Theorem htpyco1 21346

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 19635	. . 3 |- ( ( P e. ( J Cn K ) /\ F e. ( K Cn L ) ) -> ( F o. P ) e. ( J Cn L ) )
5	2, 3, 4	syl2anc 661	. 2 |- ( ph -> ( F o. P ) e. ( J Cn L ) )
6		htpyco1.g	. . 3 |- ( ph -> G e. ( K Cn L ) )
7		cnco 19635	. . 3 |- ( ( P e. ( J Cn K ) /\ G e. ( K Cn L ) ) -> ( G o. P ) e. ( J Cn L ) )
8	2, 6, 7	syl2anc 661	. 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 21251	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
11	10	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
12	1, 11	cnmpt1st 20037	. . . . 5 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> x ) e. ( ( J tX II ) Cn J ) )
13	1, 11, 12, 2	cnmpt21f 20041	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( P ` x ) ) e. ( ( J tX II ) Cn K ) )
14	1, 11	cnmpt2nd 20038	. . . 4 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> y ) e. ( ( J tX II ) Cn II ) )
15		cntop2 19610	. . . . . . . 8 |- ( P e. ( J Cn K ) -> K e. Top )
16	2, 15	syl 16	. . . . . . 7 |- ( ph -> K e. Top )
17		eqid 2467	. . . . . . . 8 |- U. K = U. K
18	17	toptopon 19303	. . . . . . 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 21341	. . . . 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 3510	. . . 4 |- ( ph -> H e. ( ( K tX II ) Cn L ) )
23	1, 11, 13, 14, 22	cnmpt22f 20044	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> ( ( P ` x ) H y ) ) e. ( ( J tX II ) Cn L ) )
24	9, 23	syl5eqel 2559	. 2 |- ( ph -> N e. ( ( J tX II ) Cn L ) )
25		cnf2 19618	. . . . . . 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 1228	. . . . . 6 |- ( ph -> P : X --> U. K )
27	26	ffvelrnda 6032	. . . . 5 |- ( ( ph /\ s e. X ) -> ( P ` s ) e. U. K )
28	19, 3, 6, 21	htpyi 21342	. . . . 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 470	. . . 4 |- ( ( ph /\ s e. X ) -> ( ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) /\ ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) ) )
30	29	simpld 459	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 0 ) = ( F ` ( P ` s ) ) )
31		simpr 461	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
32		0elunit 11650	. . . 4 |- 0 e. ( 0 [,] 1 )
33		fveq2 5872	. . . . . 6 |- ( x = s -> ( P ` x ) = ( P ` s ) )
34		id 22	. . . . . 6 |- ( y = 0 -> y = 0 )
35	33, 34	oveqan12d 6314	. . . . 5 |- ( ( x = s /\ y = 0 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 0 ) )
36		ovex 6320	. . . . 5 |- ( ( P ` s ) H 0 ) e. _V
37	35, 9, 36	ovmpt2a 6428	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
38	31, 32, 37	sylancl 662	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( P ` s ) H 0 ) )
39		fvco3 5951	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
40	26, 39	sylan 471	. . 3 |- ( ( ph /\ s e. X ) -> ( ( F o. P ) ` s ) = ( F ` ( P ` s ) ) )
41	30, 38, 40	3eqtr4d 2518	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( ( F o. P ) ` s ) )
42	29	simprd 463	. . 3 |- ( ( ph /\ s e. X ) -> ( ( P ` s ) H 1 ) = ( G ` ( P ` s ) ) )
43		1elunit 11651	. . . 4 |- 1 e. ( 0 [,] 1 )
44		id 22	. . . . . 6 |- ( y = 1 -> y = 1 )
45	33, 44	oveqan12d 6314	. . . . 5 |- ( ( x = s /\ y = 1 ) -> ( ( P ` x ) H y ) = ( ( P ` s ) H 1 ) )
46		ovex 6320	. . . . 5 |- ( ( P ` s ) H 1 ) e. _V
47	45, 9, 46	ovmpt2a 6428	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
48	31, 43, 47	sylancl 662	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( P ` s ) H 1 ) )
49		fvco3 5951	. . . 4 |- ( ( P : X --> U. K /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
50	26, 49	sylan 471	. . 3 |- ( ( ph /\ s e. X ) -> ( ( G o. P ) ` s ) = ( G ` ( P ` s ) ) )
51	42, 48, 50	3eqtr4d 2518	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( ( G o. P ) ` s ) )
52	1, 5, 8, 24, 41, 51	ishtpyd 21343	1 |- ( ph -> N e. ( ( F o. P ) ( J Htpy L ) ( G o. P ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   U.cuni 4251   o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   |-> cmpt2 6297   0cc0 9504   1c1 9505   [,]cicc 11544   Topctop 19263  TopOnctopon 19264   Cn ccn 19593   tX ctx 19929   IIcii 21247   Htpy chtpy 21335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-topgen 14716  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-top 19268  df-bases 19270  df-topon 19271  df-cn 19596  df-tx 19931  df-ii 21249  df-htpy 21338

This theorem is referenced by: (None)