Theorem htpyco1 22087

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 20359	. . 3 |- ( ( P e. ( J Cn K ) /\ F e. ( K Cn L ) ) -> ( F o. P ) e. ( J Cn L ) )
5	2, 3, 4	syl2anc 673	. 2 |- ( ph -> ( F o. P ) e. ( J Cn L ) )
6		htpyco1.g	. . 3 |- ( ph -> G e. ( K Cn L ) )
7		cnco 20359	. . 3 |- ( ( P e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. P ) e. ( J Cn L ) )
8	2, 6, 7	syl2anc 673	. 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 21989	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
11	10	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
12	1, 11	cnmpt1st 20760	. . . . 5 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> x ) e. ( ( J tX II ) Cn J ) )
13	1, 11, 12, 2	cnmpt21f 20764	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( P ` x ) ) e. ( ( J tX II ) Cn K ) )
14	1, 11	cnmpt2nd 20761	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> y ) e. ( ( J tX II ) Cn II ) )
15		cntop2 20334	. . . . . . . 8 |- ( P e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 17	. . . . . . 7 |- ( ph -> K e. Top )
17		eqid 2471	. . . . . . . 8 |- U. K = U. K
18	17	toptopon 20025	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
19	16, 18	sylib 201	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
20	19, 3, 6	htpycn 22082	. . . . 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 3419	. . . 4 |- ( ph -> H e. ( ( K tX II ) Cn L ) )
23	1, 11, 13, 14, 22	cnmpt22f 20767	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) ) e. ( ( J tX II ) Cn L ) )
24	9, 23	syl5eqel 2553	. 2 |- ( ph -> N e. ( ( J tX II ) Cn L ) )
25		cnf2 20342	. . . . . . 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 1292	. . . . . 6 |- ( ph -> P : X --> U. K )
27	26	ffvelrnda 6037	. . . . 5 |- ( ( ph /\ s e. X ) -> ( P ` s ) e. U. K )
28	19, 3, 6, 21	htpyi 22083	. . . . 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 478	. . . 4 |- ( ( ph /\ s e. X ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
30	29	simpld 466	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) )
31		simpr 468	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
32		0elunit 11776	. . . 4 |- 0 e. ( 0 [,] 1 )
33		fveq2 5879	. . . . . 6 |- ( x = s -> ( P ` x ) = ( P ` s ) )
34		id 22	. . . . . 6 |- ( y = 0 -> y = 0 )
35	33, 34	oveqan12d 6327	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 0 ) )
36		ovex 6336	. . . . 5 |- ( ( P ` s ) H 0 ) e. _V
37	35, 9, 36	ovmpt2a 6446	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
38	31, 32, 37	sylancl 675	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
39		fvco3 5957	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
40	26, 39	sylan 479	. . 3 |- ( ( ph /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
41	30, 38, 40	3eqtr4d 2515	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( F o. P ) ` s ) )
42	29	simprd 470	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) )
43		1elunit 11777	. . . 4 |- 1 e. ( 0 [,] 1 )
44		id 22	. . . . . 6 |- ( y = 1 -> y = 1 )
45	33, 44	oveqan12d 6327	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 1 ) )
46		ovex 6336	. . . . 5 |- ( ( P ` s ) H 1 ) e. _V
47	45, 9, 46	ovmpt2a 6446	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
48	31, 43, 47	sylancl 675	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
49		fvco3 5957	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
50	26, 49	sylan 479	. . 3 |- ( ( ph /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
51	42, 48, 50	3eqtr4d 2515	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( G o. P ) ` s ) )
52	1, 5, 8, 24, 41, 51	ishtpyd 22084	1 |- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   U.cuni 4190   o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   0cc0 9557   1c1 9558   [,]cicc 11663   Topctop 19994  TopOnctopon 19995   Cn ccn 20317   tX ctx 20652   IIcii 21985   Htpy chtpy 22076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-icc 11667  df-seq 12252  df-exp 12311  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-topgen 15420  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-tx 20654  df-ii 21987  df-htpy 22079

This theorem is referenced by: (None)