Theorem htpyco1 22777

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

Hypotheses

Ref	Expression
htpyco1.n	|- N = ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) )
htpyco1.j	|- ( ph -> J e. (TopOn ` X ) )
htpyco1.p	|- ( ph -> P e. ( J Cn K ) )
htpyco1.f	|- ( ph -> F e. ( K Cn L ) )
htpyco1.g	|- ( ph -> G e. ( K Cn L ) )
htpyco1.h	|- ( ph -> H e. ( F ( K Htpy L ) G ) )

Assertion

Ref	Expression
htpyco1	|- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Distinct variable groups:   x, y, H   x, K, y   x, L, y   ph, x, y   x, J, y   x, P, y   x, X, y

Allowed substitution hints:   F( x, y)   G( x, y)   N( x, y)

Proof of Theorem htpyco1

Dummy variable s is distinct from all other variables.

Step	Hyp	Ref	Expression
1		htpyco1.j	. 2 |- ( ph -> J e. (TopOn ` X ) )
2		htpyco1.p	. . 3 |- ( ph -> P e. ( J Cn K ) )
3		htpyco1.f	. . 3 |- ( ph -> F e. ( K Cn L ) )
4		cnco 21070	. . 3 |- ( ( P e. ( J Cn K ) /\ F e. ( K Cn L ) ) -> ( F o. P ) e. ( J Cn L ) )
5	2, 3, 4	syl2anc 693	. 2 |- ( ph -> ( F o. P ) e. ( J Cn L ) )
6		htpyco1.g	. . 3 |- ( ph -> G e. ( K Cn L ) )
7		cnco 21070	. . 3 |- ( ( P e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. P ) e. ( J Cn L ) )
8	2, 6, 7	syl2anc 693	. 2 |- ( ph -> ( G o. P ) e. ( J Cn L ) )
9		htpyco1.n	. . 3 |- N = ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) )
10		iitopon 22682	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
11	10	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
12	1, 11	cnmpt1st 21471	. . . . 5 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> x ) e. ( ( J tX II ) Cn J ) )
13	1, 11, 12, 2	cnmpt21f 21475	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( P ` x ) ) e. ( ( J tX II ) Cn K ) )
14	1, 11	cnmpt2nd 21472	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> y ) e. ( ( J tX II ) Cn II ) )
15		cntop2 21045	. . . . . . . 8 |- ( P e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . . . 7 |- ( ph -> K e. Top )
17		eqid 2622	. . . . . . . 8 |- U. K = U. K
18	17	toptopon 20722	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
19	16, 18	sylib 208	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
20	19, 3, 6	htpycn 22772	. . . . 5 |- ( ph -> ( F ( K Htpy L ) G ) C_ ( ( K tX II ) Cn L ) )
21		htpyco1.h	. . . . 5 |- ( ph -> H e. ( F ( K Htpy L ) G ) )
22	20, 21	sseldd 3604	. . . 4 |- ( ph -> H e. ( ( K tX II ) Cn L ) )
23	1, 11, 13, 14, 22	cnmpt22f 21478	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) ) e. ( ( J tX II ) Cn L ) )
24	9, 23	syl5eqel 2705	. 2 |- ( ph -> N e. ( ( J tX II ) Cn L ) )
25		cnf2 21053	. . . . . . 7 |- ( ( J e. (TopOn ` X ) /\ K e. (TopOn ` U. K ) /\ P e. ( J Cn K ) ) -> P : X --> U. K )
26	1, 19, 2, 25	syl3anc 1326	. . . . . 6 |- ( ph -> P : X --> U. K )
27	26	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ s e. X ) -> ( P ` s ) e. U. K )
28	19, 3, 6, 21	htpyi 22773	. . . . 5 |- ( ( ph /\ ( P ` s ) e. U. K ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
29	27, 28	syldan 487	. . . 4 |- ( ( ph /\ s e. X ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
30	29	simpld 475	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) )
31		simpr 477	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
32		0elunit 12290	. . . 4 |- 0 e. ( 0 [,] 1 )
33		fveq2 6191	. . . . . 6 |- ( x = s -> ( P ` x ) = ( P ` s ) )
34		id 22	. . . . . 6 |- ( y = 0 -> y = 0 )
35	33, 34	oveqan12d 6669	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 0 ) )
36		ovex 6678	. . . . 5 |- ( ( P ` s ) H 0 ) e. _V
37	35, 9, 36	ovmpt2a 6791	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
38	31, 32, 37	sylancl 694	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
39		fvco3 6275	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
40	26, 39	sylan 488	. . 3 |- ( ( ph /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
41	30, 38, 40	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( F o. P ) ` s ) )
42	29	simprd 479	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) )
43		1elunit 12291	. . . 4 |- 1 e. ( 0 [,] 1 )
44		id 22	. . . . . 6 |- ( y = 1 -> y = 1 )
45	33, 44	oveqan12d 6669	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 1 ) )
46		ovex 6678	. . . . 5 |- ( ( P ` s ) H 1 ) e. _V
47	45, 9, 46	ovmpt2a 6791	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
48	31, 43, 47	sylancl 694	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
49		fvco3 6275	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
50	26, 49	sylan 488	. . 3 |- ( ( ph /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
51	42, 48, 50	3eqtr4d 2666	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( G o. P ) ` s ) )
52	1, 5, 8, 24, 41, 51	ishtpyd 22774	1 |- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   [,]cicc 12178   Topctop 20698  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   IIcii 22678   Htpy chtpy 22766

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-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

This theorem is referenced by: (None)