Theorem phtpyco2 22789

Description: Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
phtpyco2.f	|- ( ph -> F e. ( II Cn J ) )
phtpyco2.g	|- ( ph -> G e. ( II Cn J ) )
phtpyco2.p	|- ( ph -> P e. ( J Cn K ) )
phtpyco2.h	|- ( ph -> H e. ( F ( PHtpy ` J ) G ) )

Assertion

Ref	Expression
phtpyco2	|- ( ph -> ( P o. H ) e. ( ( P o. F ) ( PHtpy ` K ) ( P o. G ) ) )

Proof of Theorem phtpyco2

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		phtpyco2.f	. . 3 |- ( ph -> F e. ( II Cn J ) )
2		phtpyco2.p	. . 3 |- ( ph -> P e. ( J Cn K ) )
3		cnco 21070	. . 3 |- ( ( F e. ( II Cn J ) /\ P e. ( J Cn K ) ) -> ( P o. F ) e. ( II Cn K ) )
4	1, 2, 3	syl2anc 693	. 2 |- ( ph -> ( P o. F ) e. ( II Cn K ) )
5		phtpyco2.g	. . 3 |- ( ph -> G e. ( II Cn J ) )
6		cnco 21070	. . 3 |- ( ( G e. ( II Cn J ) /\ P e. ( J Cn K ) ) -> ( P o. G ) e. ( II Cn K ) )
7	5, 2, 6	syl2anc 693	. 2 |- ( ph -> ( P o. G ) e. ( II Cn K ) )
8	1, 5	phtpyhtpy 22781	. . . 4 |- ( ph -> ( F ( PHtpy ` J ) G ) C_ ( F ( II Htpy J ) G ) )
9		phtpyco2.h	. . . 4 |- ( ph -> H e. ( F ( PHtpy ` J ) G ) )
10	8, 9	sseldd 3604	. . 3 |- ( ph -> H e. ( F ( II Htpy J ) G ) )
11	1, 5, 2, 10	htpyco2 22778	. 2 |- ( ph -> ( P o. H ) e. ( ( P o. F ) ( II Htpy K ) ( P o. G ) ) )
12	1, 5, 9	phtpyi 22783	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( 0 H s ) = ( F ` 0 ) /\ ( 1 H s ) = ( F ` 1 ) ) )
13	12	simpld 475	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 H s ) = ( F ` 0 ) )
14	13	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( P ` ( 0 H s ) ) = ( P ` ( F ` 0 ) ) )
15		0elunit 12290	. . . . . 6 |- 0 e. ( 0 [,] 1 )
16		simpr 477	. . . . . 6 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> s e. ( 0 [,] 1 ) )
17		opelxpi 5148	. . . . . 6 |- ( ( 0 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
18	15, 16, 17	sylancr 695	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
19		iitopon 22682	. . . . . . . . 9 |- II e. (TopOn ` ( 0 [,] 1 ) )
20		txtopon 21394	. . . . . . . . 9 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ II e. (TopOn ` ( 0 [,] 1 ) ) ) -> ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) )
21	19, 19, 20	mp2an 708	. . . . . . . 8 |- ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
22	21	a1i 11	. . . . . . 7 |- ( ph -> ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) )
23		cntop2 21045	. . . . . . . . 9 |- ( F e. ( II Cn J ) -> J e. Top )
24	1, 23	syl 17	. . . . . . . 8 |- ( ph -> J e. Top )
25		eqid 2622	. . . . . . . . 9 |- U. J = U. J
26	25	toptopon 20722	. . . . . . . 8 |- ( J e. Top <-> J e. (TopOn ` U. J ) )
27	24, 26	sylib 208	. . . . . . 7 |- ( ph -> J e. (TopOn ` U. J ) )
28	1, 5	phtpycn 22782	. . . . . . . 8 |- ( ph -> ( F ( PHtpy ` J ) G ) C_ ( ( II tX II ) Cn J ) )
29	28, 9	sseldd 3604	. . . . . . 7 |- ( ph -> H e. ( ( II tX II ) Cn J ) )
30		cnf2 21053	. . . . . . 7 |- ( ( ( II tX II ) e. (TopOn ` ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) /\ J e. (TopOn ` U. J ) /\ H e. ( ( II tX II ) Cn J ) ) -> H : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
31	22, 27, 29, 30	syl3anc 1326	. . . . . 6 |- ( ph -> H : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J )
32		fvco3 6275	. . . . . 6 |- ( ( H : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J /\ <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( P o. H ) ` <. 0 , s >. ) = ( P ` ( H ` <. 0 , s >. ) ) )
33	31, 32	sylan 488	. . . . 5 |- ( ( ph /\ <. 0 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( P o. H ) ` <. 0 , s >. ) = ( P ` ( H ` <. 0 , s >. ) ) )
