Theorem cvmliftpht 28952

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 21579	. . . . . 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 1006	. . . 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 28935	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
11	10	simp1d 1006	. 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 1007	. . . 4 |- ( ph -> H e. ( II Cn J ) )
14		phtpc01 21581	. . . . . . 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 457	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
17	9, 16	eqtrd 2423	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
18	1, 12, 3, 13, 8, 17	cvmliftiota 28935	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
19	18	simp1d 1006	. 2 |- ( ph -> N e. ( II Cn C ) )
20	6	simp3d 1008	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) =/= (/) )
21		n0 3721	. . . 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 463	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> F e. ( C CovMap J ) )
24	7, 13	phtpycn 21568	. . . . . . 7 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) )
25	24	sselda 3417	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( ( II tX II ) Cn J ) )
26	8	adantr 463	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> P e. B )
27	9	adantr 463	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( G ` 0 ) )
28		0elunit 11559	. . . . . . . . 9 |- 0 e. ( 0 [,] 1 )
29	7	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> G e. ( II Cn J ) )
30	13	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> H e. ( II Cn J ) )
31		simpr 459	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( G ( PHtpy ` J ) H ) )
32	29, 30, 31	phtpyi 21569	. . . . . . . . 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 669	. . . . . . . 8 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( ( 0 g 0 ) = ( G ` 0 ) /\ ( 1 g 0 ) = ( G ` 1 ) ) )
34	33	simpld 457	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( 0 g 0 ) = ( G ` 0 ) )
35	27, 34	eqtr4d 2426	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( 0 g 0 ) )
36	1, 23, 25, 26, 35	cvmlift2 28950	. . . . 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 2999	. . . . 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 723	. . . . . 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 723	. . . . . 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 723	. . . . . 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 723	. . . . . 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 723	. . . . . 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 753	. . . . . 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 754	. . . . . 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 28951	. . . . 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 3717	. . . . 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 2878	. . 3 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
52	22, 51	exlimddv 1734	. 2 |- ( ph -> ( M ( PHtpy ` C ) N ) =/= (/) )
53		isphtpc 21579	. 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 1178	1 |- ( ph -> M ( ~=ph ` C ) N )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1399   E.wex 1620   e. wcel 1826   =/= wne 2577   E.wrex 2733   E!wreu 2734   (/)c0 3711   U.cuni 4163   class class class wbr 4367   o. ccom 4917   ` cfv 5496   iota_crio 6157  (class class class)co 6196   0cc0 9403   1c1 9404   [,]cicc 11453   Cn ccn 19811   tX ctx 20146   IIcii 21464   PHtpycphtpy 21553   ~=ph cphtpc 21554   CovMap ccvm 28889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481  ax-addf 9482  ax-mulf 9483

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-ec 7231  df-map 7340  df-ixp 7389  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-fi 7786  df-sup 7816  df-oi 7850  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-q 11102  df-rp 11140  df-xneg 11239  df-xadd 11240  df-xmul 11241  df-ioo 11454  df-ico 11456  df-icc 11457  df-fz 11594  df-fzo 11718  df-fl 11828  df-seq 12011  df-exp 12070  df-hash 12308  df-cj 12934  df-re 12935  df-im 12936  df-sqrt 13070  df-abs 13071  df-clim 13313  df-sum 13511  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-sca 14718  df-vsca 14719  df-ip 14720  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-hom 14726  df-cco 14727  df-rest 14830  df-topn 14831  df-0g 14849  df-gsum 14850  df-topgen 14851  df-pt 14852  df-prds 14855  df-xrs 14909  df-qtop 14914  df-imas 14915  df-xps 14917  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-psmet 18524  df-xmet 18525  df-met 18526  df-bl 18527  df-mopn 18528  df-cnfld 18534  df-top 19484  df-bases 19486  df-topon 19487  df-topsp 19488  df-cld 19605  df-ntr 19606  df-cls 19607  df-nei 19685  df-cn 19814  df-cnp 19815  df-cmp 19973  df-con 19998  df-lly 20052  df-nlly 20053  df-tx 20148  df-hmeo 20341  df-xms 20908  df-ms 20909  df-tms 20910  df-ii 21466  df-htpy 21555  df-phtpy 21556  df-phtpc 21577  df-pcon 28855  df-scon 28856  df-cvm 28890

This theorem is referenced by:  cvmlift3lem1  28953