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Mirrors > Home > MPE Home > Th. List > htpycom | Structured version Visualization version GIF version |
Description: Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Mario Carneiro, 23-Feb-2015.) |
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)) |
Ref | Expression |
---|---|
htpycom | ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹)) |
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 23414 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
7 | 1, 6 | cnmpt1st 22204 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
8 | 1, 6 | cnmpt2nd 22205 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
9 | iirevcn 23461 | . . . . . 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 7153 | . . . . 5 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦)) | |
12 | 1, 6, 8, 6, 10, 11 | cnmpt21 22207 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((𝐽 ×t II) Cn II)) |
13 | 1, 3, 2 | htpycn 23504 | . . . . 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 3965 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾)) |
16 | 1, 6, 7, 12, 15 | cnmpt22f 22211 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) ∈ ((𝐽 ×t II) Cn 𝐾)) |
17 | 4, 16 | eqeltrid 2914 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×t II) Cn 𝐾)) |
18 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → 𝑡 ∈ 𝑋) | |
19 | 0elunit 12843 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
20 | oveq1 7152 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥𝐻(1 − 𝑦)) = (𝑡𝐻(1 − 𝑦))) | |
21 | oveq2 7153 | . . . . . . 7 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0)) | |
22 | 1m0e1 11746 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2869 | . . . . . 6 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1) |
24 | 23 | oveq2d 7161 | . . . . 5 ⊢ (𝑦 = 0 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻1)) |
25 | ovex 7178 | . . . . 5 ⊢ (𝑡𝐻1) ∈ V | |
26 | 20, 24, 4, 25 | ovmpo 7299 | . . . 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 23505 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → ((𝑡𝐻0) = (𝐹‘𝑡) ∧ (𝑡𝐻1) = (𝐺‘𝑡))) |
29 | 28 | simprd 496 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻1) = (𝐺‘𝑡)) |
30 | 27, 29 | eqtrd 2853 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝐺‘𝑡)) |
31 | 1elunit 12844 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
32 | oveq2 7153 | . . . . . . 7 ⊢ (𝑦 = 1 → (1 − 𝑦) = (1 − 1)) | |
33 | 1m1e0 11697 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
34 | 32, 33 | syl6eq 2869 | . . . . . 6 ⊢ (𝑦 = 1 → (1 − 𝑦) = 0) |
35 | 34 | oveq2d 7161 | . . . . 5 ⊢ (𝑦 = 1 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻0)) |
36 | ovex 7178 | . . . . 5 ⊢ (𝑡𝐻0) ∈ V | |
37 | 20, 35, 4, 36 | ovmpo 7299 | . . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑡𝑀1) = (𝑡𝐻0)) |
38 | 18, 31, 37 | sylancl 586 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝑡𝐻0)) |
39 | 28 | simpld 495 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻0) = (𝐹‘𝑡)) |
40 | 38, 39 | eqtrd 2853 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝐹‘𝑡)) |
41 | 1, 2, 3, 17, 30, 40 | ishtpyd 23506 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 0cc0 10525 1c1 10526 − cmin 10858 [,]cicc 12729 TopOnctopon 21446 Cn ccn 21760 ×t ctx 22096 IIcii 23410 Htpy chtpy 23498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-ii 23412 df-htpy 23501 |
This theorem is referenced by: phtpycom 23519 |
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