htpyco1 - Metamath Proof Explorer

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Theorem htpyco1 24948

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 23214	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
5	2, 3, 4	syl2anc 582	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
6		htpyco1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿))
7		cnco 23214	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
8	2, 6, 7	syl2anc 582	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
9		htpyco1.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦))
10		iitopon 24843	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
11	10	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
12	1, 11	cnmpt1st 23616	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
13	1, 11, 12, 2	cnmpt21f 23620	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×_t II) Cn 𝐾))
14	1, 11	cnmpt2nd 23617	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
15		cntop2 23189	. . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
16	2, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
17		toptopon2 22864	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
18	16, 17	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
19	18, 3, 6	htpycn 24943	. . . . 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 3977	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×_t II) Cn 𝐿))
22	1, 11, 13, 14, 21	cnmpt22f 23623	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×_t II) Cn 𝐿))
23	9, 22	eqeltrid 2829	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐿))
24		cnf2 23197	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾)
25	1, 18, 2, 24	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾)
26	25	ffvelcdmda 7093	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾)
27	18, 3, 6, 20	htpyi 24944	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
28	26, 27	syldan 589	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
29	28	simpld 493	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)))
30		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
31		0elunit 13481	. . . 4 ⊢ 0 ∈ (0[,]1)
32		fveq2 6896	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠))
33		id 22	. . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0)
34	32, 33	oveqan12d 7438	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0))
35		ovex 7452	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V
36	34, 9, 35	ovmpoa 7576	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
37	30, 31, 36	sylancl 584	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
38		fvco3 6996	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
39	25, 38	sylan 578	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
40	29, 37, 39	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠))
41	28	simprd 494	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))
42		1elunit 13482	. . . 4 ⊢ 1 ∈ (0[,]1)
43		id 22	. . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1)
44	32, 43	oveqan12d 7438	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1))
45		ovex 7452	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V
46	44, 9, 45	ovmpoa 7576	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
47	30, 42, 46	sylancl 584	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
48		fvco3 6996	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
49	25, 48	sylan 578	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
50	41, 47, 49	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠))
51	1, 5, 8, 23, 40, 50	ishtpyd 24945	1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∪ cuni 4909 ∘ ccom 5682 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 0cc0 11140 1c1 11141 [,]cicc 13362 Topctop 22839 TopOnctopon 22856 Cn ccn 23172 ×_t ctx 23508 IIcii 24839 Htpy chtpy 24937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-icc 13366 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-topgen 17428 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-top 22840 df-topon 22857 df-bases 22893 df-cn 23175 df-tx 23510 df-ii 24841 df-htpy 24940

This theorem is referenced by: (None)

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