continuity (Axiom A11). This proof indirectly refers to syl3anc mpbid simpr1 simpr2 3adantr3 axcgrid syl13anc sylbird ralrimivvva simpr3 jca32 istrkgc sylibr wrex cpw cbtwn axbtwnid imp equcomd ex simpll syl axpasch syl132anc anbi12d simplll eleqtrd rexeqbidva 3imtr3d ad2antrl - simpr wss sseqtr4d ad2antll simplrl simplrr axcont syl12anc simplr sseldd + simpr wss sseqtrrd ad2antll simplrl simplrr axcont syl12anc simplr sseldd elpwi 2ralbidva istrkgb sylanbrc elind wne simplr1 simplr2 simplr3 3anass 3jca syl5bir syl333anc 3anbi23d axsegcon syl122anc istrkgcb elntg istrkgl ax5seg df-trkg syl6eleqr ) AUNGZAUIUJZUKULUOZUMUAHZUPUJBCUBHZUUGBHZUQZURD @@ -408432,7 +408432,7 @@ segment of the walk (of length ` N ` ) forms a walk on the subgraph ( vk cfv c1 caddc co cpr cop csn cun ciedg wceq cc0 cfz eqeq12d wlkp1lem5 cv fveq2 cwlks wbr chash cn0 wcel wlkcl eqcomi eleq1i nn0fz0 3syl rspcdva sylbb fveq1i cvv wn ovex wlkp1lem1 fsnunfv mp3an2i syl5eq preq12d syl3anc - cdm cedg 3sstr4d wlkp1lem3 sseqtr4d ) ALEUJZLUKULUMZEUJZUNZBKBGUOUPUQUJZL + cdm cedg 3sstr4d wlkp1lem3 sseqtrrd ) ALEUJZLUKULUMZEUJZUNZBKBGUOUPUQUJZL JUJFURUJUJALDUJZCUNGWPWQUDAWMWRWOCAUIVDZEUJZWSDUJZUSWMWRUSUIUTLVAUMZLWSLU SWTWMXAWRWSLEVEWSLDVEVBABCDEFUIGHIJKLMNOPQRSTUAUBUCUDUEUFUGUHVCAHDIVFUJVG HVHUJZVIVJZLXBVJZUADHIVKXDLVIVJXEXCLVILXCUBVLVMLVNVQVOVPAWOWNDWNCUOUPUQZU @@ -416726,7 +416726,7 @@ cycle if and only if the second set contains exactly one vertex (in an { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) $= ( cfv c1 co wceq cc0 caddc csn cpr wss wif chash cfzo wral cs1 fveq1i cvv cv wa wcel 1wlkdlem2 elfvexd s1fv syl syl5eq fveq2d adantr eqtrd wn df-ne - wne sylan2br sseqtr4d ifpimpda wb s2fv0 s2fv1 eqeq12 eqeq2d preq12 sseq1d + wne sylan2br sseqtrrd ifpimpda wb s2fv0 s2fv1 eqeq12 eqeq2d preq12 sseq1d sneq ifpbi123d syl2anc mpbird c0ex oveq1 0p1e1 syl6eq wkslem2 mpdan ralsn cs2 sylibr fveq2i s1len eqtri oveq2i fzo01 a1i raleqdv ) ACULZBPZWPQUARZB PZSWPDPEPZWQUBSWQWSUCWTUDUEZCTDUFPZUGRZUHXACTUBZUHZATBPZQBPZSZTDPZEPZXFUB @@ -443919,7 +443919,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] supremum. (Contributed by NM, 24-Nov-2004.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) $) hsupunss $p |- ( A C_ ~P ~H -> U. A C_ ( \/H ` A ) ) $= - ( chba cpw wss cuni cort cfv chsup sspwuni ococss sylbi hsupval sseqtr4d + ( chba cpw wss cuni cort cfv chsup sspwuni ococss sylbi hsupval sseqtrrd ) ABCDZAEZOFGFGZAHGNOBDOPDABIOJKALM $. $( The union of a set of closed subspaces is smaller than its supremum. @@ -444743,7 +444743,7 @@ to a member of the subspace (Definition of complete subspace in [Beran] ( vx vy cva co wceq cfv wrex cph wcel csh wb syl syl2anc wa adantr cv cch cort cpjh chsh shocsh shsel simprr simprll simprlr rspe pjpreeq mpbir2and mpbid wss shococss sseldd chba shel ax-hvcom eqtrd choccl shless syl31anc - shscom sseqtr4d oveq12d eqtr4d exp32 rexlimdvv mpd ) ABFUAZGUAZHIZJZGCUCK + shscom sseqtrrd oveq12d eqtr4d exp32 rexlimdvv mpd ) ABFUAZGUAZHIZJZGCUCK ZLZFCLZBBCUDKKZBVPUDKKZHIZJZABCVPMIZNZVREACONZVPONZWDVRPACUBNZWEDCUEZQZAW EWFWICUFZQZFGCVPBUGRUNAVOWBFGCVPAVLCNZVMVPNZSZVOWBAWNVOSZSZBVNWAAWNVOUHZW PVSVLVTVMHWPVSVLJZWLVQAWLWMVOUIZWPWMVOVQAWLWMVOUJZWQVOGVPUKRAWRWLVQSPZWOA @@ -455639,7 +455639,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = p. 1. (Contributed by NM, 21-Jun-2004.) (New usage is discouraged.) $) ssmd2 $p |- ( ( A e. CH /\ B e. CH /\ A C_ B ) -> B MH A ) $= ( vx cch wcel wss cmd wbr wa cv chj co wi wral inss2 chub2 sstrid adantrl - cin wceq simpl sseqin2 sylib adantl oveq2d sseqtr4d exp32 ralrimdv adantr + cin wceq simpl sseqin2 sylib adantl oveq2d sseqtrrd exp32 ralrimdv adantr a1d wb mdbr2 ancoms sylibrd 3impia ) ADEZBDEZABFZBAGHZUPUQIURCJZAFZUTBKLZ ASZUTBASZKLZFZMZCDNZUSUPURVHMUQUPURVGCDUPURUTDEZVGUPURVIIZIZVFVAVKVCUTAKL ZVEUPVIVCVLFURUPVIIVCAVLVBAOAUTPQRVKVDAUTKVJVDATZUPVJURVMURVIUAABUBUCUDUE @@ -456277,7 +456277,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = sylancom eqtr3d rspceeqv rexlimdva2 syld sylibrd spansneleq eqcom sylan9r syl6ib adantlrl adantrr adantll ax-hvcom eqeq1d bitr4d adantrl cpr spanpr adantlrr oveq12 cph cun df-pr fveq2i snssi spanun syl2an spansnch spansnj - syl5eq cch sylan eqtr2d sseqtr4d 3anbi123d rspcev syl13anc expl rexlimivv + syl5eq cch sylan eqtr2d sseqtrrd 3anbi123d rspcev syl13anc expl rexlimivv sseq1 syl5bi sylbir syl2anb 3impia ) BUBHZCUBHZBCIZAUPZBIZUUFCIZUUFBCUCJZ UDZUEZAUBKZUUCDUPZUFIZBUUMUGZLUHZMZNZDOKZEUPZUFIZCUUTUGZLUHZMZNZEOKZUUEUU LPZUUDDBUIECUIUUSUVFNUURUVENZEOKDOKUVGUURUVEDEOOUJUVHUVGDEOOUVHUUNUVANZUU @@ -456648,7 +456648,7 @@ A C_ ( _|_ ` B ) ) ) -> ( ( normh ` ( S ` ( A vH B ) ) ) ^ 2 ) = ( cch wcel cat w3a chj co wss wceq wa atelch sylan2 3adant2 wpss wi ccv wbr chjcl cin c0h chub2 ancoms adantr cvpss syldan sylbid 3adant3 adantld chub1 cvp id a1d ancrd wb chlub syld3an3 sylibd syl3an adantrd jcad simp1 anim12d - imp 3jca ancomsd psssstr syl6 chcv2 cvnbtwn2 sylsyld mpd sseqtr4d ex ) ADEZ + imp 3jca ancomsd psssstr syl6 chcv2 cvnbtwn2 sylsyld mpd sseqtrrd ex ) ADEZ BFEZCFEZGZBACHIZJZABUAUBKZLZCABHIZJVSWCLZCVTWDVSCVTJZWCVPVRWFVQVRVPCDEZWFCM ZWGVPWFCAUCUDNOUEWEAWDPZWDVTJZLZWDVTKZVSWCWKVSWCWIWJVSWBWIWAVPVQWBWIQVRVPVQ LWBAWDRSZWIABULVPVQWDDEZWMWIQVQVPBDEZWNBMZABTZNAWDUFUGUHUIZUJVSWAWJWBVPVPVQ @@ -461393,7 +461393,7 @@ its graph has a given second element (that is, function value). curry2ima $p |- ( ( F Fn ( A X. B ) /\ C e. B /\ D C_ A ) -> ( G " D ) = { y | E. x e. D y = ( x F C ) } ) $= ( cxp wfn wcel wss cv wceq wrex cab cvv wf syl2anc w3a cima co wfun simp1 - cfv dffn2 sylib simp2 curry2f ffund simp3 fdmd sseqtr4d dfimafn curry2val + cfv dffn2 sylib simp2 curry2f ffund simp3 fdmd sseqtrrd dfimafn curry2val cdm 3adant3 eqeq1d eqcom syl6bb rexbidv abbidv eqtrd ) GCDJZKZEDLZFCMZUAZ HFUBZANZHUFZBNZOZAFPZBQZVMVKEGUCZOZAFPZBQVIHUDFHUQZMVJVPOVICRHVIVERGSZVGC RHSVIVFWAVFVGVHUEVEGUGUHVFVGVHUICDERGHIUJTZUKVIFCVTVFVGVHULVICRHWBUMUNABF @@ -464397,7 +464397,7 @@ its graph has a given second element (that is, function value). 13-Dec-2023.) $) pfxf1 $p |- ( ph -> ( W prefix L ) : dom ( W prefix L ) -1-1-> S ) $= ( cpfx co cdm wf1 cc0 cfzo wss wf cfv wcel wceq syl syl2anc chash cfz cuz - cres elfzuz3 fzoss2 3syl cword wrddm sseqtr4d wrdf fssresd f1resf1 pfxres + cres elfzuz3 fzoss2 3syl cword wrddm sseqtrrd wrdf fssresd f1resf1 pfxres syl3anc wfn pfxfn fndmd eqidd f1eq123d mpbird ) ADCHIZJZBVBKLCMIZBDVDUDZK ZADJZBDKVDVGNVDBVEOVFFAVDLDUAPZMIZVGACLVHUBIQZVHCUCPQVDVINGCLVHUECLVHUFUG ZADBUHQZVGVIREBDUISUJAVIBVDDAVLVIBDOEBDUKSVKULVGBVDBDUMUOAVCVDBBVBVEAVLVJ @@ -466573,7 +466573,7 @@ real number multiplication operation (this has to be defined in the main ( cid cdif cdm cres wfn wss cin wceq syl wcel wf symgbasf fnresi difssd ffnd a1i ssidd nfpconfp inres wrel reli ax-mp eqsstrrd relssres sylancr relin2 syl5eq dmeqd eqtr4d sseqtrd w3a fnreseql syl31anc resabs1d eqtrd - biimpar difss dmss fdm 3syl sseqtrid c0 wb reldisj mpbid sseqtr4d difid + biimpar difss dmss fdm 3syl sseqtrid c0 wb reldisj mpbid sseqtrrd difid difin2 syl5reqr cun undif1 ssequn2 sylib symgcom ) ABCBELMZNZMZWGDEFGHI JAEWHOZLBOZWHOZLWHOAEBPZWJBPZWHBQZWHEWJRZNZQZWIWKSZABBEAECUAZBBEUBZIBCE DGHUCZTUFZWMABUDUGZABWGUEZAWHWHWPAWHUHAWHELRZNZWPAWLWHXFSXBBEUITZAWOXEA @@ -466720,7 +466720,7 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $= eqeltrid syl6eleqr a1i wfn f1of ffnd difss dmss ax-mp sseldi fdmd eleqtrd wf wa fnelnfp biimpa syl21anc eqnetrd eldifsn sylanbrc snssd dfss4 syl5eq sylib sseqtrd sssn simpr pmtrcnel2 ssdif0 adantr ex cun eqsstrrid ssundif - eqdif sylibr ssidd sseqtr4d difss2d ssequn1 eqssd orim12d mpd ) ABGIJUAZU + eqdif sylibr ssidd sseqtrrd difss2d ssequn1 eqssd orim12d mpd ) ABGIJUAZU BZUBZUCUDZXKJUEZUDZUFZBXJUDZBGIUEZUBZUDZUFAXKXMUGXOAXKXRXJUBZXMABXRXJAXIF UHHUOUIUBUJZHUIUBZUJZXQUBZBXRACDEFHIJKLMNOPQRUKTGYCXQSULUMZUNAXTXRXRXMUBZ UBZXMXJYFXRGIJUPZUQAXMXRUGYGXMUDAJXRAJGURJIUSJXRURAJYCGAJYCXQAJIHUHZYDOAD @@ -467670,7 +467670,7 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $= crab cbs wf eqid tocycf syl fdmd eleqtrd sseldi crn eldifad s1cld splcl syl2anc eqeltrid cycpmco2f1 cc0 cfzo chash cmin cfz cuz wf1o wa wceq id dmeq eqidd f1eq123d elrab sylib simprd f1cnv f1of 3syl ffvelrnd fzofzp1 - wss elfzuz3 fzoss2 cycpmco2lem3 oveq2d sseqtr4d sseldd cycpmfv1 f1f1orn + wss elfzuz3 fzoss2 cycpmco2lem3 oveq2d sseqtrrd sseldd cycpmfv1 f1f1orn wrddm ssun1 cycpmco2rn sseqtrrid sselda f1ocnvfv2 syl2an2r fveq2d mpdan csn cun f1ocnvfv1 fveq1i cn0 nn0fz0 splfv1 syl5eq eqtr3d oveq1d 3eqtr3d fz0ssnn0 a1i fveq1d cz nn0zd simpr elfzonn0 nn0cnd 1cnd adantr 3eqtr2rd @@ -467905,7 +467905,7 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $= $( Lemma for ~ cycpmconjv (Contributed by Thierry Arnoux, 9-Oct-2023.) $) cycpmconjvlem $p |- ( ph -> ( ( F |` ( D \ B ) ) o. `' F ) = ( _I |` ( D \ ran ( F |` B ) ) ) ) $= - ( cdif cres ccnv ccom crn wfun wceq wf1o syl sylib wss sseqtr4d wfo 3syl + ( cdif cres ccnv ccom crn wfun wceq wf1o syl sylib wss sseqtrrd wfo 3syl cid f1ofun funrel dfrel2 reseq1d cnveqd coeq2d difssd f1odm ssdmres ssidd wrel cdm eqsstrd cores2 f1ocnv fores syl2anc wb df-ima foeq3 ax-mp resdif cima syl3anc wfn f1ofn fnresdm rneqd f1ofo forn difeq1d f1oeq3d f1ococnv2 @@ -467938,7 +467938,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry cnvco simp2 ffvelrnd symgcl eqid grpsubval symginv oveq2d simp1 elsymgbas f1ocnv biimpar symgov 3eqtrd simpld tocycfv coeq2d coundi coires1 cz 1zzd a1i f1of cshco syl3anc coeq1d syl5eqr eqtrd coundir syl6eq wrdco f1of1 wb - f1co wss sseqtr4d dmcosseq f1eq2 mpbird 3eqtr4d ) BHOZEAOZIFUBZOZUCZEBIUD + f1co wss sseqtrrd dmcosseq f1eq2 mpbird 3eqtr4d ) BHOZEAOZIFUBZOZUCZEBIUD ZUEZUFZEUGZPZEIPZUHUIQZIUGZPZYNPZUJZUKBYPUDZUEZUFZYQYPUGZPZUJEIFULZCQZEGQ ZYPFULYJYOUUDYTUUFYJYOUKBEYKUFUDZUEZUFUUDYJYKBEYGYFBBEUMZYIBAEDJNUNUOZYJI