Theorem cvmliftpht 30053

Description: If G and H are path-homotopic, then their lifts M and N are also path-homotopic. (Contributed by Mario Carneiro, 6-Jul-2015.)

Hypotheses

Ref	Expression
cvmliftpht.b	|- B = U. C
cvmliftpht.m	|- M = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
cvmliftpht.n	|- N = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
cvmliftpht.f	|- ( ph -> F e. ( C CovMap J ) )
cvmliftpht.p	|- ( ph -> P e. B )
cvmliftpht.e	|- ( ph -> ( F ` P ) = ( G ` 0 ) )
cvmliftpht.g	|- ( ph -> G ( ~=ph ` J ) H )

Assertion

Ref	Expression
cvmliftpht	|- ( ph -> M ( ~=ph ` C ) N )

Distinct variable groups:   B, f   f, F   f, J   C, f   f, G   f, H   P, f

Allowed substitution hints:   ph( f)   M( f)   N( f)

Proof of Theorem cvmliftpht

Dummy variables h g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmliftpht.b	. . . 4 |- B = U. C
2		cvmliftpht.m	. . . 4 |- M = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = G /\ ( f ` 0 ) = P ) )
3		cvmliftpht.f	. . . 4 |- ( ph -> F e. ( C CovMap J ) )
4		cvmliftpht.g	. . . . . 6 |- ( ph -> G ( ~=ph ` J ) H )
5		isphtpc 22037	. . . . . 6 |- ( G ( ~=ph ` J ) H <-> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
6	4, 5	sylib 200	. . . . 5 |- ( ph -> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
7	6	simp1d 1021	. . . 4 |- ( ph -> G e. ( II Cn J ) )
8		cvmliftpht.p	. . . 4 |- ( ph -> P e. B )
9		cvmliftpht.e	. . . 4 |- ( ph -> ( F ` P ) = ( G ` 0 ) )
10	1, 2, 3, 7, 8, 9	cvmliftiota 30036	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
11	10	simp1d 1021	. 2 |- ( ph -> M e. ( II Cn C ) )
12		cvmliftpht.n	. . . 4 |- N = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = H /\ ( f ` 0 ) = P ) )
13	6	simp2d 1022	. . . 4 |- ( ph -> H e. ( II Cn J ) )
14		phtpc01 22039	. . . . . . 7 |- ( G ( ~=ph ` J ) H -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
15	4, 14	syl 17	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
16	15	simpld 461	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
17	9, 16	eqtrd 2487	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
18	1, 12, 3, 13, 8, 17	cvmliftiota 30036	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
19	18	simp1d 1021	. 2 |- ( ph -> N e. ( II Cn C ) )
20	6	simp3d 1023	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) =/= (/) )
21		n0 3743	. . . 4 |- ( ( G ( PHtpy ` J ) H ) =/= (/) <-> E. g g e. ( G ( PHtpy ` J ) H ) )
22	20, 21	sylib 200	. . 3 |- ( ph -> E. g g e. ( G ( PHtpy ` J ) H ) )
23	3	adantr 467	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> F e. ( C CovMap J ) )
24	7, 13	phtpycn 22026	. . . . . . 7 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) )
25	24	sselda 3434	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( ( II tX II ) Cn J ) )
26	8	adantr 467	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> P e. B )
27	9	adantr 467	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( G ` 0 ) )
28		0elunit 11757	. . . . . . . . 9 |- 0 e. ( 0 [,] 1 )
29	7	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> G e. ( II Cn J ) )
30	13	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> H e. ( II Cn J ) )
31		simpr 463	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( G ( PHtpy ` J ) H ) )
32	29, 30, 31	phtpyi 22027	. . . . . . . . 9 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ 0 e. ( 0 [,] 1 ) ) -> ( ( 0 g 0 ) = ( G ` 0 ) /\ ( 1 g 0 ) = ( G ` 1 ) ) )
33	28, 32	mpan2 678	. . . . . . . 8 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( ( 0 g 0 ) = ( G ` 0 ) /\ ( 1 g 0 ) = ( G ` 1 ) ) )
34	33	simpld 461	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( 0 g 0 ) = ( G ` 0 ) )
35	27, 34	eqtr4d 2490	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( 0 g 0 ) )
36	1, 23, 25, 26, 35	cvmlift2 30051	. . . . 5 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> E! h e. ( ( II tX II ) Cn C ) ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) )
37		reurex 3011	. . . . 5 |- ( E! h e. ( ( II tX II ) Cn C ) ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) -> E. h e. ( ( II tX II ) Cn C ) ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) )
38	36, 37	syl 17	. . . 4 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> E. h e. ( ( II tX II ) Cn C ) ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) )
39	3	ad2antrr 733	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> F e. ( C CovMap J ) )
40	8	ad2antrr 733	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> P e. B )
41	9	ad2antrr 733	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> ( F ` P ) = ( G ` 0 ) )
42	7	ad2antrr 733	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> G e. ( II Cn J ) )
43	13	ad2antrr 733	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> H e. ( II Cn J ) )
44		simplr 763	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> g e. ( G ( PHtpy ` J ) H ) )
45		simprl 765	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> h e. ( ( II tX II ) Cn C ) )
46		simprrl 775	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> ( F o. h ) = g )
47		simprrr 776	. . . . . 6 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> ( 0 h 0 ) = P )
48	1, 2, 12, 39, 40, 41, 42, 43, 44, 45, 46, 47	cvmliftphtlem 30052	. . . . 5 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> h e. ( M ( PHtpy ` C ) N ) )
49		ne0i 3739	. . . . 5 |- ( h e. ( M ( PHtpy ` C ) N ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
50	48, 49	syl 17	. . . 4 |- ( ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) /\ ( h e. ( ( II tX II ) Cn C ) /\ ( ( F o. h ) = g /\ ( 0 h 0 ) = P ) ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
51	38, 50	rexlimddv 2885	. . 3 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
52	22, 51	exlimddv 1783	. 2 |- ( ph -> ( M ( PHtpy ` C ) N ) =/= (/) )
53		isphtpc 22037	. 2 |- ( M ( ~=ph ` C ) N <-> ( M e. ( II Cn C ) /\ N e. ( II Cn C ) /\ ( M ( PHtpy ` C ) N ) =/= (/) ) )
54	11, 19, 52, 53	syl3anbrc 1193	1 |- ( ph -> M ( ~=ph ` C ) N )

