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Mirrors > Home > MPE Home > Th. List > pi1inv | Structured version Visualization version GIF version |
Description: An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.) |
Ref | Expression |
---|---|
pi1grp.2 | ⊢ 𝐺 = (𝐽 π1 𝑌) |
pi1inv.n | ⊢ 𝑁 = (invg‘𝐺) |
pi1inv.j | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
pi1inv.y | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
pi1inv.f | ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) |
pi1inv.0 | ⊢ (𝜑 → (𝐹‘0) = 𝑌) |
pi1inv.1 | ⊢ (𝜑 → (𝐹‘1) = 𝑌) |
pi1inv.i | ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) |
Ref | Expression |
---|---|
pi1inv | ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1grp.2 | . . . 4 ⊢ 𝐺 = (𝐽 π1 𝑌) | |
2 | eqid 2825 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | pi1inv.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | pi1inv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
5 | eqid 2825 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
6 | pi1inv.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽)) | |
7 | pi1inv.i | . . . . . . . 8 ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥))) | |
8 | 7 | pcorevcl 23201 | . . . . . . 7 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
9 | 6, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = (𝐹‘1) ∧ (𝐼‘1) = (𝐹‘0))) |
10 | 9 | simp1d 1176 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
11 | 9 | simp2d 1177 | . . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
12 | pi1inv.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌) | |
13 | 11, 12 | eqtrd 2861 | . . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌) |
14 | 9 | simp3d 1178 | . . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
15 | pi1inv.0 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌) | |
16 | 14, 15 | eqtrd 2861 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌) |
17 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) |
18 | 1, 3, 4, 17 | pi1eluni 23218 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌))) |
19 | 10, 13, 16, 18 | mpbir3and 1446 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺)) |
20 | 1, 3, 4, 17 | pi1eluni 23218 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌))) |
21 | 6, 15, 12, 20 | mpbir3and 1446 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺)) |
22 | 1, 2, 3, 4, 5, 19, 21 | pi1addval 23224 | . . 3 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽)) |
23 | phtpcer 23171 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
25 | eqid 2825 | . . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)}) | |
26 | 7, 25 | pcorev 23203 | . . . . . 6 ⊢ (𝐹 ∈ (II Cn 𝐽) → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘1)})) |
27 | 6, 26 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {(𝐹‘1)})) |
28 | 12 | sneqd 4411 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌}) |
29 | 28 | xpeq2d 5376 | . . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌})) |
30 | 27, 29 | breqtrd 4901 | . . . 4 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {𝑌})) |
31 | 24, 30 | erthi 8063 | . . 3 ⊢ (𝜑 → [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃ph‘𝐽)) |
32 | eqid 2825 | . . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌}) | |
33 | 1, 2, 3, 4, 32 | pi1grplem 23225 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺))) |
34 | 33 | simprd 491 | . . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺)) |
35 | 22, 31, 34 | 3eqtrd 2865 | . 2 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺)) |
36 | 33 | simpld 490 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
37 | 1, 2, 3, 4, 6, 15, 12 | elpi1i 23222 | . . 3 ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
38 | 1, 2, 3, 4, 10, 13, 16 | elpi1i 23222 | . . 3 ⊢ (𝜑 → [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
39 | eqid 2825 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
40 | pi1inv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
41 | 2, 5, 39, 40 | grpinvid2 17832 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺) ∧ [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) → ((𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽) ↔ ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺))) |
42 | 36, 37, 38, 41 | syl3anc 1494 | . 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽) ↔ ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺))) |
43 | 35, 42 | mpbird 249 | 1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 {csn 4399 ∪ cuni 4660 class class class wbr 4875 ↦ cmpt 4954 × cxp 5344 ‘cfv 6127 (class class class)co 6910 Er wer 8011 [cec 8012 0cc0 10259 1c1 10260 − cmin 10592 [,]cicc 12473 Basecbs 16229 +gcplusg 16312 0gc0g 16460 Grpcgrp 17783 invgcminusg 17784 TopOnctopon 21092 Cn ccn 21406 IIcii 23055 ≃phcphtpc 23145 *𝑝cpco 23176 π1 cpi1 23179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-2o 7832 df-oadd 7835 df-er 8014 df-ec 8016 df-qs 8020 df-map 8129 df-ixp 8182 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-fi 8592 df-sup 8623 df-inf 8624 df-oi 8691 df-card 9085 df-cda 9312 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ioo 12474 df-icc 12477 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-hash 13418 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-hom 16336 df-cco 16337 df-rest 16443 df-topn 16444 df-0g 16462 df-gsum 16463 df-topgen 16464 df-pt 16465 df-prds 16468 df-xrs 16522 df-qtop 16527 df-imas 16528 df-qus 16529 df-xps 16530 df-mre 16606 df-mrc 16607 df-acs 16609 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-mulg 17902 df-cntz 18107 df-cmn 18555 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-cnfld 20114 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-cld 21201 df-cn 21409 df-cnp 21410 df-tx 21743 df-hmeo 21936 df-xms 22502 df-ms 22503 df-tms 22504 df-ii 23057 df-htpy 23146 df-phtpy 23147 df-phtpc 23168 df-pco 23181 df-om1 23182 df-pi1 23184 |
This theorem is referenced by: (None) |
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