UBZBIYJUUNBIUPZUUNBIUQYJIBURZOZUUOYJIUAUSZUBZBUURUPZUAUUPVDZOUUQUUOUTYJIY @@ -468548,7 +468548,7 @@ Formula in property (b) of [Lang] p. 32. (Contributed by Thierry cid eqtr3d cun coeq2d symgov symgcl eqeltrrd 3eqtr4d symgsubg cvv c2 cmin coass wfn ffn r19.29a hashcl eqeltrid nn0zd 3z 0red zred 3pos nn0red 3lt5 5re letrd elfz4 syl32anc cycpmgcl sseldd cs2 cycpm2tr s2f1 elin1d id dmeq - s2cld eqidd f1eq123d simprd f1f frn ssconb syl21anc s2rn difeq2d sseqtr4d + s2cld eqidd f1eq123d simprd f1f frn ssconb syl21anc s2rn difeq2d sseqtrrd elrab 3eqtr3d simp-4r simpld c0 disjdif undif symgcom syl6eq cnvco coeq1i coeq12d eqtr3i symgbasf fcoi1 wf1o elsymgbas f1ococnv2 coeq1d eqtr4d cgrp 3eqtrd symggrp grpcl simp-7r 3eqtr4rd caddc 3p2e5 eqbrtrid 2re leaddsub2d @@ -471828,7 +471828,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a ( vx vw clvec wcel cfv wa cv clbs cldim cle wbr eqid syl wss chash wceq c0 wne wex lsslvec lbsex n0 sylib hashss adantll dimval ad2antrr ad5ant14 clss sylan clinds wrex simpll clmod lveclmod simplr cbs simpr lbsss lssss - 3brtr4d ressbas2 3syl sseqtr4d lbslinds sseldi lsslinds syl31anc islinds4 + 3brtr4d ressbas2 3syl sseqtrrd lbslinds sseldi lsslinds syl31anc islinds4 w3a biimpa syl2anc r19.29a exlimddv ) BGHZABUMIZHZJZEKZCLIZHZCMIZBMIZNOZE WBWDUAUBZWEEUCWBCGHZWIVTABCDVTPZUDZWDCWDPZUEQEWDUFUGWBWEJZWCFKZRZWHFBLIZW NWOWQHZJWPJWCSIZWOSIZWFWGNWRWPWSWTNOWNWOWCWQUHUIWNWFWSTZWRWPWBWJWEXAWLWCC @@ -472235,7 +472235,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a ressbas2 eqtrd lmodgrp eqidd cplusg ad6antr fveq2d r19.29a mpbid sylanbrc 3syl wrex lsslinds lspcl reslmhm reslmhm2b csubg lmghm lsssubg lmhmf ffnd resghm wf lspssv fnssres simpld simprd syl12anc elind lmhmkerlss cnvimass - fssdm sseqtr4d ineq2d lbsdiflsp0 ad5ant145 ssrdv 0ellsp fvexd ghmid elsng + fssdm sseqtrrd ineq2d lbsdiflsp0 ad5ant145 ssrdv 0ellsp fvexd ghmid elsng ex elpreimad snssd eqssd cmnd grpmnd ress0g sneqd kerf1ghm f1eq123d f1ssr mpbird f1f1orn ad8antr simplbda oveq1d sseqtrd lmhmlvec2 ad9antr ffvelrnd lspss ghmlin eqtr2d grplid 3eqtr3d fnfvimad clsm simp-7l lsmsp2 lsmelvalx @@ -472384,7 +472384,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a fssdm breq1d chvarv iunfi xpfi fveq1d cbvmptv syl6eq suppovss feqmptd subrg0 feq1d ressbas2 eqsstrrd iuneq12d xpeq12d 3sstr3d suppssfifsupp eqtrd fssd syl32anc breqtrd drgextgsum cmulr csubg subrgsubg subgsubm - csubmnd lmodvscl gsumsubm ressmulr sstrd sseqtr4d ringass syl13anc wb + csubmnd lmodvscl gsumsubm ressmulr sstrd sseqtrrd ringass syl13anc wb fmpttd cplusg breq2d rmfsupp2 gsummulc1 3eqtr4rd w3a offval22 ovmpt4g mpbid ccmn ringcmn eqsstrd simprl eleqtrd inidm off ringlz offinsupp1 gsumxp ofeq 3eqtr2rd elfvexd gsumsra 3eqtr2d jca ) ALBVFVGZVHVGZVIVJR @@ -472492,7 +472492,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a oveq2d drgextgsum wne crab cfn mptexd fmpttd ovex rgenw mpteqb eqeq1d wn ax-mp sylib 3ad2ant1 csn subsubrg simpld clbs lbsss eqsstrrd sstrd csra clss srasubrg cmulr simprd simprl ressabs oveq1i 3eqtr4g 3eqtr4d - syl5eq eleqtrrd simprr subrgmcl eqeltrrd ralrimivva syl12anc sseqtr4d + syl5eq eleqtrrd simprr subrgmcl eqeltrrd ralrimivva syl12anc sseqtrrd islss4 biimpar lbslinds sseldi resssca sraaddg eqeltrd lmodvscl 3expb ressplusg sylan ressmulr eqtr2d ressvsca lindspropd lsslinds syl31anc islinds5 wfn cfrlm xpexd frlmelbas mpbid elmapd ffnd cmnd crg ringmnd @@ -472657,7 +472657,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a co c0 wne wex clvec cdr cress csubrg wss wa biimpa syl2anc simprd sralvec eqid syl3anc lbsex syl n0 sylib adantr cmpo chash cxp wbr cbs cvv wf wral weq wi wel ad4antr simplr lbsss subrgss csra srabase eqtrd ad2antrr simpr - a1i sseldd sseqtr4d simpld eleqtrd ralrimivva cbvmpov fmpo cvsca csca c0g + a1i sseldd sseqtrrd simpld eleqtrd ralrimivva cbvmpov fmpo cvsca csca c0g anasss ad5antr simpllr srasca syl5eq fveq2d simp-4r clspn ad7antr oveq12d eqtr3d clmod cgsu lveclmod fveq2 cmap wrex cmpt ffvelrnd eleqtrrd cbvmptv cfsupp oveq1d oveq2d anbi12d ad8antr exlimddv cnzr dimval subsubrg oveq1i @@ -474035,7 +474035,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a nn0ltlem1 biimpa eqbrtrd iftrued oveq1d nncnd 1cnd pncand 3eqtrd fvmptd eqtrd idi f1ocnvfv imp fveq2d breq1d breqtrrd ad2antrr ad3antrrr simplr adantlr eqtr2d wn eqbrtrrd ex con3d an32s ifeqda wfun cdm f1ocnv f1ofun - 3syl wss cdif fzdif2 difss syl6eqssr f1odm sseqtr4d sseldd fvco syl2anc + 3syl wss cdif fzdif2 difss syl6eqssr f1odm sseqtrrd sseldd fvco syl2anc csn eqtr4d ) APUMOUMUNUOZUPUOZUQZURZPKUSUTZPPUMVAUOZVBPGVCZVHZEVHZPEUVD VDVHZUVAUVBPUVCUVFUVAUVBURZUVFUVCEVHZPUVAUVFUVIVEZUVBUVAUVEUVCEUVAUMOUP UOZUVKGVFZUVCUVKUQZUVCGVHPVEZUVEUVCVEZAUVLUUTAGUVKVGVHZVIVHZUQZUVLAOUVK @@ -475725,7 +475725,7 @@ commutative monoid (=vectors) together with a semiring (=scalars) and a bastg sstri syl5eq sseqtrrid syl unssad simplrl simplrr oran 3expa nrexdv anbi12i anandi exbii n0f df-rex necon1bbii ineq1d 0in syl6eq eldif ssconb nfin vex sylanb anass1rs mpbi2and nfun ianor equcom syl6bb syl5bbr 3expia - pm5.32d orbi12i elun andi 3bitr4ri eldifsn bicomi eqrd sseqtr4d nconnsubb + pm5.32d orbi12i elun andi 3bitr4ri eldifsn bicomi eqrd sseqtrrd nconnsubb 3bitr3g anasss adantllr rexanali reeanv necomd anbi12d pm5.32rd r19.29af biimpa con4d ) FUAMZCDUBZNZAUCZHUCZGUDZUXMBUCZGUDZNZUXMCMZUEZHDUJZBCUJZAC UJZECUFUGUHMZUXKUYBOZUYCOZUXKUYDNUXMUXLGUDZOZACPZUXOUXMGUDZOZBCPZNZUXROZN @@ -481493,7 +481493,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry = sum* n e. N ( M ` ( A \ U_ k e. ( 1 ..^ n ) B ) ) ) $= ( cfv c1 wcel wa cn wss ciun cv cfzo co cdif cesum iundisjcnt fveq2d wral cmeas com cdom wbr wdisj wceq crn cuni measbase syl adantr simpll fzossnn - csiga simpr sseqtrrid cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtr4d wo + csiga simpr sseqtrrid cuz simplr eleqtrd elfzouz2 fzoss2 3syl sseqtrrd wo mpjaodan sselda wsb sbimi sban sbv clelsb3 anbi12i bitri csb sbsbc wb cvv wsbc sbcel1g elv nfcv cbvcsb csbid eqtri eleq1i 3imtr3i syl2anc ralrimiva 3bitri sigaclfu2 difelsiga eqimss sseq1 mpbiri jaoi nnct ssct iundisj2cnt @@ -482338,7 +482338,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry <-> ( F : U. S --> U. T /\ A. a e. K ( `' F " a ) e. S ) ) ) $= ( vx vy wcel cuni cima wral wa adantr wss syl wceq cmbfm co wf ccnv csiga crn csigagen cfv cvv sgsiga eqeltrd simpr mbfmf ad2antrr simplr sssigagen - cv sseqtr4d sseldd mbfmcnvima ralrimiva jca cmap unielsiga simprl biimpar + cv sseqtrrd sseldd mbfmcnvima ralrimiva jca cmap unielsiga simprl biimpar elmapg syl21anc crab cpw cdif com wbr wi w3a simpl ssrab2 pwuni sstri a1i cdom fimacnv ad2antrl imaeq2 eleq1d elrab elrabi adantl difelsiga syl3anc sylanbrc wfun simplrl ffun difpreima 3syl difeq1d simprbi ad3antrrr sspwb @@ -482382,7 +482382,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cnmbfm $p |- ( ph -> F e. ( S MblFnM T ) ) $= ( va co wcel cuni wf eqid syl csigagen cfv ctop 3syl cmbfm ccnv cima wral ccn cnf unieqd wceq cntop1 unisg eqtrd cntop2 feq23d mpbird wss sssigagen - cv wa sseqtr4d adantr cnima sylan sseldd ralrimiva elex csiga sigagensiga + cv wa sseqtrrd adantr cnima sylan sseldd ralrimiva elex csiga sigagensiga cvv crn eqeltrd elrnsiga imambfm mpbir2and ) ADBCUAKLBMZCMZDNZDUBJUQZUCZB LZJFUDAVPEMZFMZDNZADEFUEKLZWBGDEFVTWAVTOWAOUFPAVNVOVTWADAVNEQRZMZVTABWDHU GAWCESLZWEVTUHGDEFUIZESUJTUKAVOFQRZMZWAACWHIUGAWCFSLZWIWAUHGDEFULZFSUJTUK @@ -482788,7 +482788,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry vex simpl simprr eleqtrd 3jca c1st cfv c2nd simpr xp1st adantl simpll 3ad2ant1 3ad2ant3 dya2icoseg2 syl3anc xp2nd simplr reeanv xpeq1 xpeq2 sylanbrc eqeq2d rspc2ev mp3an3 sylibr ad2antrl cvv xpss simpl1 sseldi - simprrl simpld simprrr syl12anc simprd xpss12 syl2anc simpl2 sseqtr4d + simprrl simpld simprrr syl12anc simprd xpss12 syl2anc simpl2 sseqtrrd elxp7 eleq2 sseq1 anbi12d rspcev rexlimdvv sylc sylan2 rexlimivv 3syl exp32 ex ) LUCUCUDZRZDERZLDRZUEZDGUFZHUFZUDZUGZHUHUIZSGYESZXQXSTZTZYD YGTZHYESGYESZLMUFZRZYKDUJZTZMFUIZSZXTYFXQXSXRXQYFXSXRDGHYEYEYCUKZUIZR @@ -483089,7 +483089,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry ( va wcel cc0 cpnf cuni wss cv wa cmpt clt cvv wceq cxr cicc w3a com cdom co wf wbr cdm cpw crab cfv cesum crn cinf coms simp2 simp1 syl2anc omsval fex syl simpr sseq1d anbi1d rabbidv mpteq1d rneqd infeq1d 3ad2ant2 unieqd - simp3 fdm sseqtr4d wb elex uniexb biimpi ssexg elpwg mpbird wor xrltso wi + simp3 fdm sseqtrrd wb elex uniexb biimpi ssexg elpwg mpbird wor xrltso wi 3syl iccssxr soss ax-mp mp1i infexd fvmptd ) EGIZEJKUAUEZFUFZDELZMZUBZHDA HNZCNZLZMZWRUCUDUGZOZCFUHZUIZUJZANBNFUKBULZPZUMZWLQUNZADWSMZXAOZCXDUJZXFP ZUMZWLQUNXCLZUIZFUOUKZRWPFRIZXQHXPXIPSWPWMWKXRWKWMWOUPWKWMWOUQZEWLGFUTURA @@ -483206,7 +483206,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry /\ z ~<_ _om ) } sum* w e. x ( R ` w ) < ( ( M ` A ) + E ) ) $= ( vu clt wbr wcel ve vt cv cfv caddc co cuni wss com cdom cdm cpw cesum crab cmpt crn wrex cc0 cpnf cicc cinf cxr cle rpred readdcld rexrd coms - wa wf omsf syl2anc feq1i sylibr fdmd unieqd sseqtr4d cvv uniexg syl jca + wa wf omsf syl2anc feq1i sylibr fdmd unieqd sseqtrrd cvv uniexg syl jca ssexg elpwg 3syl mpbird ffvelrnd elxrge0 simprbi rpge0d sylanbrc fveq1i wb addge0d wceq omsfval syl3anc syl5req ltaddrpd eqbrtrd iccssxr xrltso wor soss mp2 a1i wn wral omscl xrge0infss infglb mp2and wex eqid esumex @@ -483245,7 +483245,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry 2rp a1i syl3anc cxr wn wi imp syl21anc breq1 ax-mp unieq sseq2d anbi12d eqid elrab simprbi simpld mpd ex ralrimi ad2antrr ralrimiva 3syl sselda esumcl rexrd simpllr sylib ralimi c1 cxmu oveq2d cc xrletrd wf1 domentr - cen nnenom ensymi brdomi cdm nfesum1 nfel cicc coms omsf feq1i sseqtr4d + cen nnenom ensymi brdomi cdm nfesum1 nfel cicc coms omsf feq1i sseqtrrd fdmd unieqd ssexg esumcvgre df-f1 simplbi ffvelrnda rpdivcld cioo dfrp2 rexadd ioossicc eqsstri xrge0addcld eqeltrrd rpgt0d 2re adantllr expgt0 cz 2pos divgt0d ltaddposd mpbid fveq1i omsfval syl5eq eqcomd breq1d jca @@ -486618,7 +486618,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry $( Law of total probability, deduction form. (Contributed by Thierry Arnoux, 25-Dec-2016.) $) totprobd $p |- ( ph -> ( P ` A ) = sum* b e. B ( P ` ( b i^i A ) ) ) $= - ( vc cuni cin cfv wceq wcel syl syl2anc adantr cesum wss elssuni sseqtr4d + ( vc cuni cin cfv wceq wcel syl syl2anc adantr cesum wss elssuni sseqtrrd cv cdm sseqin2 sylib fveq2d cmpt cmeas cpw com cdom wbr wdisj domprobmeas cprb measinb measvun syl112anc cc0 c1 cicc co eqidd wa simpr ineq1d csiga crn domprobsiga sigaclcu syl3anc inelsiga prob01 fvmptd elelpwi esumeq2dv @@ -489288,7 +489288,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry interval. (Contributed by Thierry Arnoux, 8-Oct-2018.) $) fzssfzo $p |- ( K e. ( M ..