Theorem cvmlift3 31310

Description: A general version of cvmlift 31281. If K is simply connected and weakly locally path-connected, then there is a unique lift of functions on K which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	|- B = U. C
cvmlift3.y	|- Y = U. K
cvmlift3.f	|- ( ph -> F e. ( C CovMap J ) )
cvmlift3.k	|- ( ph -> K e. SConn )
cvmlift3.l	|- ( ph -> K e. 𝑛Locally PConn )
cvmlift3.o	|- ( ph -> O e. Y )
cvmlift3.g	|- ( ph -> G e. ( K Cn J ) )
cvmlift3.p	|- ( ph -> P e. B )
cvmlift3.e	|- ( ph -> ( F ` P ) = ( G ` O ) )

Assertion

Ref	Expression
cvmlift3	|- ( ph -> E! f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) )

Distinct variable groups:   f, J   f, F   B, f   f, G   C, f   ph, f   f, K   P, f   f, O   f, Y

Proof of Theorem cvmlift3

Dummy variables b c d k s z g a u v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . 3 |- B = U. C
2		cvmlift3.y	. . 3 |- Y = U. K
3		cvmlift3.f	. . 3 |- ( ph -> F e. ( C CovMap J ) )
4		cvmlift3.k	. . 3 |- ( ph -> K e. SConn )
5		cvmlift3.l	. . 3 |- ( ph -> K e. 𝑛Locally PConn )
6		cvmlift3.o	. . 3 |- ( ph -> O e. Y )
7		cvmlift3.g	. . 3 |- ( ph -> G e. ( K Cn J ) )
8		cvmlift3.p	. . 3 |- ( ph -> P e. B )
9		cvmlift3.e	. . 3 |- ( ph -> ( F ` P ) = ( G ` O ) )
10		eqeq2 2633	. . . . . . . 8 |- ( b = z -> ( ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b <-> ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) )
11	10	3anbi3d 1405	. . . . . . 7 |- ( b = z -> ( ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b ) <-> ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) ) )
12	11	rexbidv 3052	. . . . . 6 |- ( b = z -> ( E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b ) <-> E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) ) )
13	12	cbvriotav 6622	. . . . 5 |- ( iota_ b e. B E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b ) ) = ( iota_ z e. B E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) )
14		fveq1 6190	. . . . . . . . . 10 |- ( c = f -> ( c ` 0 ) = ( f ` 0 ) )
15	14	eqeq1d 2624	. . . . . . . . 9 |- ( c = f -> ( ( c ` 0 ) = O <-> ( f ` 0 ) = O ) )
16		fveq1 6190	. . . . . . . . . 10 |- ( c = f -> ( c ` 1 ) = ( f ` 1 ) )
17	16	eqeq1d 2624	. . . . . . . . 9 |- ( c = f -> ( ( c ` 1 ) = a <-> ( f ` 1 ) = a ) )
18		coeq2 5280	. . . . . . . . . . . . . . 15 |- ( d = g -> ( F o. d ) = ( F o. g ) )
19	18	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( d = g -> ( ( F o. d ) = ( G o. c ) <-> ( F o. g ) = ( G o. c ) ) )
20		fveq1 6190	. . . . . . . . . . . . . . 15 |- ( d = g -> ( d ` 0 ) = ( g ` 0 ) )
21	20	eqeq1d 2624	. . . . . . . . . . . . . 14 |- ( d = g -> ( ( d ` 0 ) = P <-> ( g ` 0 ) = P ) )
22	19, 21	anbi12d 747	. . . . . . . . . . . . 13 |- ( d = g -> ( ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) <-> ( ( F o. g ) = ( G o. c ) /\ ( g ` 0 ) = P ) ) )
23	22	cbvriotav 6622	. . . . . . . . . . . 12 |- ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) = ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. c ) /\ ( g ` 0 ) = P ) )
24		coeq2 5280	. . . . . . . . . . . . . . 15 |- ( c = f -> ( G o. c ) = ( G o. f ) )
25	24	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( c = f -> ( ( F o. g ) = ( G o. c ) <-> ( F o. g ) = ( G o. f ) ) )
26	25	anbi1d 741	. . . . . . . . . . . . 13 |- ( c = f -> ( ( ( F o. g ) = ( G o. c ) /\ ( g ` 0 ) = P ) <-> ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) )
27	26	riotabidv 6613	. . . . . . . . . . . 12 |- ( c = f -> ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. c ) /\ ( g ` 0 ) = P ) ) = ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) )
28	23, 27	syl5eq 2668	. . . . . . . . . . 11 |- ( c = f -> ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) = ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) )
29	28	fveq1d 6193	. . . . . . . . . 10 |- ( c = f -> ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) )
30	29	eqeq1d 2624	. . . . . . . . 9 |- ( c = f -> ( ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z <-> ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) )
