Theorem cvmliftpht 28595

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 21360	. . . . . 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 1008	. . . 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 28578	. . 3 |- ( ph -> ( M e. ( II Cn C ) /\ ( F o. M ) = G /\ ( M ` 0 ) = P ) )
11	10	simp1d 1008	. 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 1009	. . . 4 |- ( ph -> H e. ( II Cn J ) )
14		phtpc01 21362	. . . . . . 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 2508	. . . 4 |- ( ph -> ( F ` P ) = ( H ` 0 ) )
18	1, 12, 3, 13, 8, 17	cvmliftiota 28578	. . 3 |- ( ph -> ( N e. ( II Cn C ) /\ ( F o. N ) = H /\ ( N ` 0 ) = P ) )
19	18	simp1d 1008	. 2 |- ( ph -> N e. ( II Cn C ) )
20	6	simp3d 1010	. . . 4 |- ( ph -> ( G ( PHtpy ` J ) H ) =/= (/) )
21		n0 3799	. . . 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 21349	. . . . . . 7 |- ( ph -> ( G ( PHtpy ` J ) H ) C_ ( ( II tX II ) Cn J ) )
25	24	sselda 3509	. . . . . 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 11650	. . . . . . . . 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 21350	. . . . . . . . 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 2511	. . . . . 6 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( F ` P ) = ( 0 g 0 ) )
36	1, 23, 25, 26, 35	cvmlift2 28593	. . . . 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 3083	. . . . 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 28594	. . . . 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 3796	. . . . 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 2963	. . 3 |- ( ( ph /\ g e. ( G ( PHtpy ` J ) H ) ) -> ( M ( PHtpy ` C ) N ) =/= (/) )
52	22, 51	exlimddv 1702	. 2 |- ( ph -> ( M ( PHtpy ` C ) N ) =/= (/) )
53		isphtpc 21360	. 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 1180	1 |- ( ph -> M ( ~=ph ` C ) N )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   =/= wne 2662   E.wrex 2818   E!wreu 2819   (/)c0 3790   U.cuni 4251   class class class wbr 4453   o. ccom 5009   ` cfv 5594   iota_crio 6255  (class class class)co 6295   0cc0 9504   1c1 9505   [,]cicc 11544   Cn ccn 19591   tX ctx 19927   IIcii 21245   PHtpycphtpy 21334   ~=ph cphtpc 21335   CovMap ccvm 28532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-submnd 15839  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-cn 19594  df-cnp 19595  df-cmp 19753  df-con 19779  df-lly 19833  df-nlly 19834  df-tx 19929  df-hmeo 20122  df-xms 20689  df-ms 20690  df-tms 20691  df-ii 21247  df-htpy 21336  df-phtpy 21337  df-phtpc 21358  df-pcon 28498  df-scon 28499  df-cvm 28533

This theorem is referenced by:  cvmlift3lem1  28596