^ N ) -> ( M ... K ) C_ ( M ..^ N ) ) $= ( cfzo co wcel cfz cmin cuz cfv wss wceq elfzoel2 fzoval syl eleq2d elfzuz3 - c1 cz ibi fzss2 3syl sseqtr4d ) ABCDEZFZBAGEZBCRHEZGEZUDUEAUHFZUGAIJFUFUHKU + c1 cz ibi fzss2 3syl sseqtrrd ) ABCDEZFZBAGEZBCRHEZGEZUDUEAUHFZUGAIJFUFUHKU EUIUEUDUHAUECSFUDUHLABCMBCNOZPTABUGQABUGUAUBUJUC $. ${ @@ -489936,7 +489936,7 @@ strict in the case where the sets B(x) overlap. (Contributed by Thierry cmnd wne lencl syl eldifsn hasheq0 necon3bid biimpar sylbi elnnne0 adantr sylanbrc nnm1nn0 nn0uz syl6eleq cxr cfzo wf ccatws1cl wrdf wceq cz fzoval nn0zd fzossfz syl6eqssr s1cl ccatlen sylan2 oveq2i syl6eq oveq2d peano2zd - s1len nn0cnd 1cnd pncand 3eqtrd sseqtr4d sselda sylanl1 rexrd signswplusg + s1len nn0cnd 1cnd pncand 3eqtrd sseqtrrd sselda sylanl1 rexrd signswplusg ffvelrnd sgncl rexr adantl id npcand sylan9eqr fveq2d ccatws1ls gsumnunsn eqtrd mpteq1d simpll ad2antlr eleq2d ccatval1 syl3anc mpteq2dva oveq1d wo sylan 3eqtr3d eqid olci wb fzosplitsni mpbiri eleqtrrd signstfval syl2anc @@ -498533,7 +498533,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a -> _pred ( y , A , R ) C_ ( G ` suc i ) ) $= ( cv com wcel cfv w-bnj17 c-bnj14 wss w3a wa wfn vex bnj919 bnj918 bnj976 csuc bnj1254 anim1i bnj252 3imtr4i ciun ssiun2 bnj708 3simpa ancomd simp3 - wceq wi bnj539 bnj965 bnj228 sylc bnj721 sseqtr4d syl ) UAJUMZUNUOZWGVGZO + wceq wi bnj539 bnj965 bnj228 sylc bnj721 sseqtrrd syl ) UAJUMZUNUOZWGVGZO UMZUOZDUMZWGMUPZUOZUQZTWHWKWNUQZEHWLURZWIMUPZUSUAWHWKWNUTZVATWSVAWOWPUATW SUAWJGUOMWJVBSTPQRGIMWJSTUAABCGWJLIUMZPQRUDUEUFUGOVCZVDUHUIUJFILMULVEVFVH VIUAWHWKWNVJTWHWKWNVJVKWPWQDWMWQVLZWRTWHWKWNWQXBUSDWMWQVMVNTWHWKWNWRXBVRZ @@ -500397,7 +500397,7 @@ have become an indirect lemma of the theorem in question (i.e. a lemma nf5ri bnj1521 3ad2ant1 bnj1147 simp3 bnj1213 simp2 bnj1125 syl3anc ssiun2 nfun nfan 3ad2ant2 bnj593 nfss bnj1397 wo bnj1138 sylib mpjaodan df-bnj19 ralrimiva sylibr bnj931 a1i syl3anbrc bnj1124 iunss1 3syl 3sstr4g bnj1136 - unss2 sseqtr4d eqssd ) DGUAZHDNZOZDGHPZEXEABXFEQAXEKUBXEEUCNDEGUFZDGHUGZE + unss2 sseqtrrd eqssd ) DGUAZHDNZOZDGHPZEXEABXFEQAXEKUBXEEUCNDEGUFZDGHUGZE QZBCDEGHIUHXEDGMUIZUGZEQZMEUDXGXEXLMEXEXJENZOZXJXHNZXLXJCXHDGCUIZPZUEZNZX NXOOZXKDGXJPZEXTXCXJDNZXKYAQZXCXDXMXOUJXOYBXNXHDXJDGHUKZULUMDGXJUNZRXOYAE QZXNXOYAXRQZYFCXHXQXJYADGXPXJUOUQYGYAXHXRUPZEYAXRXHURIUSUTUMSXNXSOZXLCYIY @@ -500809,7 +500809,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a ( cfv csn wcel wceq wfun cop c-bnj18 cun bnj930 opex elun2 ax-mp eleqtrri cv snid funopfv mpisyl bnj832 elsni simplbiim fveq2d c-bnj14 cres reseq2d bnj602 syl wss bnj931 a1i cdm w-bnj15 wbr wral wne simplbi bnj835 bnj1212 - wn wex bnj906 syl2anc fndm sseqtr4d bnj1503 eqtrd opeq12d 3eqtr4g 3eqtr4d + wn wex bnj906 syl2anc fndm sseqtrrd bnj1503 eqtrd opeq12d 3eqtr4g 3eqtr4d c0 wfn ) EFVPZNVCZUBRVCZHVPZNVCTRVCCXPXMVDZVEZXNXOVFZEVBBXPQVEZXSCVABNVGX MXOVHZNVEXSBXQIOXMVIZVJNUTVKZYAMYAVDZVJZNYAYDVEYAYEVEYAXMXOVLVQYAYDMVMVNU PVOXMXONVRVSVTVTEXPXMNECXRXPXMVFZVBXPXMWAWBZWCETUBREXPNIOXPWDZWEZVHXMMIOX @@ -500861,7 +500861,7 @@ become an indirect lemma of the theorem in question (i.e. a lemma of a sylib bitri bnj1444 bnj1340 wfn wral bnj771 bnj1445 fveq2 cres cop bnj602 wa id reseq2d opeq12d 3eqtr4g fveq2d eqeq12d bnj1254 simp3bi fndm eleqtrd bnj769 rspcdva bnj930 bnj835 wss simp2bi elssuni syl6sseqr bnj593 bnj1397 - wfun ssun3 bnj1502 sseq1d bnj1517 bnj770 sseqtr4d bnj1503 3eqtr4d bnj1446 + wfun ssun3 bnj1502 sseq1d bnj1517 bnj770 sseqtrrd bnj1503 3eqtr4d bnj1446 3syl opeq2d bnj1447 bnj1448 ) GLVNZRVOZUDUBVOZVPZTGIUUQTUUNTVNZVQZVRZGITU CGTUCUUSUUNGUUNUCVSZVQZTUCUUSVTGUUNQVQZUVBGUUNMSJVNZWAZUVCGDUUNUVEVRZVHWB DUVFUVCUVEVPZGVHCUUNUAVRZUVGDVFCUVEQVDWCWDWDWEQUVAUSWQWFTVMUCUIKMSUVDWGZW @@ -501399,7 +501399,7 @@ have become an indirect lemma of the theorem in question (i.e. a lemma eqtri bnj1405 bnj1209 bnj1436 bnj1299 bnj31 bnj836 bnj1518 bnj1521 bnj930 wfun bnj835 wss elssuni syl6sseqr simp3bi bnj1502 bnj1514 bnj1294 bnj1517 eqtrd eleqtrd bnj1503 opeq2d syl6eqr fveq2d eqtr4d bnj593 bnj1519 bnj1397 - sseqtr4d bnj1520 bnj1459 ) EHUBZADUCZJUDZXAJEHXAUEZUFZUGZKUDZUHZDERAXGIAB + sseqtrrd bnj1520 bnj1459 ) EHUBZADUCZJUDZXAJEHXAUEZUFZUGZKUDZUHZDERAXGIAB XGIXAIUCZUIZUJZABIGAIGXIXAAXAJUIZIGXIUKZAXAEXKAWTXAEUJZRULWTXMXKEUHZARWTJ EUMXNDEFGHIJKLMNOPQUNZEJUOUPUQURXKGUSZUIXLJXPQUTIUAGXHMUCZUMZXAXHUDZLKUDZ UHZDXQVAZVBMFVCZIUAGPVDVEVGVFVHSVIBXGMBCXGMXIXQUHZBCMFAXHGUJZXJYDMFVCBSYE @@ -501987,7 +501987,7 @@ have become an indirect lemma of the theorem in question (i.e. a lemma fzval3 3eqtr4rd feq2d mpbird eleq2d biimpa wa c2 revfv sylan wlklenvm1 cc lencl nn0cnd sub1m1 fvoveq1d fveq2d fzonn0p1p1 eleqtrrd syl2an2r elfzoelz adantl zcnd 1cnd addcomd subsub4d 3eqtr2d sneqd sneq eqcom cn0 fzossfzop1 - subcld sseqtr4d sselda sub32d npcand 3eqtr3d eqeq12d syl5bb wi wn wkslem1 + subcld sseqtrrd sselda sub32d npcand 3eqtr3d eqeq12d syl5bb wi wn wkslem1 wlkprop simp3d ubmelm1fzo eqeltrrd rspcdva dfifp2 sylib simpld sylbid imp notbid simprd prcom preq12d syl5eqr 3sstr4d ifpimpda syldan ralrimiva cvv w3a wb wlkv simp1d iswlkg mpbir3and ) BACUAFZUBZBUCFZAUCFZYPUBZYRCUDFZUEZ @@ -505590,7 +505590,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by simpl1 cvmtop1 syl adantr cvmsss adantl cvmcn cnima syl2anc inopn syl3anc sselda ccn fmpttd frnd cvmsn0 cdm dmmptg inex1g mprg eqeq1i dm0rn0 bitr3i cvv necon3bii sylib jca cpw inss2 wb elpw2g mpbiri sspwuni simpl3 cvmsuni - imass2 sseqtr4d wrex eqid ineq1 rspceeqv mpan2 ad2antrl vex inex1 elrnmpt + imass2 sseqtrrd wrex eqid ineq1 rspceeqv mpan2 ad2antrl vex inex1 elrnmpt ax-mp sylibr simprr simplr elind rspcev rexlimdvaa eluni2 3imtr4g eqelssd eleq2 mpd eldifsn wi weq wn equcoms necon3ai simpllr simpr cvmsdisj inss1 wo ord sseq0 eqeq1d syl5bi restabs mpan syl56 neeq1 ineq2 inindir syl6eqr @@ -505779,7 +505779,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by syl sylan 3eqtr3d simplbda fvres crest cuni eqid cconn wss cnvimass fssdm sstrd cnrest syl2anc ctopon ctop ccvm cvmtop1 toptopon df-ima crn sylib elssuni cvmsuni sseqtrd imass2 cnveqd cnvco 3eqtr3g imaco - imaeq1d wfun ffund fdmd sseqtr4d funimass3 eqsstrrid cvmcn sseqtrid + imaeq1d wfun ffund fdmd sseqtrrd funimass3 eqsstrrid cvmcn sseqtrid mpbird cnrest2 syl3anc mpbid eqeltrrd eqeq12d elrab3 3imtr4d eleq2d cvv fveq2 df-ss topopn ssexd cvmsss elrestr cvmscld conntop restuni ccld eleqtrd simpr eqeltrd conncn ffvelrnd 3eqtr4d wf1 wf1o cvmsf1o @@ -505962,7 +505962,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ( wa cima cfv c1st c1 cfz co wcel cmin cdiv cicc wss wral adantr wceq cv oveq1 oveq1d oveq12d syl6eqr imaeq2d 2fveq3 sseq12d rspcv sylc cdm wfun wi cc0 wf cii ccn iiuni cnf syl ffund cvmliftlem2 fdmd funfvima2 - sseqtr4d syl2anc mpd sseldd ) ABUPZNSUQZQJURUSURZENURZWSQUTRVAVBZVCNL + sseqtrrd syl2anc mpd sseldd ) ABUPZNSUQZQJURUSURZENURZWSQUTRVAVBZVCNL VKZUTVDVBZRVEVBZXDRVEVBZVFVBZUQZXDJURUSURZVGZLXCVHZWTXAVGZUMAXLBUKVIX KXMLQXCXDQVJZXIWTXJXAXNXHSNXNXHQUTVDVBZRVEVBZQRVEVBZVFVBSXNXFXPXGXQVF XNXEXORVEXDQUTVDVLVMXDQRVEVLVNUNVOVPXDQUSJVQVRVSVTWSESVCZXBWTVCZUOWSN @@ -506763,7 +506763,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ccvm fveq2i eleq1i anbi2i xpss12 wral ad2antrl simprr ctopon sselda cv adantrr cvmlift2lem6 syldan cnrest resabs1d ovex restabs 3eltr3d xpex oveq1d cvmtop1 toptopon simprl imaco cnvco cnveqd imaeq1d wfun - sseqtr4d cdm ffund funimass3 cnvimass cvmcn cnf fdm sseqtrid cvmsss + sseqtrrd cdm ffund funimass3 cnvimass cvmcn cnf fdm sseqtrid cvmsss fdmd simpld cvmsuni cvmsrcl cnima restopn2 mpbir2and cvmscld eqtr4d ccld mpdan simprd eqeltrrd ralrimivva funimassov cvmlift2lem9a ) AR EFHILMQPVJVJVKVLZJSVMZTUAUBVNZUUAUUBUUCVLZUXNVMZUDUFUGUHVJVJUXNUXNV @@ -506913,7 +506913,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by toponunii syl3anc ralrimiva wb resttopon sylancr ctop ccvm toptopon cvmtop1 sylib cncnp mpbir2and wceq sneq xpeq2d oveq2d oveq1d rspcev reseq2d eleq12d imp syldan xpss2 txtopi restuni sseqtrd sselda eqid - iitop cncnpi cnt a1i txopn syl22anc isopn3i sseqtr4d cnprest mpbird + iitop cncnpi cnt a1i txopn syl22anc isopn3i sseqtrrd cnprest mpbird ad2antrr ssrabdv ex ) AIRUMZUNZPUOZISUMZUNZPUOAUUBUPZUUDODUQZURURUS VHZGUTVHVAVBZDVCVIVDVHZUUIUNZVEZPUUEUUHDUUJUUDUUEIUUIUOZUUCUUIUOUUD UUJUOUUEIURVBZUULAUUMUUBUHVFZUUMIURVGZUUIIURVJVKVLVRZUUESUUIASUUIVB @@ -507388,7 +507388,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by ffvelrnd 3eqtr4d wreu cvmlift syl22anc anbi12d riota2 eqeq2d anbi1d mpbi2and riotabidv 3anbi123d rspcev cvmlift3lem4 mpdan cconn iiconn cvmtop1 rneqd 3eqtr3g iitopon resttopon cnf2 funimass3 eqsstrd fdmd - syl6ss sseqtr4d mpbid cnrest2 cvmsss cvmsiota elssuni cvmsuni cnima + syl6ss sseqtrrd mpbid cnrest2 cvmsss cvmsiota elssuni cvmsuni cnima sseqtrd cvmsrcl restopn2 mpbir2and cvmscld conncn 