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Mirrors > Home > MPE Home > Th. List > frgpinv | Structured version Visualization version GIF version |
Description: The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
frgpadd.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) |
frgpadd.g | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgpadd.r | ⊢ ∼ = ( ~FG ‘𝐼) |
frgpinv.n | ⊢ 𝑁 = (invg‘𝐺) |
frgpinv.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) |
Ref | Expression |
---|---|
frgpinv | ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpadd.w | . . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
2 | fviss 6745 | . . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
3 | 1, 2 | eqsstri 3911 | . . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
4 | 3 | sseli 3873 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o)) |
5 | revcl 14212 | . . . . . . 7 ⊢ (𝐴 ∈ Word (𝐼 × 2o) → (reverse‘𝐴) ∈ Word (𝐼 × 2o)) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (reverse‘𝐴) ∈ Word (𝐼 × 2o)) |
7 | frgpinv.m | . . . . . . 7 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
8 | 7 | efgmf 18957 | . . . . . 6 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
9 | wrdco 14282 | . . . . . 6 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) | |
10 | 6, 8, 9 | sylancl 589 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) |
11 | 1 | efgrcl 18959 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
12 | 11 | simprd 499 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
13 | 10, 12 | eleqtrrd 2836 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
14 | frgpadd.g | . . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼) | |
15 | frgpadd.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
16 | eqid 2738 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 1, 14, 15, 16 | frgpadd 19007 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
18 | 13, 17 | mpdan 687 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
19 | 1, 15 | efger 18962 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊) |
21 | eqid 2738 | . . . . 5 ⊢ (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
22 | 1, 15, 7, 21 | efginvrel2 18971 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) |
23 | 20, 22 | erthi 8371 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ ) |
24 | 14, 15 | frgp0 19004 | . . . . . 6 ⊢ (𝐼 ∈ V → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
25 | 24 | adantr 484 | . . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)) → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
26 | 11, 25 | syl 17 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
27 | 26 | simprd 499 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [∅] ∼ = (0g‘𝐺)) |
28 | 18, 23, 27 | 3eqtrd 2777 | . 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺)) |
29 | 26 | simpld 498 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp) |
30 | eqid 2738 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
31 | 14, 15, 1, 30 | frgpeccl 19005 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺)) |
32 | 14, 15, 1, 30 | frgpeccl 19005 | . . . 4 ⊢ ((𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊 → [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) |
33 | 13, 32 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) |
34 | eqid 2738 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
35 | frgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
36 | 30, 16, 34, 35 | grpinvid1 18272 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ [𝐴] ∼ ∈ (Base‘𝐺) ∧ [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
37 | 29, 31, 33, 36 | syl3anc 1372 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
38 | 28, 37 | mpbird 260 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∖ cdif 3840 ∅c0 4211 〈cop 4522 〈cotp 4524 ↦ cmpt 5110 I cid 5428 × cxp 5523 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 1oc1o 8124 2oc2o 8125 Er wer 8317 [cec 8318 0cc0 10615 ...cfz 12981 ♯chash 13782 Word cword 13955 ++ cconcat 14011 splice csplice 14200 reversecreverse 14209 〈“cs2 14292 Basecbs 16586 +gcplusg 16668 0gc0g 16816 Grpcgrp 18219 invgcminusg 18220 ~FG cefg 18950 freeGrpcfrgp 18951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-ec 8322 df-qs 8326 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-xnn0 12049 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-fzo 13125 df-hash 13783 df-word 13956 df-lsw 14004 df-concat 14012 df-s1 14039 df-substr 14092 df-pfx 14122 df-splice 14201 df-reverse 14210 df-s2 14299 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-0g 16818 df-imas 16884 df-qus 16885 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-frmd 18130 df-grp 18222 df-minusg 18223 df-efg 18953 df-frgp 18954 |
This theorem is referenced by: vrgpinv 19013 |
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