Theorem htpyco1 20509

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 18829	. . 3 |- ( ( P e. ( J Cn K ) /\ F e. ( K Cn L ) ) -> ( F o. P ) e. ( J Cn L ) )
5	2, 3, 4	syl2anc 656	. 2 |- ( ph -> ( F o. P ) e. ( J Cn L ) )
6		htpyco1.g	. . 3 |- ( ph -> G e. ( K Cn L ) )
7		cnco 18829	. . 3 |- ( ( P e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. P ) e. ( J Cn L ) )
8	2, 6, 7	syl2anc 656	. 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 20414	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
11	10	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
12	1, 11	cnmpt1st 19200	. . . . 5 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> x ) e. ( ( J tX II ) Cn J ) )
13	1, 11, 12, 2	cnmpt21f 19204	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( P ` x ) ) e. ( ( J tX II ) Cn K ) )
14	1, 11	cnmpt2nd 19201	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> y ) e. ( ( J tX II ) Cn II ) )
15		cntop2 18804	. . . . . . . 8 |- ( P e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 16	. . . . . . 7 |- ( ph -> K e. Top )
17		eqid 2441	. . . . . . . 8 |- U. K = U. K
18	17	toptopon 18497	. . . . . . 7 |- ( K e. Top <-> K e. (TopOn ` U. K ) )
19	16, 18	sylib 196	. . . . . 6 |- ( ph -> K e. (TopOn ` U. K ) )
20	19, 3, 6	htpycn 20504	. . . . 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 3354	. . . 4 |- ( ph -> H e. ( ( K tX II ) Cn L ) )
23	1, 11, 13, 14, 22	cnmpt22f 19207	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) ) e. ( ( J tX II ) Cn L ) )
24	9, 23	syl5eqel 2525	. 2 |- ( ph -> N e. ( ( J tX II ) Cn L ) )
25		cnf2 18812	. . . . . . 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 1213	. . . . . 6 |- ( ph -> P : X --> U. K )
27	26	ffvelrnda 5840	. . . . 5 |- ( ( ph /\ s e. X ) -> ( P ` s ) e. U. K )
28	19, 3, 6, 21	htpyi 20505	. . . . 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 467	. . . 4 |- ( ( ph /\ s e. X ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
30	29	simpld 456	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) )
31		simpr 458	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
32		0elunit 11399	. . . 4 |- 0 e. ( 0 [,] 1 )
33		fveq2 5688	. . . . . 6 |- ( x = s -> ( P ` x ) = ( P ` s ) )
34		id 22	. . . . . 6 |- ( y = 0 -> y = 0 )
35	33, 34	oveqan12d 6109	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 0 ) )
36		ovex 6115	. . . . 5 |- ( ( P ` s ) H 0 ) e. _V
37	35, 9, 36	ovmpt2a 6220	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
38	31, 32, 37	sylancl 657	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
39		fvco3 5765	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
40	26, 39	sylan 468	. . 3 |- ( ( ph /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
41	30, 38, 40	3eqtr4d 2483	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( F o. P ) ` s ) )
42	29	simprd 460	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) )
43		1elunit 11400	. . . 4 |- 1 e. ( 0 [,] 1 )
44		id 22	. . . . . 6 |- ( y = 1 -> y = 1 )
45	33, 44	oveqan12d 6109	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 1 ) )
46		ovex 6115	. . . . 5 |- ( ( P ` s ) H 1 ) e. _V
47	45, 9, 46	ovmpt2a 6220	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
48	31, 43, 47	sylancl 657	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
49		fvco3 5765	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
50	26, 49	sylan 468	. . 3 |- ( ( ph /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
51	42, 48, 50	3eqtr4d 2483	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( G o. P ) ` s ) )
52	1, 5, 8, 24, 41, 51	ishtpyd 20506	1 |- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1364   e. wcel 1761   U.cuni 4088   o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090   e. cmpt2 6092   0cc0 9278   1c1 9279   [,]cicc 11299   Topctop 18457  TopOnctopon 18458   Cn ccn 18787   tX ctx 19092   IIcii 20410   Htpy chtpy 20498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-n0 10576  df-z 10643  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-icc 11303  df-seq 11803  df-exp 11862  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-topgen 14378  df-psmet 17768  df-xmet 17769  df-met 17770  df-bl 17771  df-mopn 17772  df-top 18462  df-bases 18464  df-topon 18465  df-cn 18790  df-tx 19094  df-ii 20412  df-htpy 20501

This theorem is referenced by: (None)