1elunit ffvelrn ccld ) AUGQVOZVPRVOZUDAUWQUWRVQZVRLVSZVOZUCVQZVPUWTVOZUGVQZVPOMVSZV TZPUWTVTZVQZVRUXEVOZGVQZWAZMWBFWCWDZWEZVOZUWRVQZWFZLWBTWCWDZUUAZAHU @@ -507453,7 +507453,7 @@ property of an acyclic graph (see also ~ acycgr0v ). (Contributed by fveq1 coeq2 anbi12d cbvriotav syl6eqr 3anbi123d cbvrexv sylib restuni c1 cpconn ad3antrrr wb mpbird syl22anc eleqtrd eleq2d pconncn syl3anc biimpa reeanv cnlly simpllr simplrl simprl cvmlift3lem6 ex rexlimdvva - simplrr simprr syl5bir mp2and ralrimiva wfun ffund sseqtr4d funimass4 + simplrr simprr syl5bir mp2and ralrimiva wfun ffund sseqtrrd funimass4 fdmd cvmlift2lem9a cncnpi cnt ssntr cnprest ) AOUBQFVKWKVLVMZORVNZUBQ RVOWKZFVKWKVLVMZAUVJUVKFVSWKVMUBUVKVPZVMZUVLADEFHILMOPQRUAUBUCUDUFUGU HUIURUJABCEFGJKMNOPQSUCUHUIUJUKULUMUNUOUPUQVQZAMOVTZNQPVSWKZABCEFGJKM @@ -511720,7 +511720,7 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership ssmclslem $p |- ( ph -> ( B u. ran H ) C_ ( K C B ) ) $= ( vc cv wss cfv wal vm vo vp vs vx vy crn cun cotp cmax wcel cima cmvrs wbr cxp wi wa cmsub wral cab cint simpl a1i alrimiv ssintab sylibr eqid - co mclsval sseqtr4d ) ABGUGZUHZVLPQZRZUAQZUBQZUCQZUIEUJSZUKUDQZVPVKUHUL + co mclsval sseqtrrd ) ABGUGZUHZVLPQZRZUAQZUBQZUCQZUIEUJSZUKUDQZVPVKUHUL VMRUEQZUFQZVOUNVTGSVSSEUMSZSWAGSVSSWBSUOHRUPUFTUETUQVQVSSVMUKUPUDEURSZU GUSUPUCTUBTUATZUQZPUTVAZHBCVHAWEVNUPZPTVLWFRAWGPWGAVNWDVBVCVDWEPVLVEVFA UEUFVRBCDWCEUAUBFGHWBUDUCPIJKLMNOVRVGWCVGWBVGVIVJ $. @@ -511782,7 +511782,7 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership w3a cmsta mstapst cmfs maxsta syl sseldd mpstrcl wceq simp1 simp2 simp3 sseldi eleq1d uneq1d imaeq2d breqd imbi1d 2albidv anbi12d fveq2d spc3gv oteq123d imbi12d ralbidv 3syl wo elun ralrimiva wf wfn mvhf fveq2 ralrn - wb ffn mpbird r19.21bi sseqtr4d sylibr fveq1 jaodan sylan2b msubf ffund + wb ffn mpbird r19.21bi sseqtrrd sylibr fveq1 jaodan sylan2b msubf ffund wfun cdm cfn ccnv elmpst sylib simp2d fdmd frnd unssd funimass4 syl2anc simpld 3exp2 imp4b ralrimivv dfss3 cop eleq1 df-br ralxp bitri alrimivv syl6bbr jca imaeq1 xpeq12d imbi2d rspcv mpid embantd 3syld alrimiv fvex @@ -512155,7 +512155,7 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership wfun cdm wb ffund simpld 3syl elpreima simplbda wbr wrex ciun cxp ssbrd cid imp brxp fveq2d msubvrs syl3anc eqtrd eleq2d eliun syl6bb wi breq12 wal cvv simpl simpr syl5bir vex simp2bi mvhf unssd fdmd funimass3 mpbid - frn sseqtr4d cnvco imaeq1i imaco eqtri syl6sseqr unssad sselda fnfvelrn + frn sseqtrrd cnvco imaeq1i imaco eqtri syl6sseqr unssad sselda fnfvelrn unssbd ffn sylan simp1d cdif mdvval difss eqsstri simprd anbi12d reeanv simpll xpeq12d sseq1d imbi12d spc2gv el2v 3anbi123d anbi2d imbi1d 3exp2 vtocl2 imp4b rexlimdvva sylbid exp4b 3imp2 mclsax eqeltrrd mpbir2and ) @@ -512211,7 +512211,7 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership mclspps $p |- ( ph -> ( S ` P ) e. ( K C B ) ) $= ( vz vw vm vo vs vp wfn ccnv co cima wcel cfv crn msubf syl ffnd cmax wss wf wceq cfn cotp cmpst w3a eqid mppspst sseldi elmpst sylib simp1d simpld - wa simp2d cv wral ralrimiva wfun wb ffund fdmd sseqtr4d funimass5 syl2anc + wa simp2d cv wral ralrimiva wfun wb ffund fdmd sseqtrrd funimass5 syl2anc cdm mpbird cmfs mvhf ffvelrnda elpreima adantr mpbir2and cun wbr 3ad2ant1 cxp wi wal 3ad2antl1 simp21 simp22 simp23 simp3 mclsppslem mclsind elmpps simprbi sseldd simplbda ) AIKVDZHIVENEFVFZVGZVHZHIVIYGVHZAKKIAIOVJZVHZKKI @@ -515398,7 +515398,7 @@ Set induction (or epsilon induction) vi unieqd eqeq12d cdm crn cmpt crecs cres df-rdg dfrecs3 wb w3a vex resex eqeq1 wrel relres reldm0 ax-mp syl6bb dmeq limeq syl rneq syl6eqr fveq12d df-ima id ifbieq12d eqid imaexg uniex fvex fvmpt cin dmres wss onelss imp - ifex 3adant2 fndm 3ad2ant2 sseqtr4d df-ss sylib syl5eq unieq onelon eloni + ifex 3adant2 fndm 3ad2ant2 sseqtrrd df-ss sylib syl5eq unieq onelon eloni word csuc ordzsl iftrue eqtr4d sucid fvres ordunisuc 3eqtr4a nsuceq0 neii w3o iffalsei wn nlimsucg iffalse eqtri 3eqtr4g reseq2 syl5ibrcom rexlimiv mp2b wne df-lim simp2bi neneqd iffalsed 3jaoi sylbi sylan9eqr mpdan 3expa @@ -515763,7 +515763,7 @@ Set induction (or epsilon induction) ctrpred cmpt crdg cres wrex wse eltrpred csuc peano2 simpr predeq3 sseq2d ssid rspcev ssiun sylancl wral setlikespec trpredlem1 sseld expcom adantl fvex wi syld ralrimiv ad2antrr iunexg sylancr nfcv cbviunv iuneq1 cbvmptv - syl5eq rdgeq1 reseq1 frsucmpt syl2anc sseqtr4d fveq2 dftrpred2 rexlimdva2 + syl5eq rdgeq1 reseq1 frsucmpt syl2anc sseqtrrd fveq2 dftrpred2 rexlimdva2 mp2b syl6sseqr syl5bi ) DABCUCZKDELZFMGFLZABGLZNZOZUDZABCNZUEZPUFZUAZKZEP UGCAKZABUHZQZABDNZWHRZGABECDFUIXBWSXDEPXBWIPKZQZWSQZWIUJZPKZXCXHWQUAZRZXD XGXEXIXBXEWSUBZWIUKSXGXCHWRABHLZNZOZXJXGWSXCXCRZXCXORZXFWSULXCUOWSXPQXCXN @@ -516877,7 +516877,7 @@ Set induction (or epsilon induction) frrlem8 $p |- ( z e. dom F -> Pred ( R , A , z ) C_ dom F ) $= ( vw vg va cv wcel wex wss wa syl cdm cop cpred vex eldm2 wfn wral cfv co cres wceq w3a cab cuni frrlem5 frrlem1 unieqi eqtri eleq2i eluniab wi wel - bitri simpr2r opeldm adantr simpr1 fndm eleqtrd rsp sseqtr4d 19.8a abeq2i + bitri simpr2r opeldm adantr simpr1 fndm eleqtrd rsp sseqtrrd 19.8a abeq2i sylc sylibr adantl elssuni syl6sseqr sstrd expcom exlimiv impcom sylbi dmss ) COZHUAZPWELOZUBZHPZLQDFWEUCZWFRZLWEHCUDZUEWIWKLWIWHMOZPZWMNOZUFZWO DRZWJWORZCWOUGZSZWEWMUHWEWMWJUJIUIUKCWOUGZULZNQZSZMQZWKWIWHXCMUMZUNZPXEHX @@ -516923,7 +516923,7 @@ Set induction (or epsilon induction) ( vz wcel wceq wss cv cdm cfv cpred cres co cop wex vex eldm2 wfn wral wa w3a cuni frrlem5 unieqi eqtri eleq2i eluniab bitri wi 19.8a 3ad2ant2 abid cab sylibr elssuni syl wel simpl23 simpl3 opeldm simpl21 fndm eleqtrd rsp - syl6sseqr sylc wfun simpl1 frrlem9 simpr funssfv syl3anc simp22r sseqtr4d + syl6sseqr sylc wfun simpl1 frrlem9 simpr funssfv syl3anc simp22r sseqtrrd adantr fun2ssres oveq2d 3eqtr4d mpdan 3exp exlimdv impcomd syl5bi imp ) A CUAZLUBRZWRLUCZWRLFHWRUDZUEZMUFZSZWSWRQUAZUGZLRZQUHAXDQWRLCUIZUJAXGXDQXGX FIUAZRZXIBUAZUKZXKFTZXAXKTZCXKULZUMZWRXIUCZWRXIXAUEZMUFZSZCXKULZUNZBUHZUM @@ -525967,7 +525967,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and B ) ) -> ( M i^i N ) e. ( ( nei ` J ) ` ( A i^i B ) ) ) $= ( cfv cin wss wa simpr wb simpl neiss2 neii1 neiint syl3anc ssinss1 3adant3 wcel syl ctop cnei w3a cnt cuni eqid mpbid inss2 3adant2 sstrid ssind simp1 - wceq ntrin sseqtr4d mpbird ) CUASZDACUBFZFSZEBURFSZUCZDEGZABGZURFSZVCVBCUDF + wceq ntrin sseqtrrd mpbird ) CUASZDACUBFZFSZEBURFSZUCZDEGZABGZURFSZVCVBCUDF ZFZHZVAVCDVEFZEVEFZGZVFVAVCVHVIUQUSVCVHHZUTUQUSIZAVHHZVKVLUSVMUQUSJVLUQACUE ZHZDVNHZUSVMKUQUSLZACDVNVNUFZMZACDVNVRNZACDVNVROPUGABVHQTRVAVCBVIABUHUQUTBV IHZUSUQUTIZUTWAUQUTJWBUQBVNHEVNHZUTWAKUQUTLBCEVNVRMBCEVNVRNZBCEVNVROPUGUIUJ @@ -526507,7 +526507,7 @@ conditions of the Five Segment Axiom ( ~ ax5seg ). See ~ brofs and ( N C_ X /\ ( ( F ` P ) i^i ~P N ) =/= (/) ) ) ) $= ( wcel vs vu vz vg va vb vn vf wa csn cnei cfv wss cpw cin c0 cuni ctopon wne ctop neibastop1 topontop syl adantr eqid neii1 wceq toponuni ad2antrr - sylan sseqtr4d cv wrex neii2 wral wi pweq ineq2d neeq1d raleqbi1dv elrab2 + sylan sseqtrrd cv wrex neii2 wral wi pweq ineq2d neeq1d raleqbi1dv elrab2 weq simprrr sspwb sylib sslin wb snssg ad3antlr mpbird fveq2 ineq1d rspcv simprrl ssn0 syl6an expr com23 expimpd syl5bi rexlimdv mpd jca ex n0 elin wex simprl sseqtrd cvv ciun cmpt crdg com cres crab cdif wf simpll simplr @@ -526925,7 +526925,7 @@ A Fne if ( S = (/) , { X } , U. S ) ) $= wf co fssres2 sylancr filtop xpexg mpdan ssexd fex syl2anc dirdm simpli syl syl6reqr feq2d mpbid cfg cima c1st wral cpw wfn tailf mpbird adantr eqid ffn imaeq2 sseq1d rexrn 3syl wfun wfo fo2nd fofn ax-mp ssv fnssres - mp2an fnfun ffvelrnda elpwid ad2antrr sseqtr4d fndm syl6sseqr funimass4 + mp2an fnfun ffvelrnda elpwid ad2antrr sseqtrrd fndm syl6sseqr funimass4 wb wbr simpr eleqtrd vex a1i eltail syl3anc biantrurd anbi1d filnetlem1 syl6bbr bitr4d imbi1d eleq1d bitri rexbidva csn ciun cop op1std imbi12d weq wne sseq1 cfbas wn sylib ss0b dmss eqeq2d anbi12d ralbidv2 raliunxp @@ -538721,7 +538721,7 @@ coordinates of a barycenter of two points in one dimension (complex fvex unex nfcv cmpt nfmpt1 nfcxfr nfrdg nffv nfcsb1 nfun ax-mp id csbeq1a rdgeq1 uneq12d rdgsucmptf mpan2 sseqtrrid sstr2 syl5com imim2d imp sseq1d syl5ibrcom adantr jaod syl5bi ralimdv2 cab df-sbc sucex sseq2d raleqbi1dv - cbvabv elab2 bitri syl6ibr wlim ciun ssiun2 adantl rdglim2a mpan sseqtr4d + cbvabv elab2 bitri syl6ibr wlim ciun ssiun2 adantl rdglim2a mpan sseqtrrd ex ralrimiva eleq2i abid sylibr a1d tfindes rsp syl eleq1 sseq12d imbi12d wb eleq12 mpbii vtocleg com12 pm2.43b ) ELMZFEMZFDBUAZNZEYDNZOZYBYCYGYCYB YCYGPZYCYBYHPZPIELYCIUBZEQZYIYKYIPJFEJUBZFQZYKYIYMYKUCZYJLMZYLYJMZYLYDNZY @@ -542085,7 +542085,7 @@ can be simplified (see ~ wl-dfrexf , ~ wl-dfrexv ). crg drngring frlmlmod lssmre csn cdif wral cacs clvec csca simpl eqeltrrd frlmsca islvec sylanbrc lssacsex simprd wss dif0 linds1 3ad2ant3 wf uvcff frnd syl6sseqr wceq fveq2i clspn mrclsp fveq1d clbs frlmlbs eqtr3d syl5eq - un0 lbssp sseqtr4d cnzr drngnzr adantr jca lindsind2 3expa sylanl1 eleq2d + un0 lbssp sseqtrrd cnzr drngnzr adantr jca lindsind2 3expa sylanl1 eleq2d wn wb ad2antrr mtbid ralrimiva 3impa ismri2 syl2an eqeltrid wo simpr enfi mpbird uvcendim mpbid olcd mreexexd cvv ovex elpwi ssdomg endomtr syl2anr rnex mpsyl rexlimiva ensymd domentr syl2anc ) AHIZBJIZCABUBUAZUCKIZUDZCAB @@ -543685,7 +543685,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= cvv ifbid oveq12d eqtrd mpteq2dv eqeq2d elrab2 breq12 sylan2 ancoms wb oveq1 cc ifbieq2d cz nnzd cn0 nn0red adantr oveq2d adantl csbied mpbid ovexd fvmptd fveq1d wfn elmapfn 1ex fnconstg ax-mp c0ex imain - pm3.2i ima0 fnun syldan wi eldif cab cfzo cmap crab sseqtr4d ssdifd + pm3.2i ima0 