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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 24819 | . . . 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 24928 | . . . 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 3983 | . . 3 β’ (π β π» β (πΉ(II Htpy π½)πΊ)) |
9 | 4, 2, 1, 5, 8 | htpycom 24922 | . 2 β’ (π β πΎ β (πΊ(II Htpy π½)πΉ)) |
10 | 0elunit 13486 | . . . 4 β’ 0 β (0[,]1) | |
11 | simpr 483 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β π‘ β (0[,]1)) | |
12 | oveq1 7433 | . . . . 5 β’ (π₯ = 0 β (π₯π»(1 β π¦)) = (0π»(1 β π¦))) | |
13 | oveq2 7434 | . . . . . 6 β’ (π¦ = π‘ β (1 β π¦) = (1 β π‘)) | |
14 | 13 | oveq2d 7442 | . . . . 5 β’ (π¦ = π‘ β (0π»(1 β π¦)) = (0π»(1 β π‘))) |
15 | ovex 7459 | . . . . 5 β’ (0π»(1 β π‘)) β V | |
16 | 12, 14, 5, 15 | ovmpo 7587 | . . . 4 β’ ((0 β (0[,]1) β§ π‘ β (0[,]1)) β (0πΎπ‘) = (0π»(1 β π‘))) |
17 | 10, 11, 16 | sylancr 585 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (0πΎπ‘) = (0π»(1 β π‘))) |
18 | iirev 24870 | . . . . 5 β’ (π‘ β (0[,]1) β (1 β π‘) β (0[,]1)) | |
19 | 2, 1, 7 | phtpyi 24930 | . . . . 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 591 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β ((0π»(1 β π‘)) = (πΉβ0) β§ (1π»(1 β π‘)) = (πΉβ1))) |
21 | 20 | simpld 493 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (0π»(1 β π‘)) = (πΉβ0)) |
22 | 2, 1, 7 | phtpy01 24931 | . . . . 5 β’ (π β ((πΉβ0) = (πΊβ0) β§ (πΉβ1) = (πΊβ1))) |
23 | 22 | adantr 479 | . . . 4 β’ ((π β§ π‘ β (0[,]1)) β ((πΉβ0) = (πΊβ0) β§ (πΉβ1) = (πΊβ1))) |
24 | 23 | simpld 493 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (πΉβ0) = (πΊβ0)) |
25 | 17, 21, 24 | 3eqtrd 2772 | . 2 β’ ((π β§ π‘ β (0[,]1)) β (0πΎπ‘) = (πΊβ0)) |
26 | 1elunit 13487 | . . . 4 β’ 1 β (0[,]1) | |
27 | oveq1 7433 | . . . . 5 β’ (π₯ = 1 β (π₯π»(1 β π¦)) = (1π»(1 β π¦))) | |
28 | 13 | oveq2d 7442 | . . . . 5 β’ (π¦ = π‘ β (1π»(1 β π¦)) = (1π»(1 β π‘))) |
29 | ovex 7459 | . . . . 5 β’ (1π»(1 β π‘)) β V | |
30 | 27, 28, 5, 29 | ovmpo 7587 | . . . 4 β’ ((1 β (0[,]1) β§ π‘ β (0[,]1)) β (1πΎπ‘) = (1π»(1 β π‘))) |
31 | 26, 11, 30 | sylancr 585 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (1πΎπ‘) = (1π»(1 β π‘))) |
32 | 20 | simprd 494 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (1π»(1 β π‘)) = (πΉβ1)) |
33 | 23 | simprd 494 | . . 3 β’ ((π β§ π‘ β (0[,]1)) β (πΉβ1) = (πΊβ1)) |
34 | 31, 32, 33 | 3eqtrd 2772 | . 2 β’ ((π β§ π‘ β (0[,]1)) β (1πΎπ‘) = (πΊβ1)) |
35 | 1, 2, 9, 25, 34 | isphtpyd 24932 | 1 β’ (π β πΎ β (πΊ(PHtpyβπ½)πΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6553 (class class class)co 7426 β cmpo 7428 0cc0 11146 1c1 11147 β cmin 11482 [,]cicc 13367 TopOnctopon 22832 Cn ccn 23148 IIcii 24815 Htpy chtpy 24913 PHtpycphtpy 24914 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-icc 13371 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cn 23151 df-cnp 23152 df-tx 23486 df-hmeo 23679 df-xms 24246 df-ms 24247 df-tms 24248 df-ii 24817 df-htpy 24916 df-phtpy 24917 |
This theorem is referenced by: phtpcer 24941 |
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