Syntax hints:   -> wi 4   /\ wa 371   /\ w3a 986   = wceq 1446   E.wex 1665   e. wcel 1889   =/= wne 2624   E.wrex 2740   E!wreu 2741   (/)c0 3733   U.cuni 4201   class class class wbr 4405   o. ccom 4841   ` cfv 5585   iota_crio 6256  (class class class)co 6295   0cc0 9544   1c1 9545   [,]cicc 11645   Cn ccn 20252   tX ctx 20587   IIcii 21919   PHtpycphtpy 22011   ~=ph cphtpc 22012   CovMap ccvm 29990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-inf2 8151  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622  ax-addf 9623  ax-mulf 9624

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-isom 5594  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-supp 6920  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-ec 7370  df-map 7479  df-ixp 7528  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-fsupp 7889  df-fi 7930  df-sup 7961  df-inf 7962  df-oi 8030  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-7 10680  df-8 10681  df-9 10682  df-10 10683  df-n0 10877  df-z 10945  df-dec 11059  df-uz 11167  df-q 11272  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-ioo 11646  df-ico 11648  df-icc 11649  df-fz 11792  df-fzo 11923  df-fl 12035  df-seq 12221  df-exp 12280  df-hash 12523  df-cj 13174  df-re 13175  df-im 13176  df-sqrt 13310  df-abs 13311  df-clim 13564  df-sum 13765  df-struct 15135  df-ndx 15136  df-slot 15137  df-base 15138  df-sets 15139  df-ress 15140  df-plusg 15215  df-mulr 15216  df-starv 15217  df-sca 15218  df-vsca 15219  df-ip 15220  df-tset 15221  df-ple 15222  df-ds 15224  df-unif 15225  df-hom 15226  df-cco 15227  df-rest 15333  df-topn 15334  df-0g 15352  df-gsum 15353  df-topgen 15354  df-pt 15355  df-prds 15358  df-xrs 15412  df-qtop 15418  df-imas 15419  df-xps 15422  df-mre 15504  df-mrc 15505  df-acs 15507  df-mgm 16500  df-sgrp 16539  df-mnd 16549  df-submnd 16595  df-mulg 16688  df-cntz 16983  df-cmn 17444  df-psmet 18974  df-xmet 18975  df-met 18976  df-bl 18977  df-mopn 18978  df-cnfld 18983  df-top 19933  df-bases 19934  df-topon 19935  df-topsp 19936  df-cld 20046  df-ntr 20047  df-cls 20048  df-nei 20126  df-cn 20255  df-cnp 20256  df-cmp 20414  df-con 20439  df-lly 20493  df-nlly 20494  df-tx 20589  df-hmeo 20782  df-xms 21347  df-ms 21348  df-tms 21349  df-ii 21921  df-htpy 22013  df-phtpy 22014  df-phtpc 22035  df-pcon 29956  df-scon 29957  df-cvm 29991

This theorem is referenced by:  cvmlift3lem1  30054