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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 2823 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | pi1inv.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | pi1inv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
5 | eqid 2823 | . . . 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 23631 | . . . . . . 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 1138 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
11 | 9 | simp2d 1139 | . . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
12 | pi1inv.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌) | |
13 | 11, 12 | eqtrd 2858 | . . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌) |
14 | 9 | simp3d 1140 | . . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
15 | pi1inv.0 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌) | |
16 | 14, 15 | eqtrd 2858 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌) |
17 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) |
18 | 1, 3, 4, 17 | pi1eluni 23648 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌))) |
19 | 10, 13, 16, 18 | mpbir3and 1338 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺)) |
20 | 1, 3, 4, 17 | pi1eluni 23648 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌))) |
21 | 6, 15, 12, 20 | mpbir3and 1338 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺)) |
22 | 1, 2, 3, 4, 5, 19, 21 | pi1addval 23654 | . . 3 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽)) |
23 | phtpcer 23601 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
25 | eqid 2823 | . . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)}) | |
26 | 7, 25 | pcorev 23633 | . . . . . 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 4581 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌}) |
29 | 28 | xpeq2d 5587 | . . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌})) |
30 | 27, 29 | breqtrd 5094 | . . . 4 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {𝑌})) |
31 | 24, 30 | erthi 8342 | . . 3 ⊢ (𝜑 → [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃ph‘𝐽)) |
32 | eqid 2823 | . . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌}) | |
33 | 1, 2, 3, 4, 32 | pi1grplem 23655 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺))) |
34 | 33 | simprd 498 | . . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺)) |
35 | 22, 31, 34 | 3eqtrd 2862 | . 2 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺)) |
36 | 33 | simpld 497 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
37 | 1, 2, 3, 4, 6, 15, 12 | elpi1i 23652 | . . 3 ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
38 | 1, 2, 3, 4, 10, 13, 16 | elpi1i 23652 | . . 3 ⊢ (𝜑 → [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
39 | eqid 2823 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
40 | pi1inv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
41 | 2, 5, 39, 40 | grpinvid2 18157 | . . 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 1367 | . 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽) ↔ ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺))) |
43 | 35, 42 | mpbird 259 | 1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 {csn 4569 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ‘cfv 6357 (class class class)co 7158 Er wer 8288 [cec 8289 0cc0 10539 1c1 10540 − cmin 10872 [,]cicc 12744 Basecbs 16485 +gcplusg 16567 0gc0g 16715 Grpcgrp 18105 invgcminusg 18106 TopOnctopon 21520 Cn ccn 21834 IIcii 23485 ≃phcphtpc 23575 *𝑝cpco 23606 π1 cpi1 23609 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-qus 16784 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-cn 21837 df-cnp 21838 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 df-ii 23487 df-htpy 23576 df-phtpy 23577 df-phtpc 23598 df-pco 23611 df-om1 23612 df-pi1 23614 |
This theorem is referenced by: (None) |
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