fnun syldan wi eldif cab cfzo cmap crab sseqtrrd ssdifd imadif difun2 fzsplit uncom difeq1d incom elfznn nnred fzdisj disj3 cn syl5eq 3eqtr4a sseqtrd sylan2br npcan1 sylan9eqr elfzm1b syl2anc nncnd cle elfzle1 0red nnm1nn0 lenltd iffalsed oveq2 oveq1d sylancr @@ -543783,7 +543783,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= oveq12d cr iftrued sylan9eqr oveq2d adantl csbied adantr ovexd fvmptd fveq1d wfn elmapfn fnconstg ax-mp c0ex pm3.2i imain ima0 fnun sylancr 1ex imaundi syldan wi eldif cab cfzo cmap cmin cof crab ssdifd imadif - sseqtr4d difun2 elfznn0 nn0p1nn nnuz syl6eleq cc nncnd npcan1 elfzuz3 + sseqtrrd difun2 elfznn0 nn0p1nn nnuz syl6eleq cc nncnd npcan1 elfzuz3 peano2uz eqeltrrd fzsplit2 syl2anc uncom difeq1d incom fzdisj 3eqtr4a syl5eq disj3 sseqtrd sylan2br nnred peano2rem cle elfzle2 ltm1d oveq2 id lelttrd oveq1 oveq1d syl5eqr mpbid inidm eqidd fvun2 mp3an12 sylan @@ -548257,7 +548257,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= 0mbl a1i ltpnf imaeq2d 3bitr2d ltadd1d ltaddsubd ltdivmul2d ltnled 3eqtrd elicopnf bitri covol mblvol eldifsn anim1d syl5bi ineq12d biimpri intnand eqeltrid iffalsed eqtr mpancom simpll eqnetrd ex orcom necon3bbii imbi12i - necomd imor ad2antll sseqtri reldisj wfun ffun difpreima difeq1d sseqtr4d + necomd imor ad2antll sseqtri reldisj wfun ffun difpreima difeq1d sseqtrrd dmmpti fdm neldifsn i1fima2 ovolsscl i1fd ) ADUEZUUCUBIZJZCUEZUCIZJZHBKBU EZEUFZWUHUGUDLZUDLZUHUFZUSUILZWUMUJLZWUKWUEUFZMNZWUROUKZJZWUQWURUUAZULZWU JBKWVBKWUJWUKKIZJZWVAWUQWURKWVEWUPWUMWVEWUOKIZWUPKIZWVEWUNKIZWVFWVEWULWUM @@ -552577,7 +552577,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= ( ctop wcel wa wss ccn co cfv crest syl2anc wb cv wral cres simp3l simp2l w3a cnrest ctopon crn simp1r toptopon sylib cima df-ima simp3r cdm wf cnf wfun ffun 3syl wceq fdm funimass4 mpbird eqsstrrid simp2r cnrest2 syl3anc - sseqtr4d mpbid ) EKLZFKLZMZBGNZCHNZMZDEFOPLZAUADQCLABUBZMZUFZDBUCZEBRPZFO + sseqtrrd mpbid ) EKLZFKLZMZBGNZCHNZMZDEFOPLZAUADQCLABUBZMZUFZDBUCZEBRPZFO PLZWBWCFCRPOPLZWAVRVOWDVNVQVRVSUDZVNVOVPVTUEZBDEFGIUGSWAFHUHQLZWBUIZCNVPW DWETWAVMWHVLVMVQVTUJFHJUKULWAWIDBUMZCDBUNWAWJCNZVSVNVQVRVSUOWADUSZBDUPZNW KVSTWAVRGHDUQZWLWFDEFGHIJURZGHDUTVAWABGWMWGWAVRWNWMGVBWFWOGHDVCVAVJABCDVD @@ -552744,7 +552744,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= sstrd syld eleq1 ac6sfi cdm ccnv w3a fdm feq2d ffn baibd ralbidva ralrab2 3jca simpr2 frnd ffnd simpr1 fnfi rnfi sylanbrc rpxr blssm iunin1 simplrr crn syl6sseq 0ss sseq1 mpbiri a1i simpr3 imbi12d rspccva ad5antr cnvimass - cr sstrid rpred simplbda syl22anc sseli ffvelrn simp-5r sseqtr4d fnfvelrn + cr sstrid rpred simplbda syl22anc sseli ffvelrn simp-5r sseqtrrd fnfvelrn blhalf ssiun2s adantlr pm2.61dne eqssd iuneq1 eqeq1d rspcev mpd ralrimdva exlimdv rexlimdvaa sylibrd impbid ) CEUDUEKZFELZMZDFUFUEKZFABUGZAUGZGUGZC UHUEZNZUIZLZBEUJZUKOZULZGPQZUXAUXBAUXCUXDUXEDUHUEZNZUIZFRZBFUJZUKOZULZGPQ @@ -552804,7 +552804,7 @@ curry M LIndF ( R freeLMod I ) ) ) $= unieq syl6eqr sseq2d ssabral sseq1 syl5bbr anbi12d syl12anc rexlimdvaa wf rspcev wex oveq1 eqeq2d ac6sfi adantrl adantl frn wfo simplrl dffn4 sylib wfn ffn fofi syl2anc sylanbrc simprrl adantr uniiun iuneq2 syl5eq sseqtrd - ad2antll eleq2d rexrn eliun 3bitr4g eqrdv sseqtr4d iuneq1 exlimddv impbid + ad2antll eleq2d rexrn eliun 3bitr4g eqrdv sseqtrrd iuneq1 exlimddv impbid ralbidv bitrd ) CEUBLMFENOZDFUCLMFAJPZAPZHPZCUDLZUGZQZNZJEUERUFZSZHUHUIFB PZUJZNZGPZYBTZAESZGYGUIZOZBRSZHUHUIAJCDEFHIUKXQYFYOHUHXQYFYOXQYDYOJYEXQXR YEMZYDOOZAXRYBULZUMZRMZYDYSYLGUNZNZYOYPYTXQYDYPXRRMZYRRMYTYPXRENZUUCXREUO @@ -553283,7 +553283,7 @@ counterexample is the discrete extended metric (assigning distinct fvexd wfn dffn5 sylib oveq2d syl5eq fveq2d 3eltr4d wex adantr istotbnd3 simprbi r19.21bi df-rex rexv bitr4i an32s ralrimiva eleq1 iuneq1 eqeq1d wf anbi12d ac6sfi syl2anc cixp wss elfpw simplbi ralimi ad2antll ss2ixp - fnfi fndm prdsbas rgenw ixpeq2 ax-mp syl6eqr ad2antrr sseqtr4d sylanbrc + fnfi fndm prdsbas rgenw ixpeq2 ax-mp syl6eqr ad2antrr sseqtrrd sylanbrc cdm ixpfi cxmet cxr metxmet rpxr blssm 3expa syl2an ssralv iunss sylibr eleq2d vex elixp wi eliun 3bitr4i eleq2 syl5bbr biimprd adantl eleqtrrd sylc ex syl6ibr ral2imi sylbid imp oveq1 ffn simpl anim12i biimpa ixpfn @@ -553781,7 +553781,7 @@ counterexample is the discrete extended metric (assigning distinct ne0i 0ex zex pwex frn ssexi abrexex elfi sylnibr cmptop cmpfi ibi fveq2 notbid neeq1d syl6bb rspccv syl3c wrel lmrel r19.23v albii eleq2 ralbii ceqsalv ralcom4 bitr3i elintab rspceeqv elab mpbir intss1 clsss3 sselda - 3bitr4i sseqtr4d c2 cdiv w3a cpm ccau 1zzd iscau3 simprd simp3 rphalfcl + 3bitr4i sseqtrrd c2 cdiv w3a cpm ccau 1zzd iscau3 simprd simp3 rphalfcl cc reximi breq2 2ralbidv rspccva syl2an cbl ffund simplr rspcv ad2antrl nnz cxr blopn blcntr clsndisj syl32anc fvelima sylan2 adantl reximdv ex syl5bi r19.29 uznnssnn simprlr simplrl elbl3 simprr lt2add rpcnd breq2d @@ -558163,7 +558163,7 @@ the next (since the empty set has a finite subcover, the ( vi wcel wss wa wceq cv wi wral w3a igenmin adantr sseq2 crngo cigen cfv co cidl igenidl igenss 3expia ralrimiva 3jca eleq1 sseq1 imbi2d 3anbi123d ralbidv syl5ibcom 3adant3r3 crab cint ssint ralrab sylbbr 3ad2ant3 adantl - adantlr igenval sseqtr4d eqssd ex impbid ) AUAJZBFKZLZABUBUDZEMZEAUEUCZJZ + adantlr igenval sseqtrrd eqssd ex impbid ) AUAJZBFKZLZABUBUDZEMZEAUEUCZJZ BEKZBCNZKZEVSKZOZCVPPZQZVMVNVPJZBVNKZVTVNVSKZOZCVPPZQVOWDVMWEWFWIABDFGHUF ABDFGHUGVKWIVLVKWHCVPVKVSVPJVTWGABVSRUHUISUJVOWEVQWFVRWIWCVNEVPUKVNEBTVOW HWBCVPVOWGWAVTVNEVSULUMUOUNUPVMWDVOVMWDLZVNEVKWDVNEKZVLVKVQVRWKWCABERUQVE @@ -566465,7 +566465,7 @@ inference form (e.g., cv w3a lveclmod syl islshpsm mpbid simp3d wpss simp1l simp2 csubg lshplss lsssssubg sseldd lspsncl syl2anc lsmub1 lsmub2 nelne1 sylan necomd df-pss lspsnid sylanbrc 3ad2ant1 lsmcl syl3anc lssss simpr adantlr 3adant2 lsmcv - sseqtr4d syl211anc simp3 eqtrd rexlimdv3a mpd lshpnel impbida ) AHCQUAZCH + sseqtrrd syl211anc simp3 eqtrd rexlimdv3a mpd lshpnel impbida ) AHCQUAZCH UBERZBUCZFUDZAWLUEZCPULZUBERBUCZFUDZPFUFZWOAWTWLACGUGRZQZCFUHZWTACDQZXBXC WTUMNAPBXACDEFGIJXAUIZKLAGUJQZGUKQZMGUNUOZUPUQURSWPWSWOPFWPWQFQZWSUMZWNWR FXJAXICWNUSZWNWRTZWNWRUDAWLXIWSUTWPXIWSVAWPXIXKWSWPCWNTZCWNUHXKAXMWLACGVB @@ -567446,7 +567446,7 @@ inference form (e.g., lcvexchlem4 $p |- ( ph -> ( T i^i U ) C U ) $= ( wss wa wceq wi wcel syl3anc vs vr cin wbr wpss cv wo wral clmod lsmcl co lcvpss lcvexchlem1 mpbid w3a csubg cfv 3ad2ant1 lsssssubg syl sseldd - simp2 lsmub2 syl2anc simp3r lsmless1 cabl lsmcom sseqtr4d lcvbr3 adantr + simp2 lsmub2 syl2anc simp3r lsmless1 cabl lsmcom sseqtrrd lcvbr3 adantr lmodabl wb simpr sseq2 sseq1 anbi12d eqeq1 orbi12d imbi12d rspcv sylbid adantld 3adant3 mpd mp2and ineq1 simp3l lcvexchlem2 eqeq1d syl5ib sylib sseqin2 eqeq12d orim12d 3exp ralrimiv lssincl mpbir2and ) AEFUCZFBUDWTF @@ -567574,7 +567574,7 @@ inference form (e.g., lsatexch $p |- ( ph -> R C_ ( U .(+) Q ) ) $= ( wcel co csubg cfv wss clmod clvec syl lsssssubg sseldd lsatlssel lsmub2 lveclmod syl2anc clcv eqid lsmcl syl3anc wpss wbr cin csn wceq lcvp mpbid - lcvpss lsmub1 wa wb lsmlub mpbi2and psssstrd lcv2 lcvnbtwn2 sseqtr4d ) AE + lcvpss lsmub1 wa wb lsmlub mpbi2and psssstrd lcv2 lcvnbtwn2 sseqtrrd ) AE GECUAZGDCUAZAGHUBUCZTZEVQTZEVOUDAFVQGAHUETZFVQUDAHUFTVTNHULUGZFHJUHUGZOUI ZAFVQEWBABFEHJMWAQUJZUIZCGEHKUKUMAHUNUCZGFVOVPHUFJWFUOZNOAVTGFTZEFTVOFTWA OWDCFGEHJKUPUQZAVTWHDFTVPFTWAOABFDHJMWAPUJZCFGDHJKUPUQZAGVOURGVOWFUSAGVPV @@ -567843,7 +567843,7 @@ inference form (e.g., l1cvat $p |- ( ph -> ( ( Q .(+) R ) i^i U ) e. A ) $= ( cin cabl wcel csubg cfv wceq clmod clvec lveclmod syl lmodabl lsssssubg co wss lsatlssel sseldd lsmcom syl3anc ineq1d incom syl6eq necomd lsatssv - l1cvpat sseqtr4d lsatcvat3 eqeltrd ) AEFDUOZHUCZHFEDUOZUCZBAVKVLHUCVMAVJV + l1cvpat sseqtrrd lsatcvat3 eqeltrd ) AEFDUOZHUCZHFEDUOZUCZBAVKVLHUCVMAVJV LHAJUDUEZEJUFUGZUEFVOUEVJVLUHAJUIUEZVNAJUJUEVPPJUKULZJUMULAGVOEAVPGVOUPVQ GJLUNULZABGEJLNVQRUQURAGVOFVRABGFJLNVQSUQURDEFJMUSUTVAVLHVBVCABDFEGHJLMNP QSRAEFTVDUBAFIHEDUOABFIJKNVQSVEABCDEGHIJKLMNOPQRUAUBVFVGVHVI $. @@ -568648,7 +568648,7 @@ Functionals and kernels of a left vector space (or module) lkrscss $p |- ( ph -> ( L ` G ) C_ ( L ` ( G oF .x. ( V X. { R } ) ) ) ) $= ( cfv csn cxp cof co wss c0g wceq wa clvec wcel clmod lveclmod syl lkrssv eqid lfl0sc fveq2d lfl0f lkr0f syl2anc2 mpbiri eqtr2d sseqtrd adantr sneq - wb xpeq2d oveq2d adantl sseqtr4d wne simpr lkrsc eqimss2 pm2.61dane ) AFH + wb xpeq2d oveq2d adantl sseqtrrd wne simpr lkrsc eqimss2 pm2.61dane ) AFH TZFICUAZUBZDUCZUDZHTZUEZCBUFTZACWCUGZUHVPFIWCUAZUBZVSUDZHTZWAAVPWHUEWDAVP IWHAEFHIJKOPAJUIUJZJUKUJZQJULUMZRUNAWHWFHTZIAWGWFHABDEFGIJWCKLOMNWCUOZWKR UPUQAWLIUGZWFWFUGZWFUOAWJWFEUJWNWOVFWKBEIJWCLWMKOURBEWFHIJWCLWMKOPUSUTVAV @@ -570184,7 +570184,7 @@ Functionals and kernels of a left vector space (or module) $( The kernel of a scalar product of a functional includes the kernel of the functional. (Contributed by NM, 27-Jan-2015.) $) lkrss $p |- ( ph -> ( L ` G ) C_ ( L ` ( X .x. G ) ) ) $= - ( cfv cbs csn cxp cmulr cof co eqid lkrscss clvec ldualvs fveq2d sseqtr4d + ( cfv cbs csn cxp cmulr cof co eqid lkrscss clvec ldualvs fveq2d sseqtrrd ) AFHTFIUATZJUBUCCUDTZUEUFZHTJFDUFZHTACJUNEFGHUMIUMUGZKLUNUGZMNQRSUHAUPUO HABCDUNEFGUMIJUIMUQKLUROPQSRUJUKUL $. $} @@ -570236,7 +570236,7 @@ Functionals and kernels of a left vector space (or module) clvec ldualsbase eleqtrrd ldualelvbase lvecvs0or wcel lveclmod syl ldual0 clmod eqeq2d cdif eldifsni a1d necon4d sylbid idd jaod syl6ibr con3d orrd nne ianor sylibr wss wpss df-pss ldualvscl eqeltrd lkrpssN syl5rbbr lkrss - fveq2d sseqtr4d biantrurd bitr4d necon2bbid mpbird eqcomd ) AIJUDZHJUDZAW + fveq2d sseqtrrd biantrurd bitr4d necon2bbid mpbird eqcomd ) AIJUDZHJUDZAW RWSUEICUFUDZUGZHWTUEZUHZUIZAXAUIZXBUIZUJXDAXEXFAXBXEAXBIWTUEZXEAXBBIFUKZW TUEZXGAHXHWTUCULAXIBCUMUDZUFUDZUEZXGUJXGABFXJXJUNUDZXKCUNUDZCIWTXNUOZSXJU OZXMUOZXKUOZWTUOZACKRTUPABDXMABDLUQZUAURZACXJEXMDUSKMNRXPXQTUTVAACGIXNKUS @@ -583352,7 +583352,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ). simp1 paddssat paddunssN polcon3N club cpmap cple wbr cbs hlclat 3ad2ant1 ccla unss biimpi 3adant1 atssbase pmapssbaN polssatN 3adant3 3adant2 3jca cjn 2polssN paddss12 sylc 2polvalN oveq12d sseqtrd clat hllat simp2 simp3 - wa jca pmapjoin sstrd lubun fveq2d sseqtr4d lubss pmaple 2pmaplubN eqtr4d + wa jca pmapjoin sstrd lubun fveq2d sseqtrrd lubss pmaple 2pmaplubN eqtr4d wb mpbid 3sstr4d 2polcon4bN eqssd ) EJKZCALZDALZUAZCDBMZFNZCDUBZFNZWNWKWO ALZWQWOLWPWRLWKWLWMUCZAJBECDGHUDZAJBECDGHUEAEFWQWOGIUFOWNWPFNZWRFNZLZWRWP LZWNWOEUGNZNZEUHNZNZWQXFNZXHNZXFNZXHNZXBXCWNXGXLEUINZUJZXIXMLZWNEUNKZXKEU @@ -583832,7 +583832,7 @@ one element is a lattice line (expressed as the join ` P .