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Mirrors > Home > MPE Home > Th. List > phtpycom | Structured version Visualization version GIF version |
Description: Given a homotopy from 𝐹 to 𝐺, produce a homotopy from 𝐺 to 𝐹. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
isphtpy.2 | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
isphtpy.3 | ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
phtpycom.6 | ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) |
phtpycom.7 | ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) |
Ref | Expression |
---|---|
phtpycom | ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isphtpy.3 | . 2 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) | |
2 | isphtpy.2 | . 2 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
3 | iitopon 23624 | . . . 4 ⊢ II ∈ (TopOn‘(0[,]1)) | |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1))) |
5 | phtpycom.6 | . . 3 ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑥𝐻(1 − 𝑦))) | |
6 | 2, 1 | phtpyhtpy 23727 | . . . 4 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) ⊆ (𝐹(II Htpy 𝐽)𝐺)) |
7 | phtpycom.7 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺)) | |
8 | 6, 7 | sseldd 3876 | . . 3 ⊢ (𝜑 → 𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺)) |
9 | 4, 2, 1, 5, 8 | htpycom 23721 | . 2 ⊢ (𝜑 → 𝐾 ∈ (𝐺(II Htpy 𝐽)𝐹)) |
10 | 0elunit 12936 | . . . 4 ⊢ 0 ∈ (0[,]1) | |
11 | simpr 488 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → 𝑡 ∈ (0[,]1)) | |
12 | oveq1 7171 | . . . . 5 ⊢ (𝑥 = 0 → (𝑥𝐻(1 − 𝑦)) = (0𝐻(1 − 𝑦))) | |
13 | oveq2 7172 | . . . . . 6 ⊢ (𝑦 = 𝑡 → (1 − 𝑦) = (1 − 𝑡)) | |
14 | 13 | oveq2d 7180 | . . . . 5 ⊢ (𝑦 = 𝑡 → (0𝐻(1 − 𝑦)) = (0𝐻(1 − 𝑡))) |
15 | ovex 7197 | . . . . 5 ⊢ (0𝐻(1 − 𝑡)) ∈ V | |
16 | 12, 14, 5, 15 | ovmpo 7319 | . . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1)) → (0𝐾𝑡) = (0𝐻(1 − 𝑡))) |
17 | 10, 11, 16 | sylancr 590 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (0𝐾𝑡) = (0𝐻(1 − 𝑡))) |
18 | iirev 23674 | . . . . 5 ⊢ (𝑡 ∈ (0[,]1) → (1 − 𝑡) ∈ (0[,]1)) | |
19 | 2, 1, 7 | phtpyi 23729 | . . . . 5 ⊢ ((𝜑 ∧ (1 − 𝑡) ∈ (0[,]1)) → ((0𝐻(1 − 𝑡)) = (𝐹‘0) ∧ (1𝐻(1 − 𝑡)) = (𝐹‘1))) |
20 | 18, 19 | sylan2 596 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → ((0𝐻(1 − 𝑡)) = (𝐹‘0) ∧ (1𝐻(1 − 𝑡)) = (𝐹‘1))) |
21 | 20 | simpld 498 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (0𝐻(1 − 𝑡)) = (𝐹‘0)) |
22 | 2, 1, 7 | phtpy01 23730 | . . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1))) |
23 | 22 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → ((𝐹‘0) = (𝐺‘0) ∧ (𝐹‘1) = (𝐺‘1))) |
24 | 23 | simpld 498 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (𝐹‘0) = (𝐺‘0)) |
25 | 17, 21, 24 | 3eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (0𝐾𝑡) = (𝐺‘0)) |
26 | 1elunit 12937 | . . . 4 ⊢ 1 ∈ (0[,]1) | |
27 | oveq1 7171 | . . . . 5 ⊢ (𝑥 = 1 → (𝑥𝐻(1 − 𝑦)) = (1𝐻(1 − 𝑦))) | |
28 | 13 | oveq2d 7180 | . . . . 5 ⊢ (𝑦 = 𝑡 → (1𝐻(1 − 𝑦)) = (1𝐻(1 − 𝑡))) |
29 | ovex 7197 | . . . . 5 ⊢ (1𝐻(1 − 𝑡)) ∈ V | |
30 | 27, 28, 5, 29 | ovmpo 7319 | . . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑡 ∈ (0[,]1)) → (1𝐾𝑡) = (1𝐻(1 − 𝑡))) |
31 | 26, 11, 30 | sylancr 590 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (1𝐾𝑡) = (1𝐻(1 − 𝑡))) |
32 | 20 | simprd 499 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (1𝐻(1 − 𝑡)) = (𝐹‘1)) |
33 | 23 | simprd 499 | . . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (𝐹‘1) = (𝐺‘1)) |
34 | 31, 32, 33 | 3eqtrd 2777 | . 2 ⊢ ((𝜑 ∧ 𝑡 ∈ (0[,]1)) → (1𝐾𝑡) = (𝐺‘1)) |
35 | 1, 2, 9, 25, 34 | isphtpyd 23731 | 1 ⊢ (𝜑 → 𝐾 ∈ (𝐺(PHtpy‘𝐽)𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 0cc0 10608 1c1 10609 − cmin 10941 [,]cicc 12817 TopOnctopon 21654 Cn ccn 21968 IIcii 23620 Htpy chtpy 23712 PHtpycphtpy 23713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-mulg 18336 df-cntz 18558 df-cmn 19019 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cn 21971 df-cnp 21972 df-tx 22306 df-hmeo 22499 df-xms 23066 df-ms 23067 df-tms 23068 df-ii 23622 df-htpy 23715 df-phtpy 23716 |
This theorem is referenced by: phtpcer 23740 |
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