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Theorem htpycom 24931

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 24828	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7	1, 6	cnmpt1st 23612	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×_t II) Cn 𝐽))
8	1, 6	cnmpt2nd 23613	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×_t II) Cn II))
9		iirevcn 24880	. . . . . 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 7366	. . . . 5 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
12	1, 6, 8, 6, 10, 11	cnmpt21 23615	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((𝐽 ×_t II) Cn II))
13	1, 3, 2	htpycn 24928	. . . . 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 3934	. . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×_t II) Cn 𝐾))
16	1, 6, 7, 12, 15	cnmpt22f 23619	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) ∈ ((𝐽 ×_t II) Cn 𝐾))
17	4, 16	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
18		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → 𝑡 ∈ 𝑋)
19		0elunit 13385	. . . 4 ⊢ 0 ∈ (0[,]1)
20		oveq1 7365	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥𝐻(1 − 𝑦)) = (𝑡𝐻(1 − 𝑦)))
21		oveq2 7366	. . . . . . 7 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0))
22		1m0e1 12261	. . . . . . 7 ⊢ (1 − 0) = 1
23	21, 22	eqtrdi 2787	. . . . . 6 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1)
24	23	oveq2d 7374	. . . . 5 ⊢ (𝑦 = 0 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻1))
25		ovex 7391	. . . . 5 ⊢ (𝑡𝐻1) ∈ V
26	20, 24, 4, 25	ovmpo 7518	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑡𝑀0) = (𝑡𝐻1))
27	18, 19, 26	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝑡𝐻1))
28	1, 3, 2, 14	htpyi 24929	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → ((𝑡𝐻0) = (𝐹‘𝑡) ∧ (𝑡𝐻1) = (𝐺‘𝑡)))
29	28	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻1) = (𝐺‘𝑡))
30	27, 29	eqtrd 2771	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝐺‘𝑡))
31		1elunit 13386	. . . 4 ⊢ 1 ∈ (0[,]1)
32		oveq2 7366	. . . . . . 7 ⊢ (𝑦 = 1 → (1 − 𝑦) = (1 − 1))
33		1m1e0 12217	. . . . . . 7 ⊢ (1 − 1) = 0
34	32, 33	eqtrdi 2787	. . . . . 6 ⊢ (𝑦 = 1 → (1 − 𝑦) = 0)
35	34	oveq2d 7374	. . . . 5 ⊢ (𝑦 = 1 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻0))
36		ovex 7391	. . . . 5 ⊢ (𝑡𝐻0) ∈ V
37	20, 35, 4, 36	ovmpo 7518	. . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑡𝑀1) = (𝑡𝐻0))
38	18, 31, 37	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝑡𝐻0))
39	28	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻0) = (𝐹‘𝑡))
40	38, 39	eqtrd 2771	. 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝐹‘𝑡))
41	1, 2, 3, 17, 30, 40	ishtpyd 24930	1 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 0cc0 11026 1c1 11027 − cmin 11364 [,]cicc 13264 TopOnctopon 22854 Cn ccn 23168 ×_t ctx 23504 IIcii 24824 Htpy chtpy 24922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-icc 13268 df-fz 13424 df-fzo 13571 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cn 23171 df-cnp 23172 df-tx 23506 df-hmeo 23699 df-xms 24264 df-ms 24265 df-tms 24266 df-ii 24826 df-htpy 24925

This theorem is referenced by: phtpycom 24943

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