htpyco1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > htpyco1		Structured version Visualization version GIF version

Theorem htpyco1 23275

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 21568	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐹 ∈ (𝐾 Cn 𝐿)) → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
5	2, 3, 4	syl2anc 576	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
6		htpyco1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐿))
7		cnco 21568	. . 3 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐾 Cn 𝐿)) → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
8	2, 6, 7	syl2anc 576	. 2 ⊢ (𝜑 → (𝐺 ∘ 𝑃) ∈ (𝐽 Cn 𝐿))
9		htpyco1.n	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦))
10		iitopon 23180	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
11	10	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
12	1, 11	cnmpt1st 21970	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
13	1, 11, 12, 2	cnmpt21f 21974	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑃‘𝑥)) ∈ ((𝐽 ×_t II) Cn 𝐾))
14	1, 11	cnmpt2nd 21971	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
15		cntop2 21543	. . . . . . . 8 ⊢ (𝑃 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
16	2, 15	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
17		toptopon2 21220	. . . . . . 7 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
18	16, 17	sylib 210	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘∪ 𝐾))
19	18, 3, 6	htpycn 23270	. . . . 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 3855	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐾 ×_t II) Cn 𝐿))
22	1, 11, 13, 14, 21	cnmpt22f 21977	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ ((𝑃‘𝑥)𝐻𝑦)) ∈ ((𝐽 ×_t II) Cn 𝐿))
23	9, 22	syl5eqel 2864	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐿))
24		cnf2 21551	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ (𝐽 Cn 𝐾)) → 𝑃:𝑋⟶∪ 𝐾)
25	1, 18, 2, 24	syl3anc 1351	. . . . . 6 ⊢ (𝜑 → 𝑃:𝑋⟶∪ 𝐾)
26	25	ffvelrnda 6670	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑃‘𝑠) ∈ ∪ 𝐾)
27	18, 3, 6, 20	htpyi 23271	. . . . 5 ⊢ ((𝜑 ∧ (𝑃‘𝑠) ∈ ∪ 𝐾) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
28	26, 27	syldan 582	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)) ∧ ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠))))
29	28	simpld 487	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻0) = (𝐹‘(𝑃‘𝑠)))
30		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
31		0elunit 12664	. . . 4 ⊢ 0 ∈ (0[,]1)
32		fveq2 6493	. . . . . 6 ⊢ (𝑥 = 𝑠 → (𝑃‘𝑥) = (𝑃‘𝑠))
33		id 22	. . . . . 6 ⊢ (𝑦 = 0 → 𝑦 = 0)
34	32, 33	oveqan12d 6989	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻0))
35		ovex 7002	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻0) ∈ V
36	34, 9, 35	ovmpoa 7115	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
37	30, 31, 36	sylancl 577	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝑃‘𝑠)𝐻0))
38		fvco3 6582	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
39	25, 38	sylan 572	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐹 ∘ 𝑃)‘𝑠) = (𝐹‘(𝑃‘𝑠)))
40	29, 37, 39	3eqtr4d 2818	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = ((𝐹 ∘ 𝑃)‘𝑠))
41	28	simprd 488	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑃‘𝑠)𝐻1) = (𝐺‘(𝑃‘𝑠)))
42		1elunit 12665	. . . 4 ⊢ 1 ∈ (0[,]1)
43		id 22	. . . . . 6 ⊢ (𝑦 = 1 → 𝑦 = 1)
44	32, 43	oveqan12d 6989	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((𝑃‘𝑥)𝐻𝑦) = ((𝑃‘𝑠)𝐻1))
45		ovex 7002	. . . . 5 ⊢ ((𝑃‘𝑠)𝐻1) ∈ V
46	44, 9, 45	ovmpoa 7115	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
47	30, 42, 46	sylancl 577	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝑃‘𝑠)𝐻1))
48		fvco3 6582	. . . 4 ⊢ ((𝑃:𝑋⟶∪ 𝐾 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
49	25, 48	sylan 572	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝐺 ∘ 𝑃)‘𝑠) = (𝐺‘(𝑃‘𝑠)))
50	41, 47, 49	3eqtr4d 2818	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = ((𝐺 ∘ 𝑃)‘𝑠))
51	1, 5, 8, 23, 40, 50	ishtpyd 23272	1 ⊢ (𝜑 → 𝑁 ∈ ((𝐹 ∘ 𝑃)(𝐽 Htpy 𝐿)(𝐺 ∘ 𝑃)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ∪ cuni 4706 ∘ ccom 5404 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ∈ cmpo 6972 0cc0 10327 1c1 10328 [,]cicc 12550 Topctop 21195 TopOnctopon 21212 Cn ccn 21526 ×_t ctx 21862 IIcii 23176 Htpy chtpy 23264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-map 8200 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-inf 8694 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-icc 12554 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-topgen 16563 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-top 21196 df-topon 21213 df-bases 21248 df-cn 21529 df-tx 21864 df-ii 23178 df-htpy 23267

This theorem is referenced by: (None)

W3C validator