31	15, 17, 30	3anbi123d 1399	. . . . . . . 8 |- ( c = f -> ( ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) <-> ( ( f ` 0 ) = O /\ ( f ` 1 ) = a /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
32	31	cbvrexv 3172	. . . . . . 7 |- ( E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) <-> E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = a /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) )
33		eqeq2 2633	. . . . . . . . 9 |- ( a = x -> ( ( f ` 1 ) = a <-> ( f ` 1 ) = x ) )
34	33	3anbi2d 1404	. . . . . . . 8 |- ( a = x -> ( ( ( f ` 0 ) = O /\ ( f ` 1 ) = a /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) <-> ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
35	34	rexbidv 3052	. . . . . . 7 |- ( a = x -> ( E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = a /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) <-> E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
36	32, 35	syl5bb 272	. . . . . 6 |- ( a = x -> ( E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) <-> E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
37	36	riotabidv 6613	. . . . 5 |- ( a = x -> ( iota_ z e. B E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = z ) ) = ( iota_ z e. B E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
38	13, 37	syl5eq 2668	. . . 4 |- ( a = x -> ( iota_ b e. B E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b ) ) = ( iota_ z e. B E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
39	38	cbvmptv 4750	. . 3 |- ( a e. Y |-> ( iota_ b e. B E. c e. ( II Cn K ) ( ( c ` 0 ) = O /\ ( c ` 1 ) = a /\ ( ( iota_ d e. ( II Cn C ) ( ( F o. d ) = ( G o. c ) /\ ( d ` 0 ) = P ) ) ` 1 ) = b ) ) ) = ( x e. Y |-> ( iota_ z e. B E. f e. ( II Cn K ) ( ( f ` 0 ) = O /\ ( f ` 1 ) = x /\ ( ( iota_ g e. ( II Cn C ) ( ( F o. g ) = ( G o. f ) /\ ( g ` 0 ) = P ) ) ` 1 ) = z ) ) )
40		eqid 2622	. . . 4 |- ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. c e. s ( A. d e. ( s \ { c } ) ( c i^i d ) = (/) /\ ( F |` c ) e. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } ) = ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. c e. s ( A. d e. ( s \ { c } ) ( c i^i d ) = (/) /\ ( F |` c ) e. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } )
41	40	cvmscbv 31240	. . 3 |- ( k e. J |-> { s e. ( ~P C \ { (/) } ) | ( U. s = ( `' F " k ) /\ A. c e. s ( A. d e. ( s \ { c } ) ( c i^i d ) = (/) /\ ( F |` c ) e. ( ( C ↾t c ) Homeo ( J ↾t k ) ) ) ) } ) = ( a e. J |-> { b e. ( ~P C \ { (/) } ) | ( U. b = ( `' F " a ) /\ A. v e. b ( A. u e. ( b \ { v } ) ( v i^i u ) = (/) /\ ( F |` v ) e. ( ( C ↾t v ) Homeo ( J ↾t a ) ) ) ) } )
42	1, 2, 3, 4, 5, 6, 7, 8, 9, 39, 41	cvmlift3lem9 31309	. 2 |- ( ph -> E. f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) )
43		sconnpconn 31209	. . . 4 |- ( K e. SConn -> K e. PConn )
44		pconnconn 31213	. . . 4 |- ( K e. PConn -> K e. Conn )
45	4, 43, 44	3syl 18	. . 3 |- ( ph -> K e. Conn )
46		pconnconn 31213	. . . . . 6 |- ( x e. PConn -> x e. Conn )
47	46	ssriv 3607	. . . . 5 |- PConn C_ Conn
48		nllyss 21283	. . . . 5 |- (PConn C_ Conn -> 𝑛Locally PConn C_ 𝑛Locally Conn )
49	47, 48	ax-mp 5	. . . 4 |- 𝑛Locally PConn C_ 𝑛Locally Conn
50	49, 5	sseldi 3601	. . 3 |- ( ph -> K e. 𝑛Locally Conn )
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 31266	. 2 |- ( ph -> E* f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) )
52		reu5 3159	. 2 |- ( E! f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) <-> ( E. f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) /\ E* f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) ) )
53	42, 51, 52	sylanbrc 698	1 |- ( ph -> E! f e. ( K Cn C ) ( ( F o. f ) = G /\ ( f ` O ) = P ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   0cc0 9936   1c1 9937   ↾_t crest 16081   Cn ccn 21028  Conncconn 21214  𝑛Locally cnlly 21268   Homeochmeo 21556   IIcii 22678  PConncpconn 31201  SConncsconn 31202   CovMap ccvm 31237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-cn 21031  df-cnp 21032  df-cmp 21190  df-conn 21215  df-lly 21269  df-nlly 21270  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-pconn 31203  df-sconn 31204  df-cvm 31238

This theorem is referenced by: (None)