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Mirrors > Home > MPE Home > Th. List > phtpycom | Structured version Visualization version GIF version |
Description: Given a homotopy from πΉ to πΊ, produce a homotopy from πΊ to πΉ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
isphtpy.2 | β’ (π β πΉ β (II Cn π½)) |
isphtpy.3 | β’ (π β πΊ β (II Cn π½)) |
phtpycom.6 | β’ πΎ = (π₯ β (0[,]1), π¦ β (0[,]1) β¦ (π₯π»(1 β π¦))) |
phtpycom.7 | β’ (π β π» β (πΉ(PHtpyβπ½)πΊ)) |
Ref | Expression |
---|---|
phtpycom | β’ (π β πΎ β (πΊ(PHtpyβπ½)πΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isphtpy.3 | . 2 β’ (π β πΊ β (II Cn π½)) | |
2 | isphtpy.2 | . 2 β’ (π β πΉ β (II Cn π½)) | |
3 | iitopon 24754 | . . . 4 β’ II β (TopOnβ(0[,]1)) | |
4 | 3 | a1i 11 | . . 3 β’ (π β II β (TopOnβ(0[,]1))) |
5 | phtpycom.6 | . . 3 β’ πΎ = (π₯ β (0[,]1), π¦ β (0[,]1) β¦ (π₯π»(1 β π¦))) | |
6 | 2, 1 | phtpyhtpy 24863 | . . . 4 β’ (π β (πΉ(PHtpyβπ½)πΊ) β (πΉ(II Htpy π½)πΊ)) |
7 | phtpycom.7 | . . . 4 β’ (π β π» β (πΉ(PHtpyβπ½)πΊ)) | |
8 | 6, 7 | sseldd 3978 | . . 3 β’ (π β π» β (πΉ(II Htpy π½)πΊ)) |
9 | 4, 2, 1, 5, 8 | htpycom 24857 | . 2 β’ (π β πΎ β (πΊ(II Htpy π½)πΉ)) |
10 | 0elunit 13452 | . . . 4 β’ 0 β (0[,]1) | |
11 | simpr 484 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β π‘ β (0[,]1)) | |
12 | oveq1 7412 | . . . . 5 β’ (π₯ = 0 β (π₯π»(1 β π¦)) = (0π»(1 β π¦))) | |
13 | oveq2 7413 | . . . . . 6 β’ (π¦ = π‘ β (1 β π¦) = (1 β π‘)) | |
14 | 13 | oveq2d 7421 | . . . . 5 β’ (π¦ = π‘ β (0π»(1 β π¦)) = (0π»(1 β π‘))) |
15 | ovex 7438 | . . . . 5 β’ (0π»(1 β π‘)) β V | |
16 | 12, 14, 5, 15 | ovmpo 7564 | . . . 4 β’ ((0 β (0[,]1) β§ π‘ β (0[,]1)) β (0πΎπ‘) = (0π»(1 β π‘))) |
17 | 10, 11, 16 | sylancr 586 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (0πΎπ‘) = (0π»(1 β π‘))) |
18 | iirev 24805 | . . . . 5 β’ (π‘ β (0[,]1) β (1 β π‘) β (0[,]1)) | |
19 | 2, 1, 7 | phtpyi 24865 | . . . . 5 β’ ((π β§ (1 β π‘) β (0[,]1)) β ((0π»(1 β π‘)) = (πΉβ0) β§ (1π»(1 β π‘)) = (πΉβ1))) |
20 | 18, 19 | sylan2 592 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β ((0π»(1 β π‘)) = (πΉβ0) β§ (1π»(1 β π‘)) = (πΉβ1))) |
21 | 20 | simpld 494 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (0π»(1 β π‘)) = (πΉβ0)) |
22 | 2, 1, 7 | phtpy01 24866 | . . . . 5 β’ (π β ((πΉβ0) = (πΊβ0) β§ (πΉβ1) = (πΊβ1))) |
23 | 22 | adantr 480 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β ((πΉβ0) = (πΊβ0) β§ (πΉβ1) = (πΊβ1))) |
24 | 23 | simpld 494 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (πΉβ0) = (πΊβ0)) |
25 | 17, 21, 24 | 3eqtrd 2770 | . 2 β’ ((π β§ π‘ β (0[,]1)) β (0πΎπ‘) = (πΊβ0)) |
26 | 1elunit 13453 | . . . 4 β’ 1 β (0[,]1) | |
27 | oveq1 7412 | . . . . 5 β’ (π₯ = 1 β (π₯π»(1 β π¦)) = (1π»(1 β π¦))) | |
28 | 13 | oveq2d 7421 | . . . . 5 β’ (π¦ = π‘ β (1π»(1 β π¦)) = (1π»(1 β π‘))) |
29 | ovex 7438 | . . . . 5 β’ (1π»(1 β π‘)) β V | |
30 | 27, 28, 5, 29 | ovmpo 7564 | . . . 4 β’ ((1 β (0[,]1) β§ π‘ β (0[,]1)) β (1πΎπ‘) = (1π»(1 β π‘))) |
31 | 26, 11, 30 | sylancr 586 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (1πΎπ‘) = (1π»(1 β π‘))) |
32 | 20 | simprd 495 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (1π»(1 β π‘)) = (πΉβ1)) |
33 | 23 | simprd 495 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (πΉβ1) = (πΊβ1)) |
34 | 31, 32, 33 | 3eqtrd 2770 | . 2 β’ ((π β§ π‘ β (0[,]1)) β (1πΎπ‘) = (πΊβ1)) |
35 | 1, 2, 9, 25, 34 | isphtpyd 24867 | 1 β’ (π β πΎ β (πΊ(PHtpyβπ½)πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 β cmpo 7407 0cc0 11112 1c1 11113 β cmin 11448 [,]cicc 13333 TopOnctopon 22767 Cn ccn 23083 IIcii 24750 Htpy chtpy 24848 PHtpycphtpy 24849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-icc 13337 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18714 df-mulg 18996 df-cntz 19233 df-cmn 19702 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cn 23086 df-cnp 23087 df-tx 23421 df-hmeo 23614 df-xms 24181 df-ms 24182 df-tms 24183 df-ii 24752 df-htpy 24851 df-phtpy 24852 |
This theorem is referenced by: phtpcer 24876 |
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