Theorem cvmliftpht 27222

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 20581	. . . . . 6 |- ( G ( ~=ph ` J ) H <-> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
6	4, 5	sylib 196	. . . . 5 |- ( ph -> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
7	6	simp1d 1000	. . . 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 27205	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
11	10	simp1d 1000	. 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 1001	. . . 4 |- ( ph -> H e. ( II Cn J ) )
14		phtpc01 20583	. . . . . . 7 |- ( G ( ~=ph ` J ) H -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
15	4, 14	syl 16	. . . . . 6 |- ( ph -> ( ( G ` 0 ) = ( H ` 0 ) /\ ( G ` 1 ) = ( H ` 1 ) ) )
16	15	simpld 459	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
17	9, 16	eqtrd 2475	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
18	1, 12, 3, 13, 8, 17	cvmliftiota 27205	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
19	18	simp1d 1000	. 2 |- ( ph -> N e. ( II Cn C ) )
20	6	simp3d 1002	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) =/= (/) )
21		n0 3661	. . . 4 |- ( ( G ( PHtpy ` J ) H ) =/= (/) <-> E. g g e. ( G ( PHtpy ` J ) H ) )
22	20, 21	sylib 196	. . 3 |- ( ph -> E. g g e. ( G ( PHtpy ` J ) H ) )
23	3	adantr 465	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> F e. ( C CovMap J ) )
24	7, 13	phtpycn 20570	. . . . . . 7 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) )
25	24	sselda 3371	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( ( II tX II ) Cn J ) )
26	8	adantr 465	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> P e. B )
27	9	adantr 465	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( G ` 0 ) )
28		0elunit 11418	. . . . . . . . 9 |- 0 e. ( 0 [,] 1 )
29	7	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> G e. ( II Cn J ) )
30	13	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> H e. ( II Cn J ) )
31		simpr 461	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( G ( PHtpy ` J ) H ) )
32	29, 30, 31	phtpyi 20571	. . . . . . . . 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 671	. . . . . . . 8 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( ( 0 g 0 ) = ( G ` 0 ) /\ ( 1 g 0 ) = ( G ` 1 ) ) )
34	33	simpld 459	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( 0 g 0 ) = ( G ` 0 ) )
35	27, 34	eqtr4d 2478	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( 0 g 0 ) )
36	1, 23, 25, 26, 35	cvmlift2 27220	. . . . 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 2952	. . . . 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 16	. . . 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 725	. . . . . 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 725	. . . . . 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 725	. . . . . 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 725	. . . . . 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 725	. . . . . 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 754	. . . . . 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 755	. . . . . 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 763	. . . . . 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 764	. . . . . 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 27221	. . . . 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 3658	. . . . 5 |- ( h e. ( M ( PHtpy ` C ) N ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
50	48, 49	syl 16	. . . 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 2860	. . 3 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
52	22, 51	exlimddv 1692	. 2 |- ( ph -> ( M ( PHtpy ` C ) N ) =/= (/) )
53		isphtpc 20581	. 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 1172	1 |- ( ph -> M ( ~=ph ` C ) N )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2620   E.wrex 2731   E!wreu 2732   (/)c0 3652   U.cuni 4106   class class class wbr 4307   o. ccom 4859   ` cfv 5433   iota_crio 6066  (class class class)co 6106   0cc0 9297   1c1 9298   [,]cicc 11318   Cn ccn 18843   tX ctx 19148   IIcii 20466   PHtpycphtpy 20555   ~=ph cphtpc 20556   CovMap ccvm 27159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375  ax-addf 9376  ax-mulf 9377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-se 4695  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-isom 5442  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-of 6335  df-om 6492  df-1st 6592  df-2nd 6593  df-supp 6706  df-recs 6847  df-rdg 6881  df-1o 6935  df-2o 6936  df-oadd 6939  df-er 7116  df-ec 7118  df-map 7231  df-ixp 7279  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-fsupp 7636  df-fi 7676  df-sup 7706  df-oi 7739  df-card 8124  df-cda 8352  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-q 10969  df-rp 11007  df-xneg 11104  df-xadd 11105  df-xmul 11106  df-ioo 11319  df-ico 11321  df-icc 11322  df-fz 11453  df-fzo 11564  df-fl 11657  df-seq 11822  df-exp 11881  df-hash 12119  df-cj 12603  df-re 12604  df-im 12605  df-sqr 12739  df-abs 12740  df-clim 12981  df-sum 13179  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-sca 14269  df-vsca 14270  df-ip 14271  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-hom 14277  df-cco 14278  df-rest 14376  df-topn 14377  df-0g 14395  df-gsum 14396  df-topgen 14397  df-pt 14398  df-prds 14401  df-xrs 14455  df-qtop 14460  df-imas 14461  df-xps 14463  df-mre 14539  df-mrc 14540  df-acs 14542  df-mnd 15430  df-submnd 15480  df-mulg 15563  df-cntz 15850  df-cmn 16294  df-psmet 17824  df-xmet 17825  df-met 17826  df-bl 17827  df-mopn 17828  df-cnfld 17834  df-top 18518  df-bases 18520  df-topon 18521  df-topsp 18522  df-cld 18638  df-ntr 18639  df-cls 18640  df-nei 18717  df-cn 18846  df-cnp 18847  df-cmp 19005  df-con 19031  df-lly 19085  df-nlly 19086  df-tx 19150  df-hmeo 19343  df-xms 19910  df-ms 19911  df-tms 19912  df-ii 20468  df-htpy 20557  df-phtpy 20558  df-phtpc 20579  df-pcon 27125  df-scon 27126  df-cvm 27160

This theorem is referenced by:  cvmlift3lem1  27223