34	18, 33	syldan 487	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( P o. H ) ` <. 0 , s >. ) = ( P ` ( H ` <. 0 , s >. ) ) )
35		df-ov 6653	. . . 4 |- ( 0 ( P o. H ) s ) = ( ( P o. H ) ` <. 0 , s >. )
36		df-ov 6653	. . . . 5 |- ( 0 H s ) = ( H ` <. 0 , s >. )
37	36	fveq2i 6194	. . . 4 |- ( P ` ( 0 H s ) ) = ( P ` ( H ` <. 0 , s >. ) )
38	34, 35, 37	3eqtr4g 2681	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 ( P o. H ) s ) = ( P ` ( 0 H s ) ) )
39		iiuni 22684	. . . . . . 7 |- ( 0 [,] 1 ) = U. II
40	39, 25	cnf 21050	. . . . . 6 |- ( F e. ( II Cn J ) -> F : ( 0 [,] 1 ) --> U. J )
41	1, 40	syl 17	. . . . 5 |- ( ph -> F : ( 0 [,] 1 ) --> U. J )
42	41	adantr 481	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> F : ( 0 [,] 1 ) --> U. J )
43		fvco3 6275	. . . 4 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 0 e. ( 0 [,] 1 ) ) -> ( ( P o. F ) ` 0 ) = ( P ` ( F ` 0 ) ) )
44	42, 15, 43	sylancl 694	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( P o. F ) ` 0 ) = ( P ` ( F ` 0 ) ) )
45	14, 38, 44	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 0 ( P o. H ) s ) = ( ( P o. F ) ` 0 ) )
46	12	simprd 479	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 H s ) = ( F ` 1 ) )
47	46	fveq2d 6195	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( P ` ( 1 H s ) ) = ( P ` ( F ` 1 ) ) )
48		1elunit 12291	. . . . . 6 |- 1 e. ( 0 [,] 1 )
49		opelxpi 5148	. . . . . 6 |- ( ( 1 e. ( 0 [,] 1 ) /\ s e. ( 0 [,] 1 ) ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
50	48, 16, 49	sylancr 695	. . . . 5 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) )
51		fvco3 6275	. . . . . 6 |- ( ( H : ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) --> U. J /\ <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( P o. H ) ` <. 1 , s >. ) = ( P ` ( H ` <. 1 , s >. ) ) )
52	31, 51	sylan 488	. . . . 5 |- ( ( ph /\ <. 1 , s >. e. ( ( 0 [,] 1 ) X. ( 0 [,] 1 ) ) ) -> ( ( P o. H ) ` <. 1 , s >. ) = ( P ` ( H ` <. 1 , s >. ) ) )
53	50, 52	syldan 487	. . . 4 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( P o. H ) ` <. 1 , s >. ) = ( P ` ( H ` <. 1 , s >. ) ) )
54		df-ov 6653	. . . 4 |- ( 1 ( P o. H ) s ) = ( ( P o. H ) ` <. 1 , s >. )
55		df-ov 6653	. . . . 5 |- ( 1 H s ) = ( H ` <. 1 , s >. )
56	55	fveq2i 6194	. . . 4 |- ( P ` ( 1 H s ) ) = ( P ` ( H ` <. 1 , s >. ) )
57	53, 54, 56	3eqtr4g 2681	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 ( P o. H ) s ) = ( P ` ( 1 H s ) ) )
58		fvco3 6275	. . . 4 |- ( ( F : ( 0 [,] 1 ) --> U. J /\ 1 e. ( 0 [,] 1 ) ) -> ( ( P o. F ) ` 1 ) = ( P ` ( F ` 1 ) ) )
59	42, 48, 58	sylancl 694	. . 3 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( ( P o. F ) ` 1 ) = ( P ` ( F ` 1 ) ) )
60	47, 57, 59	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. ( 0 [,] 1 ) ) -> ( 1 ( P o. H ) s ) = ( ( P o. F ) ` 1 ) )
61	4, 7, 11, 45, 60	isphtpyd 22785	1 |- ( ph -> ( P o. H ) e. ( ( P o. F ) ( PHtpy ` K ) ( P o. G ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   <.cop 4183   U.cuni 4436   X. cxp 5112   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   [,]cicc 12178   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   IIcii 22678   Htpy chtpy 22766   PHtpycphtpy 22767

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-iun 4522  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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-tx 21365  df-ii 22680  df-htpy 22769  df-phtpy 22770

This theorem is referenced by:  phtpcco2  22799