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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 6716 | . . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
3 | 1, 2 | eqsstri 3949 | . . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
4 | 3 | sseli 3911 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o)) |
5 | revcl 14114 | . . . . . . 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 18831 | . . . . . 6 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
9 | wrdco 14184 | . . . . . 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 18833 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
12 | 11 | simprd 499 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
13 | 10, 12 | eleqtrrd 2893 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
14 | frgpadd.g | . . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼) | |
15 | frgpadd.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
16 | eqid 2798 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 1, 14, 15, 16 | frgpadd 18881 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
18 | 13, 17 | mpdan 686 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
19 | 1, 15 | efger 18836 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊) |
21 | eqid 2798 | . . . . 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 18845 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) |
23 | 20, 22 | erthi 8323 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ ) |
24 | 14, 15 | frgp0 18878 | . . . . . 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 2837 | . 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺)) |
29 | 26 | simpld 498 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp) |
30 | eqid 2798 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
31 | 14, 15, 1, 30 | frgpeccl 18879 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺)) |
32 | 14, 15, 1, 30 | frgpeccl 18879 | . . . 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 2798 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
35 | frgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
36 | 30, 16, 34, 35 | grpinvid1 18146 | . . 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 1368 | . 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 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ∅c0 4243 〈cop 4531 〈cotp 4533 ↦ cmpt 5110 I cid 5424 × cxp 5517 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1oc1o 8078 2oc2o 8079 Er wer 8269 [cec 8270 0cc0 10526 ...cfz 12885 ♯chash 13686 Word cword 13857 ++ cconcat 13913 splice csplice 14102 reversecreverse 14111 〈“cs2 14194 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 invgcminusg 18096 ~FG cefg 18824 freeGrpcfrgp 18825 |
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 |
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-ot 4534 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-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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 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-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 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-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-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-splice 14103 df-reverse 14112 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-0g 16707 df-imas 16773 df-qus 16774 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-frmd 18006 df-grp 18098 df-minusg 18099 df-efg 18827 df-frgp 18828 |
This theorem is referenced by: vrgpinv 18887 |
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