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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 6735 | . . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
3 | 1, 2 | eqsstri 4000 | . . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
4 | 3 | sseli 3962 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2o)) |
5 | revcl 14117 | . . . . . . 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 18833 | . . . . . 6 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o) |
9 | wrdco 14187 | . . . . . 6 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2o) ∧ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) | |
10 | 6, 8, 9 | sylancl 588 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2o)) |
11 | 1 | efgrcl 18835 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o))) |
12 | 11 | simprd 498 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2o)) |
13 | 10, 12 | eleqtrrd 2916 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
14 | frgpadd.g | . . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼) | |
15 | frgpadd.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
16 | eqid 2821 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 1, 14, 15, 16 | frgpadd 18883 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
18 | 13, 17 | mpdan 685 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
19 | 1, 15 | efger 18838 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊) |
21 | eqid 2821 | . . . . 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 18847 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) |
23 | 20, 22 | erthi 8334 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ ) |
24 | 14, 15 | frgp0 18880 | . . . . . 6 ⊢ (𝐼 ∈ V → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
25 | 24 | adantr 483 | . . . . 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 498 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [∅] ∼ = (0g‘𝐺)) |
28 | 18, 23, 27 | 3eqtrd 2860 | . 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺)) |
29 | 26 | simpld 497 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp) |
30 | eqid 2821 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
31 | 14, 15, 1, 30 | frgpeccl 18881 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺)) |
32 | 14, 15, 1, 30 | frgpeccl 18881 | . . . 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 2821 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
35 | frgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
36 | 30, 16, 34, 35 | grpinvid1 18148 | . . 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 1367 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
38 | 28, 37 | mpbird 259 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∅c0 4290 〈cop 4566 〈cotp 4568 ↦ cmpt 5138 I cid 5453 × cxp 5547 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1oc1o 8089 2oc2o 8090 Er wer 8280 [cec 8281 0cc0 10531 ...cfz 12886 ♯chash 13684 Word cword 13855 ++ cconcat 13916 splice csplice 14105 reversecreverse 14114 〈“cs2 14197 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Grpcgrp 18097 invgcminusg 18098 ~FG cefg 18826 freeGrpcfrgp 18827 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-reverse 14115 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-frmd 18008 df-grp 18100 df-minusg 18101 df-efg 18829 df-frgp 18830 |
This theorem is referenced by: vrgpinv 18889 |
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