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Theorem htpycom 25277

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 25180	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7	1, 6	cnmpt1st 23967	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
8	1, 6	cnmpt2nd 23968	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
9		iirevcn 25231	. . . . . 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 7420	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
12	1, 6, 8, 6, 10, 11	cnmpt21 23970	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((𝐽 ×_t II) Cn II))
13	1, 3, 2	htpycn 25274	. . . . 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 3932	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×_t II) Cn 𝐾))
16	1, 6, 7, 12, 15	cnmpt22f 23974	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) ∈ ((𝐽 ×_t II) Cn 𝐾))
17	4, 16	eqeltrid 2865	. 2 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
18		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → 𝑡 ∈ 𝑋)
19		0elunit 13581	. . . 4 ⊢ 0 ∈ (0[,]1)
20		oveq1 7419	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥𝐻(1 − 𝑦)) = (𝑡𝐻(1 − 𝑦)))
21		oveq2 7420	. . . . . . 7 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
22		1m0e1 12443	. . . . . . 7 ⊢ (1 − 0) = 1
23	21, 22	eqtrdi 2812	. . . . . 6 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
24	23	oveq2d 7428	. . . . 5 ⊢ (𝑦 = 0 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻1))
25		ovex 7445	. . . . 5 ⊢ (𝑡𝐻1) ∈ V
26	20, 24, 4, 25	ovmpo 7572	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑡𝑀0) = (𝑡𝐻1))
27	18, 19, 26	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝑡𝐻1))
28	1, 3, 2, 14	htpyi 25275	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → ((𝑡𝐻0) = (𝐹‘𝑡) ∧ (𝑡𝐻1) = (𝐺‘𝑡)))
29	28	simprd 501	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻1) = (𝐺‘𝑡))
30	27, 29	eqtrd 2796	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝐺‘𝑡))
31		1elunit 13582	. . . 4 ⊢ 1 ∈ (0[,]1)
32		oveq2 7420	. . . . . . 7 ⊢ (𝑦 = 1 → (1 − 𝑦) = (1 − 1))
33		1m1e0 12396	. . . . . . 7 ⊢ (1 − 1) = 0
34	32, 33	eqtrdi 2812	. . . . . 6 ⊢ (𝑦 = 1 → (1 − 𝑦) = 0)
35	34	oveq2d 7428	. . . . 5 ⊢ (𝑦 = 1 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻0))
36		ovex 7445	. . . . 5 ⊢ (𝑡𝐻0) ∈ V
37	20, 35, 4, 36	ovmpo 7572	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑡𝑀1) = (𝑡𝐻0))
38	18, 31, 37	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝑡𝐻0))
39	28	simpld 500	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻0) = (𝐹‘𝑡))
40	38, 39	eqtrd 2796	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝐹‘𝑡))
41	1, 2, 3, 17, 30, 40	ishtpyd 25276	1 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ↦ cmpt 5186 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 0cc0 11181 1c1 11182 − cmin 11522 [,]cicc 13460 TopOnctopon 23208 Cn ccn 23522 ×_t ctx 23859 IIcii 25176 Htpy chtpy 25268

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-icc 13464 df-fz 13621 df-fzo 13769 df-seq 14125 df-exp 14185 df-hash 14455 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cn 23525 df-cnp 23526 df-tx 23861 df-hmeo 24054 df-xms 24619 df-ms 24620 df-tms 24621 df-ii 25178 df-htpy 25271

This theorem is used by: phtpycom 25289

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