htpyco1 - Metamath Proof Explorer

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Theorem htpyco1 24877

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

**Hypotheses**
Ref	Expression
htpyco1.n	⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦))
htpyco1.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
htpyco1.p	⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
htpyco1.f	⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))
htpyco1.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿))
htpyco1.h	⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺))

**Assertion**
Ref	Expression
htpyco1	⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃)))

Distinct variable groups: 𝑥,𝑦,𝐻 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝜑,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 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		htpyco1.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
3		htpyco1.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐾 Cn 𝐿))
4		cnco 23153	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
5	2, 3, 4	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
6		htpyco1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿))
7		cnco 23153	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
8	2, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
9		htpyco1.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦))
10		iitopon 24772	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
11	10	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
12	1, 11	cnmpt1st 23555	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
13	1, 11, 12, 2	cnmpt21f 23559	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×_t II) Cn 𝐾))
14	1, 11	cnmpt2nd 23556	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
15		cntop2 23128	. . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
16	2, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
17		toptopon2 22805	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
18	16, 17	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
19	18, 3, 6	htpycn 24872	. . . . 5 ⊢ (𝜑 → (𝐹(𝐾 Htpy 𝐿)𝐺) ⊆ ((𝐾 ×_t II) Cn 𝐿))
20		htpyco1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐾 Htpy 𝐿)𝐺))
21	19, 20	sseldd 3947	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×_t II) Cn 𝐿))
22	1, 11, 13, 14, 21	cnmpt22f 23562	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×_t II) Cn 𝐿))
23	9, 22	eqeltrid 2832	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐿))
24		cnf2 23136	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾)
25	1, 18, 2, 24	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾)
26	25	ffvelcdmda 7056	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾)
27	18, 3, 6, 20	htpyi 24873	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
28	26, 27	syldan 591	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
29	28	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)))
30		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
31		0elunit 13430	. . . 4 ⊢ 0 ∈ (0[,]1)
32		fveq2 6858	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠))
33		id 22	. . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0)
34	32, 33	oveqan12d 7406	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0))
35		ovex 7420	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V
36	34, 9, 35	ovmpoa 7544	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
37	30, 31, 36	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
38		fvco3 6960	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
39	25, 38	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
40	29, 37, 39	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠))
41	28	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))
42		1elunit 13431	. . . 4 ⊢ 1 ∈ (0[,]1)
43		id 22	. . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1)
44	32, 43	oveqan12d 7406	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1))
45		ovex 7420	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V
46	44, 9, 45	ovmpoa 7544	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
47	30, 42, 46	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
48		fvco3 6960	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
49	25, 48	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
50	41, 47, 49	3eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠))
51	1, 5, 8, 23, 40, 50	ishtpyd 24874	1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∪ cuni 4871 ∘ ccom 5642 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ∈ cmpo 7389 0cc0 11068 1c1 11069 [,]cicc 13309 Topctop 22780 TopOnctopon 22797 Cn ccn 23111 ×_t ctx 23447 IIcii 24768 Htpy chtpy 24866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-icc 13313 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-top 22781 df-topon 22798 df-bases 22833 df-cn 23114 df-tx 23449 df-ii 24770 df-htpy 24869

This theorem is referenced by: (None)

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