Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22490 | . . . . 5 ⊢ II ∈ (TopOn‘(0[,]1)) | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
7 | 1, 6 | cnmpt1st 21281 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((𝐽 ×t II) Cn 𝐽)) |
8 | 1, 6 | cnmpt2nd 21282 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((𝐽 ×t II) Cn II)) |
9 | iirevcn 22537 | . . . . . 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 6557 | . . . . 5 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦)) | |
12 | 1, 6, 8, 6, 10, 11 | cnmpt21 21284 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((𝐽 ×t II) Cn II)) |
13 | 1, 3, 2 | htpycn 22580 | . . . . 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 3569 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾)) |
16 | 1, 6, 7, 12, 15 | cnmpt22f 21288 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) ∈ ((𝐽 ×t II) Cn 𝐾)) |
17 | 4, 16 | syl5eqel 2692 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×t II) Cn 𝐾)) |
18 | simpr 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → 𝑡 ∈ 𝑋) | |
19 | 0elunit 12161 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
20 | oveq1 6556 | . . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥𝐻(1 − 𝑦)) = (𝑡𝐻(1 − 𝑦))) | |
21 | oveq2 6557 | . . . . . . 7 ⊢ (𝑦 = 0 → (1 − 𝑦) = (1 − 0)) | |
22 | 1m0e1 11008 | . . . . . . 7 ⊢ (1 − 0) = 1 | |
23 | 21, 22 | syl6eq 2660 | . . . . . 6 ⊢ (𝑦 = 0 → (1 − 𝑦) = 1) |
24 | 23 | oveq2d 6565 | . . . . 5 ⊢ (𝑦 = 0 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻1)) |
25 | ovex 6577 | . . . . 5 ⊢ (𝑡𝐻1) ∈ V | |
26 | 20, 24, 4, 25 | ovmpt2 6694 | . . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑡𝑀0) = (𝑡𝐻1)) |
27 | 18, 19, 26 | sylancl 693 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝑡𝐻1)) |
28 | 1, 3, 2, 14 | htpyi 22581 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → ((𝑡𝐻0) = (𝐹‘𝑡) ∧ (𝑡𝐻1) = (𝐺‘𝑡))) |
29 | 28 | simprd 478 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻1) = (𝐺‘𝑡)) |
30 | 27, 29 | eqtrd 2644 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀0) = (𝐺‘𝑡)) |
31 | 1elunit 12162 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
32 | oveq2 6557 | . . . . . . 7 ⊢ (𝑦 = 1 → (1 − 𝑦) = (1 − 1)) | |
33 | 1m1e0 10966 | . . . . . . 7 ⊢ (1 − 1) = 0 | |
34 | 32, 33 | syl6eq 2660 | . . . . . 6 ⊢ (𝑦 = 1 → (1 − 𝑦) = 0) |
35 | 34 | oveq2d 6565 | . . . . 5 ⊢ (𝑦 = 1 → (𝑡𝐻(1 − 𝑦)) = (𝑡𝐻0)) |
36 | ovex 6577 | . . . . 5 ⊢ (𝑡𝐻0) ∈ V | |
37 | 20, 35, 4, 36 | ovmpt2 6694 | . . . 4 ⊢ ((𝑡 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑡𝑀1) = (𝑡𝐻0)) |
38 | 18, 31, 37 | sylancl 693 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝑡𝐻0)) |
39 | 28 | simpld 474 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝐻0) = (𝐹‘𝑡)) |
40 | 38, 39 | eqtrd 2644 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑋) → (𝑡𝑀1) = (𝐹‘𝑡)) |
41 | 1, 2, 3, 17, 30, 40 | ishtpyd 22582 | 1 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 0cc0 9815 1c1 9816 − cmin 10145 [,]cicc 12049 TopOnctopon 20518 Cn ccn 20838 ×t ctx 21173 IIcii 22486 Htpy chtpy 22574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-ii 22488 df-htpy 22577 |
This theorem is referenced by: phtpycom 22595 |
Copyright terms: Public domain | W3C validator |