Theorem cvmliftpht 31300

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 22793	. . . . . 6 |- ( G ( ~=ph ` J ) H <-> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
6	4, 5	sylib 208	. . . . 5 |- ( ph -> ( G e. ( II Cn J ) /\ H e. ( II Cn J ) /\ ( G ( PHtpy ` J ) H ) =/= (/) ) )
7	6	simp1d 1073	. . . 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 31283	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
11	10	simp1d 1073	. 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 1074	. . . 4 |- ( ph -> H e. ( II Cn J ) )
14		phtpc01 22796	. . . . . . 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 475	. . . . 5 |- ( ph -> ( G ` 0 ) = ( H ` 0 ) )
17	9, 16	eqtrd 2656	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
18	1, 12, 3, 13, 8, 17	cvmliftiota 31283	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
19	18	simp1d 1073	. 2 |- ( ph -> N e. ( II Cn C ) )
20	6	simp3d 1075	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) =/= (/) )
21		n0 3931	. . . 4 |- ( ( G ( PHtpy ` J ) H ) =/= (/) <-> E. g g e. ( G ( PHtpy ` J ) H ) )
22	20, 21	sylib 208	. . 3 |- ( ph -> E. g g e. ( G ( PHtpy ` J ) H ) )
23	3	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> F e. ( C CovMap J ) )
24	7, 13	phtpycn 22782	. . . . . . 7 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) )
25	24	sselda 3603	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( ( II tX II ) Cn J ) )
26	8	adantr 481	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> P e. B )
27	9	adantr 481	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( G ` 0 ) )
28		0elunit 12290	. . . . . . . . 9 |- 0 e. ( 0 [,] 1 )
29	7	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> G e. ( II Cn J ) )
30	13	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> H e. ( II Cn J ) )
31		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> g e. ( G ( PHtpy ` J ) H ) )
32	29, 30, 31	phtpyi 22783	. . . . . . . . 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 707	. . . . . . . 8 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( ( 0 g 0 ) = ( G ` 0 ) /\ ( 1 g 0 ) = ( G ` 1 ) ) )
34	33	simpld 475	. . . . . . 7 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( 0 g 0 ) = ( G ` 0 ) )
35	27, 34	eqtr4d 2659	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( 0 g 0 ) )
36	1, 23, 25, 26, 35	cvmlift2 31298	. . . . 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 3160	. . . . 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 762	. . . . . 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 762	. . . . . 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 762	. . . . . 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 762	. . . . . 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 762	. . . . . 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 792	. . . . . 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 794	. . . . . 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 804	. . . . . 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 805	. . . . . 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 31299	. . . . 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 3921	. . . . 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 3035	. . 3 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
52	22, 51	exlimddv 1863	. 2 |- ( ph -> ( M ( PHtpy ` C ) N ) =/= (/) )
53		isphtpc 22793	. 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 1246	1 |- ( ph -> M ( ~=ph ` C ) N )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   E!wreu 2914   (/)c0 3915   U.cuni 4436   class class class wbr 4653   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   0cc0 9936   1c1 9937   [,]cicc 12178   Cn ccn 21028   tX ctx 21363   IIcii 22678   PHtpycphtpy 22767   ~=ph cphtpc 22768   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-pconn 31203  df-sconn 31204  df-cvm 31238

This theorem is referenced by:  cvmlift3lem1  31301