Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6166 | . . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜) | |
3 | 1, 2 | eqsstri 3598 | . . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2𝑜) |
4 | 3 | sseli 3564 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2𝑜)) |
5 | revcl 13361 | . . . . . . 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 17949 | . . . . . 6 ⊢ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜) |
9 | wrdco 13428 | . . . . . 6 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2𝑜) ∧ 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜)) | |
10 | 6, 8, 9 | sylancl 693 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2𝑜)) |
11 | 1 | efgrcl 17951 | . . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜))) |
12 | 11 | simprd 478 | . . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2𝑜)) |
13 | 10, 12 | eleqtrrd 2691 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) |
14 | frgpadd.g | . . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼) | |
15 | frgpadd.r | . . . . 5 ⊢ ∼ = ( ~FG ‘𝐼) | |
16 | eqid 2610 | . . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 1, 14, 15, 16 | frgpadd 17999 | . . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
18 | 13, 17 | mpdan 699 | . . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ ) |
19 | 1, 15 | efger 17954 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊) |
21 | eqid 2610 | . . . . 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 17963 | . . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅) |
23 | 20, 22 | erthi 7680 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ ) |
24 | 14, 15 | frgp0 17996 | . . . . . 6 ⊢ (𝐼 ∈ V → (𝐺 ∈ Grp ∧ [∅] ∼ = (0g‘𝐺))) |
25 | 24 | adantr 480 | . . . . 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 478 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [∅] ∼ = (0g‘𝐺)) |
28 | 18, 23, 27 | 3eqtrd 2648 | . 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺)) |
29 | 26 | simpld 474 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp) |
30 | eqid 2610 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
31 | 14, 15, 1, 30 | frgpeccl 17997 | . . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺)) |
32 | 14, 15, 1, 30 | frgpeccl 17997 | . . . 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 2610 | . . . 4 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
35 | frgpinv.n | . . . 4 ⊢ 𝑁 = (invg‘𝐺) | |
36 | 30, 16, 34, 35 | grpinvid1 17293 | . . 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 1318 | . 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0g‘𝐺))) |
38 | 28, 37 | mpbird 246 | 1 ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 ∅c0 3874 〈cop 4131 〈cotp 4133 ↦ cmpt 4643 I cid 4948 × cxp 5036 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1𝑜c1o 7440 2𝑜c2o 7441 Er wer 7626 [cec 7627 0cc0 9815 ...cfz 12197 #chash 12979 Word cword 13146 ++ cconcat 13148 splice csplice 13151 reversecreverse 13152 〈“cs2 13437 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 invgcminusg 17246 ~FG cefg 17942 freeGrpcfrgp 17943 |
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-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 |
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-ot 4134 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-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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 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-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 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-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-reverse 13160 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-frmd 17209 df-grp 17248 df-minusg 17249 df-efg 17945 df-frgp 17946 |
This theorem is referenced by: vrgpinv 18005 |
Copyright terms: Public domain | W3C validator |