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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 2610 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | pi1inv.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
4 | pi1inv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
5 | eqid 2610 | . . . 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 22633 | . . . . . . 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 1066 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽)) |
11 | 9 | simp2d 1067 | . . . . . 6 ⊢ (𝜑 → (𝐼‘0) = (𝐹‘1)) |
12 | pi1inv.1 | . . . . . 6 ⊢ (𝜑 → (𝐹‘1) = 𝑌) | |
13 | 11, 12 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → (𝐼‘0) = 𝑌) |
14 | 9 | simp3d 1068 | . . . . . 6 ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0)) |
15 | pi1inv.0 | . . . . . 6 ⊢ (𝜑 → (𝐹‘0) = 𝑌) | |
16 | 14, 15 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → (𝐼‘1) = 𝑌) |
17 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐺)) |
18 | 1, 3, 4, 17 | pi1eluni 22650 | . . . . 5 ⊢ (𝜑 → (𝐼 ∈ ∪ (Base‘𝐺) ↔ (𝐼 ∈ (II Cn 𝐽) ∧ (𝐼‘0) = 𝑌 ∧ (𝐼‘1) = 𝑌))) |
19 | 10, 13, 16, 18 | mpbir3and 1238 | . . . 4 ⊢ (𝜑 → 𝐼 ∈ ∪ (Base‘𝐺)) |
20 | 1, 3, 4, 17 | pi1eluni 22650 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ ∪ (Base‘𝐺) ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌))) |
21 | 6, 15, 12, 20 | mpbir3and 1238 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ ∪ (Base‘𝐺)) |
22 | 1, 2, 3, 4, 5, 19, 21 | pi1addval 22656 | . . 3 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽)) |
23 | phtpcer 22602 | . . . . 5 ⊢ ( ≃ph‘𝐽) Er (II Cn 𝐽) | |
24 | 23 | a1i 11 | . . . 4 ⊢ (𝜑 → ( ≃ph‘𝐽) Er (II Cn 𝐽)) |
25 | eqid 2610 | . . . . . . 7 ⊢ ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {(𝐹‘1)}) | |
26 | 7, 25 | pcorev 22635 | . . . . . 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 4137 | . . . . . 6 ⊢ (𝜑 → {(𝐹‘1)} = {𝑌}) |
29 | 28 | xpeq2d 5063 | . . . . 5 ⊢ (𝜑 → ((0[,]1) × {(𝐹‘1)}) = ((0[,]1) × {𝑌})) |
30 | 27, 29 | breqtrd 4609 | . . . 4 ⊢ (𝜑 → (𝐼(*𝑝‘𝐽)𝐹)( ≃ph‘𝐽)((0[,]1) × {𝑌})) |
31 | 24, 30 | erthi 7680 | . . 3 ⊢ (𝜑 → [(𝐼(*𝑝‘𝐽)𝐹)]( ≃ph‘𝐽) = [((0[,]1) × {𝑌})]( ≃ph‘𝐽)) |
32 | eqid 2610 | . . . . 5 ⊢ ((0[,]1) × {𝑌}) = ((0[,]1) × {𝑌}) | |
33 | 1, 2, 3, 4, 32 | pi1grplem 22657 | . . . 4 ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺))) |
34 | 33 | simprd 478 | . . 3 ⊢ (𝜑 → [((0[,]1) × {𝑌})]( ≃ph‘𝐽) = (0g‘𝐺)) |
35 | 22, 31, 34 | 3eqtrd 2648 | . 2 ⊢ (𝜑 → ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺)) |
36 | 33 | simpld 474 | . . 3 ⊢ (𝜑 → 𝐺 ∈ Grp) |
37 | 1, 2, 3, 4, 6, 15, 12 | elpi1i 22654 | . . 3 ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
38 | 1, 2, 3, 4, 10, 13, 16 | elpi1i 22654 | . . 3 ⊢ (𝜑 → [𝐼]( ≃ph‘𝐽) ∈ (Base‘𝐺)) |
39 | eqid 2610 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
40 | pi1inv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
41 | 2, 5, 39, 40 | grpinvid2 17294 | . . 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 1318 | . 2 ⊢ (𝜑 → ((𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽) ↔ ([𝐼]( ≃ph‘𝐽)(+g‘𝐺)[𝐹]( ≃ph‘𝐽)) = (0g‘𝐺))) |
43 | 35, 42 | mpbird 246 | 1 ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ‘cfv 5804 (class class class)co 6549 Er wer 7626 [cec 7627 0cc0 9815 1c1 9816 − cmin 10145 [,]cicc 12049 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 TopOnctopon 20518 Cn ccn 20838 IIcii 22486 ≃phcphtpc 22576 *𝑝cpco 22608 π1 cpi1 22611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-qus 15992 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-ii 22488 df-htpy 22577 df-phtpy 22578 df-phtpc 22599 df-pco 22613 df-om1 22614 df-pi1 22616 |
This theorem is referenced by: (None) |
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