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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 2798 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | pi1inv.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | pi1inv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
5 | eqid 2798 | . . . 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 23630 | . . . . . . 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 1139 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
11 | 9 | simp2d 1140 | . . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
12 | pi1inv.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌) | |
13 | 11, 12 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌) |
14 | 9 | simp3d 1141 | . . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
15 | pi1inv.0 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌) | |
16 | 14, 15 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌) |
17 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) |
18 | 1, 3, 4, 17 | pi1eluni 23647 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌))) |
19 | 10, 13, 16, 18 | mpbir3and 1339 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺)) |
20 | 1, 3, 4, 17 | pi1eluni 23647 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌))) |
21 | 6, 15, 12, 20 | mpbir3and 1339 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺)) |
22 | 1, 2, 3, 4, 5, 19, 21 | pi1addval 23653 | . . 3 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽)) |
23 | phtpcer 23600 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
25 | eqid 2798 | . . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)}) | |
26 | 7, 25 | pcorev 23632 | . . . . . 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 4537 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌}) |
29 | 28 | xpeq2d 5549 | . . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌})) |
30 | 27, 29 | breqtrd 5056 | . . . 4 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {𝑌})) |
31 | 24, 30 | erthi 8323 | . . 3 ⊢ (𝜑 → [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃ph‘𝐽)) |
32 | eqid 2798 | . . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌}) | |
33 | 1, 2, 3, 4, 32 | pi1grplem 23654 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺))) |
34 | 33 | simprd 499 | . . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺)) |
35 | 22, 31, 34 | 3eqtrd 2837 | . 2 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺)) |
36 | 33 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
37 | 1, 2, 3, 4, 6, 15, 12 | elpi1i 23651 | . . 3 ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
38 | 1, 2, 3, 4, 10, 13, 16 | elpi1i 23651 | . . 3 ⊢ (𝜑 → [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
39 | eqid 2798 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
40 | pi1inv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
41 | 2, 5, 39, 40 | grpinvid2 18147 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽) ↔ ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺))) |
43 | 35, 42 | mpbird 260 | 1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {csn 4525 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 (class class class)co 7135 Er wer 8269 [cec 8270 0cc0 10526 1c1 10527 − cmin 10859 [,]cicc 12729 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 TopOnctopon 21515 Cn ccn 21829 IIcii 23480 ≃phcphtpc 23574 *𝑝cpco 23605 π1 cpi1 23608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 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-qus 16774 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-grp 18098 df-minusg 18099 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-ii 23482 df-htpy 23575 df-phtpy 23576 df-phtpc 23597 df-pco 23610 df-om1 23611 df-pi1 23613 |
This theorem is referenced by: (None) |
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