\/ Q ` ). osumcllem2N $p |- ( ( ( K e. HL /\ X C_ A /\ Y C_ A ) /\ p e. U ) -> X C_ ( U i^i M ) ) $= ( chlt wcel wss w3a cv wa cin csn co simpl1 simpl2 simpr snssd cfv adantr - paddssat polssatN syl2anc eqsstrid sspadd1 syl6sseqr osumcllem1N sseqtr4d + paddssat polssatN syl2anc eqsstrid sspadd1 syl6sseqr osumcllem1N sseqtrrd sstrd syl3anc ) FUAUBZJAUCZKAUCZUDZLUEZDUBZUFZJHDHUGVLJJVJUHZCUIZHVLVFVGV MAUCJVNUCVFVGVHVKUJZVFVGVHVKUKVLVMDAVLVJDVIVKULUMVLDJKCUIZIUNZIUNZATVLVFV QAUCZVRAUCVOVLVFVPAUCZVSVOVIVTVKAUACFJKOPUPUOAFIVPOQUQURAFIVQOQUQURUSVDAU @@ -584005,7 +584005,7 @@ X C_ ( U i^i M ) ) $= X .<_ ( ._|_ ` Y ) ) -> ( M ` ( X .\/ Y ) ) = ( ( M ` X ) .+ ( M ` Y ) ) ) $= ( chlt wcel cfv wceq syl2anc w3a wbr wa co cpolN eqid adantr cpscN simpl1 pmapj2N wss simpl2 pmapsubclN simpl3 wb cops 3ad2ant1 simp3 opoccl pmaple - hlop syld3an3 biimpa polpmapN sseqtr4d osumclN syl31anc psubcli2N eqtrd ) + hlop syld3an3 biimpa polpmapN sseqtrrd osumclN syl31anc psubcli2N eqtrd ) DPQZHAQZIAQZUAZHIGRZEUBZUCZHICUDFRZHFRZIFRZBUDZDUERZRWARZVTVMVQWBSVOABCDF WAHIJLMOWAUFZUJUGVPVJVTDUHRZQZWBVTSVJVKVLVOUIZVPVJVRWDQZVSWDQZVRVSWARZUKW EWFVPVJVKWGWFVJVKVLVOULAWDDFHJMWDUFZUMTVPVJVLWHWFVJVKVLVOUNZAWDDFIJMWJUMT @@ -584210,7 +584210,7 @@ to be equivalent to (and derivable from) the orthomodular law ~ poml4N . C_ ( F ` ( ( X .\/ Y ) .\/ ( ( X .\/ W ) ./\ ( Y .\/ V ) ) ) ) ) $= ( chlt wcel w3a wa cfv co cin catm wss simpl1 hllatd simpl2 simpl3 latjcl clat syl3anc eqid pmapssat syl2anc simpr2 simpr3 3jca pmapjoin ss2in wceq - latmcl pmapmeet sseqtr4d jca paddss12 sylc sstrd ) EUAUBZKAUBZLAUBZUCZMAU + latmcl pmapmeet sseqtrrd jca paddss12 sylc sstrd ) EUAUBZKAUBZLAUBZUCZMAU BZJAUBZIAUBZUCZUDZKCUEZLCUEZBUFZWBJCUEBUFZWCICUEBUFZUGZBUFZKLDUFZCUEZKJDU FZLIDUFZGUFZCUEZBUFZWIWMDUFCUEZWAVMWJEUHUEZUIZWNWQUIZUCWDWJUIZWGWNUIZUDWH WOUIWAVMWRWSVMVNVOVTUJZWAVMWIAUBZWRXBWAEUOUBZVNVOXCWAEXBUKZVMVNVOVTULZVMV @@ -608557,7 +608557,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, ( vy chlt wcel wa wss cv cfv wbr wb eqid vf vs cdm wne cple cbs crab ciin vh wceq simpl simprl dibdmN sseq2d adantr mpbid simprr wrel dibvalrel wex c0 wrex n0 biimpi ad2antll a1d ancld eximdv mpd df-rex sylibr reliin cdia - syl id cltrn cid cres cmpt cop diadm sseqtr4d diaglbN syl12anc eleq2d cvv + syl id cltrn cid cres cmpt cop diadm sseqtrrd diaglbN syl12anc eleq2d cvv wral vex eliin ax-mp syl6bb anbi1d r19.27zv bitr4d hlclat ad2antrr ssrab2 ccla syl6ss clatglbcl syl2anc clat hllat ad3antrrr simplrl sselda lhpbase ad3antlr simpr clatglble syl3anc breq1 elrab sylib simprd lattrd exlimddv @@ -608689,7 +608689,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, diblss $p |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) e. S ) $= ( chlt wcel cfv eqid syl3anc vx va vb vh vs vt wa wbr ctendo cplusg cvsca - csca cltrn cxp eqidd cbs wceq dvhbase eqcomd adantr dvhvbase a1i sseqtr4d + csca cltrn cxp eqidd cbs wceq dvhbase eqcomd adantr dvhvbase a1i sseqtrrd clss dibss dibn0 cv w3a c1st c2nd ccom cop co cdia cid cres cmpt csn fvex coex op1st coeq1i simpll simpr1 simplr simpr2 dibelval1st1 tendocl simpr3 vex ctrl ltrnco simplll hllatd trlcl syl2anc latjcl simplrl trlco tendotp @@ -609098,7 +609098,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, /\ ( X e. ( I ` Q ) /\ Y e. ( I ` Q ) ) ) -> ( X .+ Y ) e. ( I ` Q ) ) $= ( wcel cfv eqid vg vs vt vh chlt wa wbr wn w3a c1st ccom c2nd csca cplusg - cop cltrn ctendo cxp wceq simp1 wss cbs dicssdvh dvhvbase eqcomd sseqtr4d + cop cltrn ctendo cxp wceq simp1 wss cbs dicssdvh dvhvbase eqcomd sseqtrrd co adantr 3adant3 simp3l sseldd simp3r dvhvadd syl12anc cv crio cmpt cmpo coc dicelval2nd 3adant3r 3adant3l lhpocnel 3ad2ant1 ltrniotacl tendospdi2 simp2 syl3anc dvhfplusr fveq1d dicelval1sta coeq12d 3eqtr4rd tendoplcl wb @@ -609134,7 +609134,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, /\ ( X e. E /\ Y e. ( I ` Q ) ) ) -> ( X .x. Y ) e. ( I ` Q ) ) $= ( cfv vg chlt wcel wa wbr wn w3a co c1st cid c2nd ccom cop cltrn cxp wceq - simp1 simp3l wss cbs eqid dicssdvh dvhvbase eqcomd adantr sseqtr4d simp3r + simp1 simp3l wss cbs eqid dicssdvh dvhvbase eqcomd adantr sseqtrrd simp3r 3adant3 sseldd dvhvsca syl12anc fvi syl coeq1d opeq2d eqtr4d dicelval1sta coc cv crio 3adant3l fveq2d wf dicelval2nd tendof lhpocnel 3ad2ant1 simp2 syl2anc ltrniotacl syl3anc fvco3 fveq1d tendococl eqeltrd fvex dicopelval @@ -609185,7 +609185,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, diclss $p |- ( ( ( K e. HL /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) e. S ) $= ( chlt wcel wa cfv eqid vx va vb wbr ctendo cplusg cvsca csca cltrn eqidd - wn cxp cbs wceq dvhbase eqcomd adantr dvhvbase clss a1i dicssdvh sseqtr4d + wn cxp cbs wceq dvhbase eqcomd adantr dvhvbase clss a1i dicssdvh sseqtrrd dicn0 cv w3a co simpll simplr simpr1 simpr2 dicvscacl syl112anc dicvaddcl simpr3 islssd ) GPQIEQRZBAQBIHUDUKRZRZUAIGUESSZDUFSZCDUGSZBFSZDUHSZIGUISS ZVSULZDUBUCVRWCUJVPVSWCUMSZUNVQVPWFVSWFDVSWCEGIPLVSTZMWCTWFTUOUPUQVPWEDUM @@ -609431,7 +609431,7 @@ all translations (for a fiducial co-atom ` W ` ). (Contributed by NM, /\ ( R e. A /\ -. R .<_ W ) /\ ( X e. B /\ X .<_ W ) ) /\ R .<_ ( Q .\/ X ) ) -> ( J ` R ) C_ ( ( J ` Q ) .(+) ( I ` X ) ) ) $= ( chlt wcel wa wbr wn w3a co cfv cid cres cop csn simp1 simp22 diclspsn - wceq syl2anc simp21 cdlemn4a syl3anc oveq1d sseqtr4d csubg clss dvhlmod + wceq syl2anc simp21 cdlemn4a syl3anc oveq1d sseqtrrd csubg clss dvhlmod clmod eqid lsssssubg diclss sseldd simp23 diblss ctrl cdlemn2a lsmless2 wss syl sstrd eqsstrd ) QVAVBUBMVBVCZEAVBEUBRVDVEVCZFAVBFUBRVDVEVCZUCBV BUCUBRVDVCZVFZFEUCPVGRVDZVFZFOVHZLVIGVJZVKVLTVHZEOVHZUCNVHZDVGZXFWTXBXG @@ -610940,7 +610940,7 @@ of phi(x) is independent of the atom q." (Contributed by NM, co latjcl cdic cdib simp11 simp3lr latlej1 simp11r lhpbase lattr syl13anc syl wi mpand mtod simp3l simp12 lhple oveq2d eqid dihvalcq syl122anc clss csubg clmod dvhlmod lsssssubg diclss sseldd latmcl diblss syl12anc lsmub1 - latmle2 simp13 simp3r syl112anc sseqtr4d fveq2d dihvalb eqtr4d eqsstrd wb + latmle2 simp13 simp3r syl112anc sseqtrrd fveq2d dihvalb eqtr4d eqsstrd wb simp2 dihlss lsmlub mpbi2and syl121anc mpbid lattrd 3expia exp4c rexlimdv dihord4 imp4a mpd ) HUDUEZKEUEZUFZLBUEZLKIUGZUFZMBUEZMKIUGUHZUFZUIZLFUJZM FUJZUKZUFZUCULZKIUGZUHZYRMKJVHZGVHMUMZUFZUCAUNZLMIUGZYQYFYLUUDYFYIYLYPUOY @@ -611579,7 +611579,7 @@ of phi(x) is independent of the atom q." (Contributed by NM, -> ( I ` ( G ` T ) ) = |^|_ x e. T ( I ` x ) ) $= ( chlt wcel wa wss c0 wne cfv wbr w3a cv ciin cdm wceq simp1 crab co wrex wi clat simp11l hllatd simp12l sseldd simp11r lhpbase syl latmle2 syl3anc - simp3 3expia biimprcd rexlimdv ss2rabdv eqsstrid dibdmN 3ad2ant1 sseqtr4d + simp3 3expia biimprcd rexlimdv ss2rabdv eqsstrid dibdmN 3ad2ant1 sseqtrrd breq1 syl6 dihglblem2aN 3adant3 syl12anc dihglblem2N fveq2d simpl1 sselda dibglbN 3adant2r elrab sylib dihvalb iineq2dv 3eqtr4rd ccla simp1l hlclat syl2anc simp2l clatglbcl 3eqtr2rd ) KUCUDZNHUDZUEZEDUFZEUGUHZUEZEGUIZNLUJ @@ -612863,7 +612863,7 @@ of phi(x) is independent of the atom q." (Contributed by NM, -> ( I ` ( G ` { x e. B | S C_ ( I ` x ) } ) ) = |^| { y e. ran I | S C_ y } ) $= ( vz wcel wa chlt wss cv cfv crab ciin crn cint wne wceq simpl ssrab2 a1i - c0 cp1 cops hlop ad2antrr eqid op1cl syl simpr dih1 adantr sseqtr4d fveq2 + c0 cp1 cops hlop ad2antrr eqid op1cl syl simpr dih1 adantr sseqtrrd fveq2 sseq2d elrab sylanbrc ne0d dihglb syl12anc wrex cab fvex dfiin2 wex dihfn wb wfn fvelrnb eqcom rexbii df-rex bitri syl6bb pm5.32rd weq anbi1i sseq2 anbi2d pm5.32ri an32 3bitr2i 19.41v 3bitrri syl6rbb abbidv df-rab syl6eqr @@ -613566,7 +613566,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= dochsat $p |- ( ph -> ( ( ._|_ ` ( ._|_ ` Q ) ) e. A <-> Q e. A ) ) $= ( cfv wcel wceq adantr wa c0g csn clsm co wpss wss wne clmod dvhlmod eqid lss0ss syl2anc simpr lsatn0 fveq2d dochoc0 eqtrd ex necon3d necomd df-pss - mpd sylanbrc chlt cbs lssss syl dochocss csubg lsatlssel lsssubg sseqtr4d + mpd sylanbrc chlt cbs lssss syl dochocss csubg lsatlssel lsssubg sseqtrrd lsm02 dvhlvec lsssn0 lsmsatcv mpd3an23 eqtr2d eqeltrrd cdih crn dih1dimat clvec sylan dochoc eqeltrd impbida ) ACHQZHQZBRZCBRZAWKUAZWJCBWMCEUBQZUCZ WJEUDQZUEZWJWMWOCUFZCWQUGCWQSWMWOCUGZWOCUHWRWMEUIRZCDRZWSAWTWKAEFGIJLOUJT @@ -613594,7 +613594,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= ( ( ._|_ ` ( ._|_ ` X ) ) =/= X <-> ( ._|_ ` ( ._|_ ` X ) ) = V ) ) $= ( cfv wcel adantr wss vv wne wceq wa cv csn clspn clsm wrex clss w3a eqid co dvhlmod islshpsm mpbid simp3d wpss id adantlr 3adant3 chlt lshplss syl - dochocss syl2anc 3ad2ant1 simp1r necomd sylanbrc dochssv sseqtr4d dvhlvec + dochocss syl2anc 3ad2ant1 simp1r necomd sylanbrc dochssv sseqtrrd dvhlvec lssss df-pss simp3 dochlss simpr lsmcv syl3anc rexlimdv3a eqnetrd impbida eqtrd mpd lshpne ) AHEQZEQZHUBZWHFUCZAWIUDZHUAUEZUFBUGQZQBUHQZUMZFUCZUAFU IZWJAWQWIAHBUJQZRZHFUBZWQAHIRWSWTWQUKPAUAWNWRHIWMFBMWMULZWRULZWNULZNABCDG @@ -613970,7 +613970,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= $( Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) $) djhspss $p |- ( ph -> ( N ` ( X u. Y ) ) C_ ( X .