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Theorem htpycom 23183

Description: Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.)

**Hypotheses**
Ref	Expression
ishtpy.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
ishtpy.3	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
htpycom.6	⊢ 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))
htpycom.7	⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))

**Assertion**
Ref	Expression
htpycom	⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))

Distinct variable groups: 𝑥,𝑦,𝐻 𝑥,𝐽,𝑦 𝜑,𝑥,𝑦 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝑀(𝑥,𝑦)

Proof of Theorem htpycom Dummy variables 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ishtpy.1	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		ishtpy.4	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
3		ishtpy.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
4		htpycom.6	. . 3 ⊢ 𝑀 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦)))
5		iitopon 23090	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7	1, 6	cnmpt1st 21880	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
8	1, 6	cnmpt2nd 21881	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
9		iirevcn 23137	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II)
10	9	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II))
11		oveq2 6930	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
12	1, 6, 8, 6, 10, 11	cnmpt21 21883	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((𝐽 ×_t II) Cn II))
13	1, 3, 2	htpycn 23180	. . . . 5 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
14		htpycom.7	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
15	13, 14	sseldd 3822	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×_t II) Cn 𝐾))
16	1, 6, 7, 12, 15	cnmpt22f 21887	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) ∈ ((𝐽 ×_t II) Cn 𝐾))
17	4, 16	syl5eqel 2863	. 2 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
18		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → 𝑡 ∈ 𝑋)
19		0elunit 12605	. . . 4 ⊢ 0 ∈ (0[,]1)
20		oveq1 6929	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥𝐻(1 − 𝑦)) = (𝑡𝐻(1 − 𝑦)))
21		oveq2 6930	. . . . . . 7 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
22		1m0e1 11503	. . . . . . 7 ⊢ (1 − 0) = 1
23	21, 22	syl6eq 2830	. . . . . 6 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
24	23	oveq2d 6938	. . . . 5 ⊢ (𝑦 = 0 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻1))
25		ovex 6954	. . . . 5 ⊢ (𝑡𝐻1) ∈ V
26	20, 24, 4, 25	ovmpt2 7073	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑡𝑀0) = (𝑡𝐻1))
27	18, 19, 26	sylancl 580	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝑡𝐻1))
28	1, 3, 2, 14	htpyi 23181	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → ((𝑡𝐻0) = (𝐹‘𝑡) ∧ (𝑡𝐻1) = (𝐺‘𝑡)))
29	28	simprd 491	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻1) = (𝐺‘𝑡))
30	27, 29	eqtrd 2814	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝐺‘𝑡))
31		1elunit 12606	. . . 4 ⊢ 1 ∈ (0[,]1)
32		oveq2 6930	. . . . . . 7 ⊢ (𝑦 = 1 → (1 − 𝑦) = (1 − 1))
33		1m1e0 11447	. . . . . . 7 ⊢ (1 − 1) = 0
34	32, 33	syl6eq 2830	. . . . . 6 ⊢ (𝑦 = 1 → (1 − 𝑦) = 0)
35	34	oveq2d 6938	. . . . 5 ⊢ (𝑦 = 1 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻0))
36		ovex 6954	. . . . 5 ⊢ (𝑡𝐻0) ∈ V
37	20, 35, 4, 36	ovmpt2 7073	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑡𝑀1) = (𝑡𝐻0))
38	18, 31, 37	sylancl 580	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝑡𝐻0))
39	28	simpld 490	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻0) = (𝐹‘𝑡))
40	38, 39	eqtrd 2814	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝐹‘𝑡))
41	1, 2, 3, 17, 30, 40	ishtpyd 23182	1 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ↦ cmpt 4965 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 0cc0 10272 1c1 10273 − cmin 10606 [,]cicc 12490 TopOnctopon 21122 Cn ccn 21436 ×_t ctx 21772 IIcii 23086 Htpy chtpy 23174

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cn 21439 df-cnp 21440 df-tx 21774 df-hmeo 21967 df-xms 22533 df-ms 22534 df-tms 22535 df-ii 23088 df-htpy 23177

This theorem is referenced by: phtpycom 23195

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