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Mirrors > Home > MPE Home > Th. List > pcohtpy | Structured version Visualization version GIF version |
Description: Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
pcohtpy.4 | ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0)) |
pcohtpy.5 | ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐻) |
pcohtpy.6 | ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐾) |
Ref | Expression |
---|---|
pcohtpy | ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcohtpy.5 | . . . . 5 ⊢ (𝜑 → 𝐹( ≃ph‘𝐽)𝐻) | |
2 | isphtpc 23590 | . . . . 5 ⊢ (𝐹( ≃ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅)) | |
3 | 1, 2 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅)) |
4 | 3 | simp1d 1137 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
5 | pcohtpy.6 | . . . . 5 ⊢ (𝜑 → 𝐺( ≃ph‘𝐽)𝐾) | |
6 | isphtpc 23590 | . . . . 5 ⊢ (𝐺( ≃ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅)) | |
7 | 5, 6 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅)) |
8 | 7 | simp1d 1137 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽)) |
9 | pcohtpy.4 | . . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0)) | |
10 | 4, 8, 9 | pcocn 23613 | . 2 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽)) |
11 | 3 | simp2d 1138 | . . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽)) |
12 | 7 | simp2d 1138 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽)) |
13 | phtpc01 23592 | . . . . . 6 ⊢ (𝐹( ≃ph‘𝐽)𝐻 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1))) | |
14 | 1, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1))) |
15 | 14 | simprd 498 | . . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐻‘1)) |
16 | phtpc01 23592 | . . . . . 6 ⊢ (𝐺( ≃ph‘𝐽)𝐾 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1))) | |
17 | 5, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1))) |
18 | 17 | simpld 497 | . . . 4 ⊢ (𝜑 → (𝐺‘0) = (𝐾‘0)) |
19 | 9, 15, 18 | 3eqtr3d 2862 | . . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0)) |
20 | 11, 12, 19 | pcocn 23613 | . 2 ⊢ (𝜑 → (𝐻(*𝑝‘𝐽)𝐾) ∈ (II Cn 𝐽)) |
21 | 3 | simp3d 1139 | . . . . 5 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅) |
22 | n0 4308 | . . . . 5 ⊢ ((𝐹(PHtpy‘𝐽)𝐻) ≠ ∅ ↔ ∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻)) | |
23 | 21, 22 | sylib 220 | . . . 4 ⊢ (𝜑 → ∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻)) |
24 | 7 | simp3d 1139 | . . . . 5 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅) |
25 | n0 4308 | . . . . 5 ⊢ ((𝐺(PHtpy‘𝐽)𝐾) ≠ ∅ ↔ ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) | |
26 | 24, 25 | sylib 220 | . . . 4 ⊢ (𝜑 → ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) |
27 | exdistrv 1950 | . . . 4 ⊢ (∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) ↔ (∃𝑚 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ ∃𝑛 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) | |
28 | 23, 26, 27 | sylanbrc 585 | . . 3 ⊢ (𝜑 → ∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) |
29 | 9 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → (𝐹‘1) = (𝐺‘0)) |
30 | 1 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝐹( ≃ph‘𝐽)𝐻) |
31 | 5 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝐺( ≃ph‘𝐽)𝐾) |
32 | eqid 2819 | . . . . . . 7 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦))) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦))) | |
33 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻)) | |
34 | simprr 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) | |
35 | 29, 30, 31, 32, 33, 34 | pcohtpylem 23615 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑚𝑦), (((2 · 𝑥) − 1)𝑛𝑦))) ∈ ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾))) |
36 | 35 | ne0d 4299 | . . . . 5 ⊢ ((𝜑 ∧ (𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾))) → ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) ≠ ∅) |
37 | 36 | ex 415 | . . . 4 ⊢ (𝜑 → ((𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) → ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) ≠ ∅)) |
38 | 37 | exlimdvv 1929 | . . 3 ⊢ (𝜑 → (∃𝑚∃𝑛(𝑚 ∈ (𝐹(PHtpy‘𝐽)𝐻) ∧ 𝑛 ∈ (𝐺(PHtpy‘𝐽)𝐾)) → ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) ≠ ∅)) |
39 | 28, 38 | mpd 15 | . 2 ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) ≠ ∅) |
40 | isphtpc 23590 | . 2 ⊢ ((𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)(𝐻(*𝑝‘𝐽)𝐾) ↔ ((𝐹(*𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽) ∧ (𝐻(*𝑝‘𝐽)𝐾) ∈ (II Cn 𝐽) ∧ ((𝐹(*𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) ≠ ∅)) | |
41 | 10, 20, 39, 40 | syl3anbrc 1338 | 1 ⊢ (𝜑 → (𝐹(*𝑝‘𝐽)𝐺)( ≃ph‘𝐽)(𝐻(*𝑝‘𝐽)𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∃wex 1774 ∈ wcel 2108 ≠ wne 3014 ∅c0 4289 ifcif 4465 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 0cc0 10529 1c1 10530 · cmul 10534 ≤ cle 10668 − cmin 10862 / cdiv 11289 2c2 11684 [,]cicc 12733 Cn ccn 21824 IIcii 23475 PHtpycphtpy 23564 ≃phcphtpc 23565 *𝑝cpco 23596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-cn 21827 df-cnp 21828 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-ii 23477 df-htpy 23566 df-phtpy 23567 df-phtpc 23588 df-pco 23601 |
This theorem is referenced by: pcophtb 23625 pi1cpbl 23640 pi1xfrf 23649 pi1xfr 23651 pi1xfrcnvlem 23652 |
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