\/ Y ) ) $= - ( cfv wcel cun coch co eqid unssd dochspss chlt wss wceq djhval2 sseqtr4d + ( cfv wcel cun coch co eqid unssd dochspss chlt wss wceq djhval2 sseqtrrd wa syl3anc ) AIJUAZFSUNHEUBSSZSUOSZIJDUCZABCEFUOGHUNKLUOUDZMNPAIJGQRUEUFA EUGTHCTULIGUHJGUHUQUPUIPQRBCDEUOGHIJKLMUROUJUMUK $. $} @@ -614007,7 +614007,7 @@ x C_ ( ( ( DIsoH ` K ) ` w ) ` y ) } ) ) ) ) ) ) $= dihsumssj $p |- ( ph -> ( ( I ` X ) .(+) ( I ` Y ) ) C_ ( I ` ( X .\/ Y ) ) ) $= ( cfv co cdjh cbs eqid chlt wcel wa wss dihss syl2anc djhsumss wceq djhlj - syl12anc sseqtr4d ) AJFUAZKFUAZCUBUQURIHUCUAUAZUBZJKGUBFUAZACDEUSHDUDUAZI + syl12anc sseqtrrd ) AJFUAZKFUAZCUBUQURIHUCUAUAZUBZJKGUBFUAZACDEUSHDUDUAZI UQURMOVBUEZPUSUEZRAHUFUGIEUGUHZJBUGZUQVBUIRSBDEFHVBIJLMQOVCUJUKAVEKBUGZUR VBUIRTBDEFHVBIKLMQOVCUJUKULAVEVFVGVAUTUMRSTBEFUSGHIJKLNMQVDUNUOUP $. $} @@ -619616,7 +619616,7 @@ Part of property (e) in [Baer] p. 40. (Contributed by NM, mapdlsm $p |- ( ph -> ( M ` ( X .(+) Y ) ) = ( ( M ` X ) .+b ( M ` Y ) ) ) $= ( co cfv ccnv wss csubg wcel clss clmod lcdlmod eqid lsssssubg syl sseldd - mapdcl2 lsmub1 syl2anc mapdcl mapdlsmcl mapdcnvid2 sseqtr4d mapdord mpbid + mapdcl2 lsmub1 syl2anc mapdcl mapdlsmcl mapdcnvid2 sseqtrrd mapdord mpbid mapdcnvcl lsmub2 wa wb dvhlmod lsmlub syl3anc mpbi2and lsmcl mpbird eqssd sseqtrd ) AKLDUCZIUDZKIUDZLIUDZCUCZAVRWAIUEUDZIUDZWAAVRWCUFVQWBUFZAKWBUFZ LWBUFZWDAVSWCUFWEAVSWAWCAVSBUGUDZUHZVTWGUHZVSWAUFABUIUDZWGVSABUJUHWJWGUFA @@ -624569,7 +624569,7 @@ fixed reference functional determined by this vector (corresponding to wcel hdmapcl eldifad lmodvacl hdmaprnlem1N lmodindp1 sylanbrc lsatlspsn syl3anc eldifsn mapdcnvatN hdmaprnlem3uN hdmaprnlem3N clsm cpr wss clss necomd dvhlmod lspsncl syl2anc mapdcl2 lsmcl csubg lsssssubg syl sseldd - lspsnid hdmap10 eleqtrrd lspsnel5 lsmelvali syl22anc lspsnel5a sseqtr4d + lspsnid hdmap10 eleqtrrd lspsnel5 lsmelvali syl22anc lspsnel5a sseqtrrd eqimss2 mpbird mapdlsm crn mapdrn2 mapdcl mapdcnvordN lsmpr mapdcnvid1N eqtr4d lsatfixedN wa simpr chlt ad2antrr hdmaprnlem4tN mapdcnv11N mpbid wn simplr ex reximdva mpd ) ACUTZJVAZTUTZHVBZVCNVAZOVDZVAZYLDUTZGVBZVCP @@ -625813,7 +625813,7 @@ fixed reference functional determined by this vector (corresponding to mapdsn eqtr3d eleqtrd wb lcdvbaselfl sseq2d elrab3 syl mpbid adantr clvec clsh dvhlvec chlt cdif anim1i eldifsn sylibr dochsnshp cxp lcd0v hdmapeq0 csca eqeq2d bitr3d biimpar lkrshp syl3anc lshpcmp eqimss2 lmod0vcl lkrssv - necon3bid dvhlmod doch0 sseqtr4d pm2.61ne eqssd ) AJBUBZKUBZJUCZGUBZAXGXI + necon3bid dvhlmod doch0 sseqtrrd pm2.61ne eqssd ) AJBUBZKUBZJUCZGUBZAXGXI UDZCUEUBZBUBZKUBZXKUCZGUBZUDJXKJXKUFZXGXMXIXOXPXFXLKJXKBUGUHXPXHXNGJXKUIU HUJAJXKUKZULZXIXGUFZXJXRXIXGUDZXSAXTXQAXFXIUAUPZKUBZUDZUACUMUBZUNZUOZXTAX FXFUCIFUQUBUBZURUBZUBZYEAYGUSUOXFYGUTUBZUOXFYIUOAYGEFILYGVAZSVBAYGYJBJCEF @@ -637829,7 +637829,7 @@ group element in (1,2), contradicting ~ pell14qrgapw . (Contributed by fnwe2lem2 $p |- ( ph -> E. b e. a A. c e. a -. c T b ) $= ( ve vd vf vg cv wbr wn cres cima wral wrex cvv wcel wfr wss c0 wf wfun wne ffun vex funimaex 3syl wwe wefr syl crn imassrn frnd sstrid cdm cin - incom wceq sseqtr4d df-ss sylib syl5eq eqnetrd imadisj necon3bii sylibr + incom wceq sseqtrrd df-ss sylib syl5eq eqnetrd imadisj necon3bii sylibr fdmd fri syl22anc cfv df-ima rexeqi wfn wb fnssres syl2anc breq2 notbid ralbidv rexrn syl5bb wel wa raleqi breq1 ralrn adantr resabs1d ad2antrr ffnd fveq1d fvres adantl eqtrd ad2antlr breq12d ralbidva bitrd rexbidva @@ -638553,7 +638553,7 @@ element in each topology (which need not be in the closed set - if it Stefan O'Rear, 1-Jan-2015.) $) lnmlsslnm $p |- ( ( M e. LNoeM /\ U e. S ) -> R e. LNoeM ) $= ( va clnm wcel wa clmod cress co clfig cfv sylan wss wceq cbs eqid clss - cv wral lnmlmod lsslmod oveq1i simplr adantl ressbas2 ad2antlr sseqtr4d + cv wral lnmlmod lsslmod oveq1i simplr adantl ressbas2 ad2antlr sseqtrrd lssss ressabs syl2anc syl5eq simpll wb lsslss simprbda lnmlssfg eqeltrd syl ralrimiva islnm sylanbrc ) DHIZCBIZJZAKIZAGUBZLMZNIZGAUAOZUCAHIVFDK IZVGVIDUDZBCDAFEUEPVHVLGVMVHVJVMIZJZVKDVJLMZNVQVKDCLMZVJLMZVRAVSVJLFUFV @@ -638617,7 +638617,7 @@ element in each topology (which need not be in the closed set - if it ( cress clfig cfv cfn wcel eqid vx cima co cv clspn wceq cbs cpw cin wrex clmod wb clmhm lmhmlmod1 syl islssfg2 syl2anc mpbid wa inss1 sseli elpwid wss lmhmlsp syl2an oveq2d lmhmlmod2 adantr crn imassrn wf lmhmf frnd cres - sstrid wfo inss2 wfun cdm ffund fdmd sseqtr4d fores fofi islssfgi syl3anc + sstrid wfo inss2 wfun cdm ffund fdmd sseqtrrd fores fofi islssfgi syl3anc adantl eqeltrd imaeq2 eleq1d syl5ibcom rexlimdva mpd eqeltrid ) AHDFBUBZO UCZPIAUAUDZCUEQZQZBUFZUACUGQZUHZRUIZUJZWPPSZAGPSZXDLACUKSZBESXFXDULAFCDUM UCSZXGNCDFUNUOMXAEBWRCGUAJKWRTZXATZUPUQURAWTXEUAXCAWQXCSZUSZDFWSUBZOUCZPS @@ -639569,7 +639569,7 @@ of ideals (the usual "pure ring theory" definition). (Contributed by simprl breq1d rexrab abbii eqtri syl6eqr adantr sseqtrd simprr fipreima wi ssrab2 sstr2 mpi adantl velpw sylibr adantrr elind cbs syl6ss lidlss ply1ring sstrd rspcl cres df-ima wral rspssid simprbi ad2antrl sylanbrc - ssrab resmptd resmpt eqtr4d syl6eqssr rnss eqsstrid sseqtr4d rspssp jca + ssrab resmptd resmpt eqtr4d syl6eqssr rnss eqsstrid sseqtrrd rspssp jca resss sseq1d anbi2d syl5ibcom sylan2b expimpd reximdv2 sseq1 syl5ibrcom mpd rexbidv rexlimdva ) AIGDRRZUAUNZCUERZRZUFZUAYOUGUHUIZUJZYOIFUNZHRZD RRZSZFGUGZUHUIZUJZACUKTZYOCULRZTZUUANACUMTZGETZIUOTZUUKAUUIUULNCUPUQZOP @@ -644154,7 +644154,7 @@ Transitive relations (not to be confused with transitive classes). $( Concrete construction of a superclass of relation ` R ` which is a transitive relation. (Contributed by RP, 25-Dec-2019.) $) trrelsuperreldg $p |- ( ph -> ( R C_ S /\ ( S o. S ) C_ S ) ) $= - ( wss ccom cdm crn cxp relssdmrn syl sseqtr4d xptrrel a1i coeq12d 3sstr4d + ( wss ccom cdm crn cxp relssdmrn syl sseqtrrd xptrrel a1i coeq12d 3sstr4d wrel jca ) ABCFCCGZCFABBHZBIZJZCABRBUCFDBKLEMAUCUCGZUCTCUDUCFAUAUBNOACUCC UCEEPEQS $. $} @@ -645114,7 +645114,7 @@ over the natural numbers (including zero) is equivalent to the ( vx vy vi vz vj wcel cv crelexp wbr wrex wa cvv cuz cfv wceq cn0 co wi w3a wal wss caddc ovexd simprlr simpll2 eleqtrd simpll3 simprll eluznn0 ccom wex syl2anc uzaddcl simplr 3eltr4d vex brcogw mp3an simprr simpll1 - simprl relexpaddss syl3anc oveq2d sseqtr4d ssbrd syl5 impr jca spcimedv + simprl relexpaddss syl3anc oveq2d sseqtrrd ssbrd syl5 impr jca spcimedv exlimdvv reeanv r2ex bitr3i df-rex 3imtr4g alrimiv briunov2uz weq oveq2 cotr breqd cbvrexv syl6bb anbi12d imbi12d albidv syl5bb biimprd 3adant3 ex mpd ) BFNZEDUAUBZUCZDUDNZUGZIOZJOZBKOZPUEZQZKERZXGLOZBMOZPUEZQZMERZS @@ -649909,7 +649909,7 @@ by analogy M(x) would be an 'adherent system'. Benoit Jubin suggested clsk1indlem2 $p |- A. s e. ~P 3o s C_ ( K ` s ) $= ( cv cfv wss c3o cpw wcel c0 csn wceq c1o cpr cif wa wn id sseq2 wo a1i snsspr1 syl6eqss ancli con3i ssid jctir orri elimif sylibr weq ifbieq2d - eqeq1 prex vex ifex fvmpt sseqtr4d rgen ) BEZVAAFZGBHIZVAVCJZVAVAKLZMZK + eqeq1 prex vex ifex fvmpt sseqtrrd rgen ) BEZVAAFZGBHIZVAVCJZVAVAKLZMZK NOZVAPZVBVDVFVAVGGZQZVFRZVAVAGZQZUAZVAVHGZVNVDVJVMVJRVKVLVFVJVFVIVFVAVE VGVFSKNUCUDUEUFVAUGUHUIUBVFVOVIVLVGVAVHVGVATVHVAVATUJUKCVACEZVEMZVGVPPV HVCACBULZVQVFVPVAVGVPVAVEUNVRSUMDVFVGVAKNUOBUPUQURUSUT $. @@ -652205,7 +652205,7 @@ base set if and only if the neighborhoods (convergents) of every point $( A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) $) funfvima2d $p |- ( ( ph /\ x e. A ) -> ( F ` x ) e. ( F " A ) ) $= - ( cv wcel cfv cima wfun cdm wss wi ffund ssidd sseqtr4d funfvima2 syl2anc + ( cv wcel cfv cima wfun cdm wss wi ffund ssidd sseqtrrd funfvima2 syl2anc fdmd imp ) ABGZCHZUBEIECJHZAEKCELZMUCUDNACDEFOACCUEACPACDEFTQCUBERSUA $. $} @@ -653734,7 +653734,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing weq eqeltrd eleqtrrd 3jca simpl2 rr-spce simp1l1 syl simp2 gruuni syl2anc rspcime simpl1 gruel 3ad2ant1 sylan rexbidva mpbird rexex cpcoll2d adantr syl3anc cxp copab cin inss2 eqsstri a1i grucollcld syl2an2r mpbid rexcom4 - wb rexlimiva exlimiv sylbi elssuni ssun2 syl6ss adantl sseqtr4d ex anim2d + wb rexlimiva exlimiv sylbi elssuni ssun2 syl6ss adantl sseqtrrd ex anim2d reximdv sylc rexlimdv3a ralrimiva jca 3expa grupw gruun ismnu ) ALMUIZBUJ ZUKZLULZYTUFUJZULZFUJZUGUJZUIZUGDUMZUNZUGLUOUUDCUJZUIZUUIUPZUUBULZUNZCDUJ ZUOZUQZFYSURZUNZUFLUOZDUSZUNZBLURZAUVABLAYSLUIZUNZUUAUUTUVDUHUJZYSULZUVEL @@ -654735,7 +654735,7 @@ collection and union ( ~ mnuop3d ), from which closure under pairing = ( S X. { 0 } ) ) $= ( cr cc cpr wcel wa csn cxp cres cdv co cc0 wss cdm wceq adantr xpssres syl wf fconst6g anim2i recnprss c0ex fconst fdmi syl6sseqr dvconst adantl dmeqd - sseqtr4d ssid jctil dvres3 syl2anc oveq2d reseq1d eqtrd 3eqtr3d ) BCDEFZADF + sseqtrrd ssid jctil dvres3 syl2anc oveq2d reseq1d eqtrd 3eqtr3d ) BCDEFZADF ZGZBDAHZIZBJZKLZDVDKLZBJZBBVCIZKLZBMHZIZVBUTDDVDTZGDDNZBVGOZNZGVFVHPVAVMUTD ADUAUBVBVPVNVBBDVKIZOZVOUTBVRNVAUTBDVRBUCZDVKVQDMUDUEUFUGQVBVGVQVAVGVQPUTAU HUIZUJUKDULUMDBVDUNUOUTVFVJPVAUTVEVIBKUTBDNZVEVIPVSDVCBRSUPQVBVHVQBJZVLVBVG @@ -682854,7 +682854,7 @@ distinct definitions for the same symbol (limit of a sequence). dmeqi fdmi eqtri csn cxp id oveq2i cnex snex xpexd mptex offval3 fconst6g cin fdmd eqid fmpti ineq12d inidm mpteq1d fvconst2g oveq12d dvmptid mptru wtru wral ax-1cn rgenw fmpt mpbi dvcmulf dmeqd ovexd fveq1i mpan2 eqeltrd - adantr dmmptd 3eqtrd dvcof ccncf coscn crn wss frnd sseqtr4d dmcosseq syl + adantr dmmptd 3eqtrd dvcof ccncf coscn crn wss frnd sseqtrrd dmcosseq syl coexg ovex dmmpti coscld simpl mulcomd coeq1i fveq1d wfun ffund syl6eleqr fvco cc0 caddc 0cnd dvmptc dvmptmul mul02d mulid2d addid2d 3eqtr4d ) BDEZ DADBAUAZFRZGHZIZJRDGADYOIZUBZJRDGJRZYRUBZDYRJRZFUCZRZADBYOKHZFRZIZYMYQYSD @@ -683303,7 +683303,7 @@ distinct definitions for the same symbol (limit of a sequence). cv wa ioossre sseldi cxr clt rexrd ioogtlb syl3anc ccncf iooltub iccssioo wss syl22anc wf cc wb ax-resscn a1i cdm fssd dvcn syl31anc syl2anc mpbird cncffvrn rescncf sylc crn ctg sstrd ccnfld ctopn eqid tgioo2 dvres iccntr - cnt reseq2d eqtrd dmeqd ltled ioossioo sseqtr4d ssdmres sylib mvth fveq1d + cnt reseq2d eqtrd dmeqd ltled ioossioo sseqtrrd ssdmres sylib mvth fveq1d w3a fvres sylan9eq eqcomd 3adant3 simp3 ubicc2 syl lbicc2 oveq1d 3ad2ant1 3eqtrd sseldd ffvelrnd resubcld recnd dvfre feq2d mpbid adantr sselda cc0 oveq12d wne posdifd gt0ne0d divmul3d fveq2d abssubge0d oveq2d abscld 0red @@ -685298,7 +685298,7 @@ distinct definitions for the same symbol (limit of a sequence). 