Theorem htpyco1 18956

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 17284	. . 3 |- ( ( P e. ( J Cn K ) /\ F e. ( K Cn L ) ) -> ( F o. P ) e. ( J Cn L ) )
5	2, 3, 4	syl2anc 643	. 2 |- ( ph -> ( F o. P ) e. ( J Cn L ) )
6		htpyco1.g	. . 3 |- ( ph -> G e. ( K Cn L ) )
7		cnco 17284	. . 3 |- ( ( P e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. P ) e. ( J Cn L ) )
8	2, 6, 7	syl2anc 643	. 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 18862	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
11	10	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
12	1, 11	cnmpt1st 17653	. . . . 5 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> x ) e. ( ( J tX II ) Cn J ) )
13	1, 11, 12, 2	cnmpt21f 17657	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( P ` x ) ) e. ( ( J tX II ) Cn K ) )
14	1, 11	cnmpt2nd 17654	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> y ) e. ( ( J tX II ) Cn II ) )
15		cntop2 17259	. . . . . . . 8 |- ( P e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 16	. . . . . . 7 |- ( ph -> K e. Top )
17		eqid 2404	. . . . . . . 8 |- U. K = U. K
18	17	toptopon 16953	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
19	16, 18	sylib 189	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
20	19, 3, 6	htpycn 18951	. . . . 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 3309	. . . 4 |- ( ph -> H e. ( ( K tX II ) Cn L ) )
23	1, 11, 13, 14, 22	cnmpt22f 17660	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) ) e. ( ( J tX II ) Cn L ) )
24	9, 23	syl5eqel 2488	. 2 |- ( ph -> N e. ( ( J tX II ) Cn L ) )
25		cnf2 17267	. . . . . . 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 1184	. . . . . 6 |- ( ph -> P : X --> U. K )
27	26	ffvelrnda 5829	. . . . 5 |- ( ( ph /\ s e. X ) -> ( P ` s ) e. U. K )
28	19, 3, 6, 21	htpyi 18952	. . . . 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 457	. . . 4 |- ( ( ph /\ s e. X ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
30	29	simpld 446	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) )
31		simpr 448	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
32		0elunit 10971	. . . 4 |- 0 e. ( 0 [,] 1 )
33		fveq2 5687	. . . . . 6 |- ( x = s -> ( P ` x ) = ( P ` s ) )
34		id 20	. . . . . 6 |- ( y = 0 -> y = 0 )
35	33, 34	oveqan12d 6059	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 0 ) )
36		ovex 6065	. . . . 5 |- ( ( P ` s ) H 0 ) e. _V
37	35, 9, 36	ovmpt2a 6163	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
38	31, 32, 37	sylancl 644	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
39		fvco3 5759	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
40	26, 39	sylan 458	. . 3 |- ( ( ph /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
41	30, 38, 40	3eqtr4d 2446	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( F o. P ) ` s ) )
42	29	simprd 450	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) )
43		1elunit 10972	. . . 4 |- 1 e. ( 0 [,] 1 )
44		id 20	. . . . . 6 |- ( y = 1 -> y = 1 )
45	33, 44	oveqan12d 6059	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 1 ) )
46		ovex 6065	. . . . 5 |- ( ( P ` s ) H 1 ) e. _V
47	45, 9, 46	ovmpt2a 6163	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
48	31, 43, 47	sylancl 644	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
49		fvco3 5759	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
50	26, 49	sylan 458	. . 3 |- ( ( ph /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
51	42, 48, 50	3eqtr4d 2446	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( G o. P ) ` s ) )
52	1, 5, 8, 24, 41, 51	ishtpyd 18953	1 |- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   U.cuni 3975   o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040   e. cmpt2 6042   0cc0 8946   1c1 8947   [,]cicc 10875   Topctop 16913  TopOnctopon 16914   Cn ccn 17242   tX ctx 17545   IIcii 18858   Htpy chtpy 18945

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-top 16918  df-bases 16920  df-topon 16921  df-cn 17245  df-tx 17547  df-ii 18860  df-htpy 18948