0expd mul02d mulassd oveq2d sqcld 3eqtrd negeqd mulneg1d cvol ioombl sinf crn ctg ccnfld ctopn cpr reelprrecn recn sincl fmptd wi wal cvv crab elex rabid sylanbrc dmmpt syl6eleqr alrimiv nfdm dfss2f dvsinexp 3eqtr4g dmeqd - sylibr sseqtr4d dvres3 syl22anc reseq1d 3eqtr3d tgioo2 cnt iccntr dvcosre + sylibr sseqtrrd dvres3 syl22anc reseq1d 3eqtr3d tgioo2 cnt iccntr dvcosre dvmptneg sin0 0cn coscl sinpi picn itgparts df-neg 3eqtr4a sqval mulneg2d ex mulcomd 3eqtr4d itgeq2dv 2nn0 itgmulc2 eqtr4d itgcl ) ABQUAUBRZBUCZUDU NZIUERZUWSUFRZUGZIUHZBUWQUWRUIUNZUJUERZUWSIUKULRZUERZUFRZUGZUFRZUHZIUXIUF @@ -686316,7 +686316,7 @@ distinct definitions for the same symbol (limit of a sequence). syl3anc cmpt eqid ccncf addccncf syl iccssred ax-resscn syl6ss sselda cle recnd wbr w3a wb elicc2 syl2anc simp2d simp3d eliccd cncfmptssg wceq wrex mpbid crab eqeq1 rexbidv oveq1 eqeq2d cbvrexv syl6bb cbvrabv ffdmd cdm wi - cres wss simp3 3adant3 eqeltrd rexlimdv3a ralrimivw rabss sylibr sseqtr4d + cres wss simp3 3adant3 eqeltrd rexlimdv3a ralrimivw rabss sylibr sseqtrrd wral fdmd cncfperiod elrab simprr nfre1 nfan 3jca 3ad2ant1 mpbird rexlimd nfv 3exp mpd sylan2b cmin resubcld pncand eqcomd lesub1dd eqbrtrd breqtrd reseq2d oveq1d cibl a1i 1cnd ssid cvol cdv resmptd eqtrd cc0 3eqtrd eqrdv @@ -688712,7 +688712,7 @@ the final h is a normalized version of G ( divided by its norm, see the wbr wa w3a chash wcel c0 wceq wne jca ssn0 unieq uni0 syl6eq necon3i 3syl wn neneqd cfn wo wi cpw cin elinel2 syl fz1f1o pm2.53 oveq2 exbidv rspcev mpd f1oeq2d f1of adantl simpll elinel1 elpwid fssd ad2antrr wfn ccnv wfun - dff1o2 simp3bi unieqd sseqtr4d cc0 cle cdif crab cmpt nfv nfan eqid simpr + dff1o2 simp3bi unieqd sseqtrrd cc0 cle cdif crab cmpt nfv nfan eqid simpr simplr sselda notnot intnand eldif sylnibr eleq2i eldifd ralrimiva sylibr crp dfss3 cvv mptfi rnfi stoweidlem31 3jca ex eximdv reximdva ) AUNNUOZUP UQZSUOZDUOZURZDUSZNUTVAZUUFQUUHVBZHUUHVCZVDZVEZUUFRBUOZVBEUOZMUOZUUPVFVFZ @@ -692408,7 +692408,7 @@ approximated is nonnegative (this assumption is removed in a later cdm fsumcncf difssd eldifsn mpbir2an divcncf eqeltrd cibl halfcld clt wbr recn zcnd 0red elfzle1 ltletrd gt0ne0d fsumcl addcld dvmptid tgioo2 reopn 0lt1 2cnd simpr coscld mp1i eqcomd fmpttd wral ralrimiva dmmptg sseqtrrid - wf dvsinax dmeqd sseqtr4d dvcnre reseq1d ax-mp syl6eq mpteq2dva dvmptfsum + wf dvsinax dmeqd sseqtrrd dvcnre reseq1d ax-mp syl6eq mpteq2dva dvmptfsum divcan3d eqtrd dvmptadd iccssred iccntr dvmptres2 syl5eq fvmpt2d itgeq2dv cnt 3eqtr4d ioosscn ssid coscn mulc1cncf csn cdif cncfmptc mp3an ioossicc halfcn cvol ioombl sselda syl6ss cniccibl syl3anc iblss idcncfg 2cn sincn @@ -695381,7 +695381,7 @@ approximated is nonnegative (this assumption is removed in a later crn ctg iooretop elrestr mnfxr vsnid sneq elun2 neqne leneltd sseli elind mnfltd elun1 pm2.61dan elinel1 elinel2 eqled simpll simplr velsn elunnel2 sylnibr ltled sylan2 impbida eqrdv ioossre rerest 3eltr4d isopn3i limcres - snssd unssd resabs1d crab feq2d mpbird sseqtr4d elrab nfrab nfcri rexlimd + snssd unssd resabs1d crab feq2d mpbird sseqtrrd elrab nfrab nfcri rexlimd simp3 limcperiod iooshift iooss1 sstrd negcld eqeq1d renegcld 3eqtr2d cif 3exp fourierdlem32 ne0i eqnetrd sylbir syl21anc rexlim2d iocssre mpteq2ia fvmpt2 mpan2 fourierdlem4 wfn ffn fvelrnb elfzelz elfzle1 simprld iocgtlb @@ -695660,7 +695660,7 @@ approximated is nonnegative (this assumption is removed in a later mpbird resubcld eqeltrid posdifd eqcomi flcld fvmptd negcld pncand addcld gt0ne0d redivcld remulcld negsubd mulneg1d znegcld simpl1 readdcld eleq1d ltsubaddd ltadd1dd 3anbi3d 3anbi2d vtoclg sylc ralrimiva 3eqtrrd renegcld - dfss3 iooshift negeqd eqeq1d limcperiod sseqtr4d eqvisset eqcoms necon3bi + dfss3 iooshift negeqd eqeq1d limcperiod sseqtrrd eqvisset eqcoms necon3bi 3sstr4d rexlimdv3a cif fourierdlem33 ne0i eqnetrd ) AMURPUSUTZVAZPVBUTZMU RPLVCZUSUTZVAZWUMVBUTZVDAWUMIVEZGVCZWUQVFVGUTZGVCZVHUTZVIZIVLOUUAUTZVJZWU LWUPVKZAWUMCDVHUTZVIZWVDAVMWVFPLABCDHLSTUAUCLBVMBVEZWVHQVCZVGUTZUUBZBVMWV @@ -697553,7 +697553,7 @@ approximated is nonnegative (this assumption is removed in a later sseldd fimaxre3 wne neqne elprn1 sylan2 ax-resscn fssd wfn fnmpti fvelrnb biimpi cres elfzofz fzofzp1 cncfioobd fvres fveq2d ralbidva mpan2 raleqdv eqtrd 3adant1 eqimss wo csup oveq12d cbvrabv supeq1i fourierdlem25 eleq2d - fveq2 rexbiia eqcomi rexeqi sylib ex orrd elun dfss3 sseqtr4d ) AUFUGLBEU + fveq2 rexbiia eqcomi rexeqi sylib ex orrd elun dfss3 sseqtrrd ) AUFUGLBEU NZIUNZUOZUPZLUQZHURZUSURZCDUTVEZUXEVAVBAUXBUXDVCVDAUXFUXEUOZVBZVFZUXGUXLU XGAUXKUXFUXIVBZUXGVGVBAUXKUXFUXBUXDVHZVBZUXMUXLUXFUXJUXNAUXKVIAUXJUXNVJZU XKAUXBVKVBZUXDVKVBUXPAEVKVBZUXQAVLKVMVEZVGEWAZUXSVKVBUXRRVLKVMVNZUXSVGVKE @@ -697644,7 +697644,7 @@ approximated is nonnegative (this assumption is removed in a later fimaxre3 neqne elprn1 fzofi rnmptfi wfn fvelrnb elfzofz fzofzp1 cncfioobd fnmpti fvres ralbidva mpan2 sylan9req 3adant1 raleqdv bitrd elun1 csup cn eqimss elinel1 cbvrabv supeq1i fourierdlem25 impbida rexbidv2 elun2 dfss3 - syl6eleqr mpbird sseqtr4d nfv nfra1 nfan sselda resubcld eqeltrid posdifd + syl6eleqr mpbird sseqtrrd nfv nfra1 nfan sselda resubcld eqeltrid posdifd breqtrrdi gt0ne0d redivcld flcld zred remulcld readdcld fvex eleq1 anbi2d oveq1 oveq2d eqeq1d imbi12d vtocl mpdan eqtr2d cioc iocssicc oveq2 oveq1d cbvralv cbvmptv eqtri fourierdlem4 sseldi rspccva eqbrtrd ralrimi reximdv @@ -699103,7 +699103,7 @@ approximated is nonnegative (this assumption is removed in a later chvarv necomd fvres 3eqtrrd mpteq2dva syl5eq syl6eq iftrued lbicc2 limccl ltled gtned neneqd ubicc2 3eqtr4rd ad4ant14 stoic1a eliccd fssresd neqned eqid ex syl5bir con3dimp subcld elex simp-4l fourierdlem8 jca vtoclg sylc - eqcom cncfiooicc cncff itgeq2dv ltned itgioo 3eqtr3d itgiccshift sseqtr4d + eqcom cncfiooicc cncff itgeq2dv ltned itgioo 3eqtr3d itgiccshift sseqtrrd eqidd crp itgioocnicc sumeq2dv ) ABCDUPUQZBURZLUSZUTBVAFUSZPFUSZUPUQZVWFU TVAPUUAUQZBJURZFUSZVWKVBVCUQZFUSZUPUQZVWFUTZJUUBZBCIVCUQZDIVCUQZUPUQZVWFU TZABVWDVWIVWFACVWGDVWHUPAVWGCAVWGCVDZVWHDVDZAVXBVXCVEZVWLVWNVFVGZJVWJUUCZ @@ -708176,7 +708176,7 @@ those for the more general case of a piecewise smooth function (see issalgend $p |- ( ph -> ( SalGen ` X ) = S ) $= ( vs csalgen cv cuni wceq wss wa csalg wcel adantl cfv eqid salgenss crab cint simpl elrabi unieq eqeq1d sseq2 anbi12d elrab biimpi simprld simprrd - wral syl13anc ralrimiva ssint sylibr salgenval syl sseqtr4d eqssd ) AELUA + wral syl13anc ralrimiva ssint sylibr salgenval syl sseqtrrd eqssd ) AELUA ZCACVEDEFVEUBGIHUCACKMZNZENZOZEVFPZQZKRUDZUEZVEACBMZPZBVLUPCVMPAVOBVLAVNV LSZQAVNRSZVNNZVHOZEVNPZVOAVPUFVPVQAVKKVNRUGTVPVSAVPVQVSVTVPVQVSVTQZQVKWAK VNRVFVNOZVIVSVJVTWBVGVRVHVFVNUHUIVFVNEUJUKULUMZUNTVPVTAVPVQVSVTWCUOTJUQUR @@ -728457,7 +728457,7 @@ Negated membership (alternative) (Contributed by AV, 31-Jul-2022.) $) f1oresf1o $p |- ( ph -> ( F |` D ) : D -1-1-onto-> { y e. B | ch } ) $= ( crab cres wf1o wss syl syl2anc cv wceq cab cima wf1 f1of1 cfv wrex wfun - f1ores f1ofun f1odm sseqtr4d dfimafn wcel wa abbidv df-rab syl6eqr eqtr2d + f1ores f1ofun f1odm sseqtrrd dfimafn wcel wa abbidv df-rab syl6eqr eqtr2d cdm f1oeq3d mpbird ) AGBDFLZHGMZNGHGUAZVBNZAEFHUBZGEOVDAEFHNZVEIEFHUCPJEF GHUGQAVAVCGVBAVCCRHUDDRZSCGUEZDTZVAAHUFZGHURZOVCVISAVFVJIEFHUHPAGEVKJAVFV KESIEFHUIPUJCDGHUKQAVIVGFULBUMZDTVAAVHVLDKUNBDFUOUPUQUSUT $. @@ -737888,7 +737888,7 @@ both being (two-sided) identity elements. (Contributed by AV, subsubmgm $p |- ( S e. ( SubMgm ` G ) -> ( A e. ( SubMgm ` H ) <-> ( A e. ( SubMgm ` G ) /\ A C_ S ) ) ) $= ( csubmgm cfv wcel wss wa cbs cress cmgm eqid submgmss adantl wceq adantr - co submgmmgm submgmbas sseqtr4d oveq1i ressabs syl5eq syldan wb submgmrcl + co submgmmgm submgmbas sseqtrrd oveq1i ressabs syl5eq syldan wb submgmrcl sstrd eqeltrrd issubmgm2 syl mpbir2and jca simprr sseqtrd adantrl eqeltrd ad2antrl impbida ) BCFGZHZADFGHZAVAHZABIZJZVBVCJZVDVEVGVDACKGZIZCALSZMHZV GABVHVGADKGZBVCAVLIZVBVLADVLNZOPVBBVLQZVCBDCEUAZRUBZVBBVHIVCVHBCVHNZORUIV @@ -743723,7 +743723,7 @@ a function into a (ring theoretic) domain equals the support of the C_ ( A supp ( 0g ` M ) ) ) $= ( vw wcel co wa cv cfv c0g wne crab csupp wceq cvv crg w3a cmap cmulr cdm cmpt oveq2 simpll1 simpll3 eqid ringrz syl2anc sylan9eqr necon3d ss2rabdv - ex elmapi adantl rabeq syl sseqtr4d weq fveq2 oveq2d cbvmptv simpl2 fvexd + ex elmapi adantl rabeq syl sseqtrrd weq fveq2 oveq2d cbvmptv simpl2 fvexd fdmd ovexd mptsuppd wfun elmapfun simpr suppval1 syl3anc 3sstr4d ) EUAJZF GJZCDJZUBZBDFUCKZJZLZCIMZBNZEUDNZKZEONZPZIFQZWEWHPZIBUEZQZAFCAMZBNZWFKZUF ZWHRKBWHRKZWCWJWKIFQZWMWCWIWKIFWCWDFJZLZWEWHWGWHXAWEWHSZWGWHSXBXAWGCWHWFK