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Theorem frgpinv 18384

Description: The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)

**Hypotheses**
Ref	Expression
frgpadd.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
frgpadd.g	⊢ 𝐺 = (freeGrp‘𝐼)
frgpadd.r	⊢ ∼ = ( ~_FG ‘𝐼)
frgpinv.n	⊢ 𝑁 = (inv_g‘𝐺)
frgpinv.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)

**Assertion**
Ref	Expression
frgpinv	⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ )

Distinct variable groups: 𝑦,𝑧,𝐼 𝑦, ∼ ,𝑧 𝑦,𝑊,𝑧 Allowed substitution hints: 𝐴(𝑦,𝑧) 𝐺(𝑦,𝑧) 𝑀(𝑦,𝑧) 𝑁(𝑦,𝑧)

Proof of Theorem frgpinv Dummy variables 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpadd.w	. . . . . . . . 9 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
2		fviss 6398	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
3	1, 2	eqsstri 3784	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
4	3	sseli 3748	. . . . . . 7 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ Word (𝐼 × 2_𝑜))
5		revcl 13719	. . . . . . 7 ⊢ (𝐴 ∈ Word (𝐼 × 2_𝑜) → (reverse‘𝐴) ∈ Word (𝐼 × 2_𝑜))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (reverse‘𝐴) ∈ Word (𝐼 × 2_𝑜))
7		frgpinv.m	. . . . . . 7 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
8	7	efgmf 18333	. . . . . 6 ⊢ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)
9		wrdco 13786	. . . . . 6 ⊢ (((reverse‘𝐴) ∈ Word (𝐼 × 2_𝑜) ∧ 𝑀:(𝐼 × 2_𝑜)⟶(𝐼 × 2_𝑜)) → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2_𝑜))
10	6, 8, 9	sylancl 574	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ Word (𝐼 × 2_𝑜))
11	1	efgrcl 18335	. . . . . 6 ⊢ (𝐴 ∈ 𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)))
12	11	simprd 483	. . . . 5 ⊢ (𝐴 ∈ 𝑊 → 𝑊 = Word (𝐼 × 2_𝑜))
13	10, 12	eleqtrrd 2853	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊)
14		frgpadd.g	. . . . 5 ⊢ 𝐺 = (freeGrp‘𝐼)
15		frgpadd.r	. . . . 5 ⊢ ∼ = ( ~_FG ‘𝐼)
16		eqid 2771	. . . . 5 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
17	1, 14, 15, 16	frgpadd 18383	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ (𝑀 ∘ (reverse‘𝐴)) ∈ 𝑊) → ([𝐴] ∼ (+_g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ )
18	13, 17	mpdan 667	. . 3 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+_g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ )
19	1, 15	efger 18338	. . . . 5 ⊢ ∼ Er 𝑊
20	19	a1i 11	. . . 4 ⊢ (𝐴 ∈ 𝑊 → ∼ Er 𝑊)
21		eqid 2771	. . . . 5 ⊢ (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩))) = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
22	1, 15, 7, 21	efginvrel2 18347	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ++ (𝑀 ∘ (reverse‘𝐴))) ∼ ∅)
23	20, 22	erthi 7945	. . 3 ⊢ (𝐴 ∈ 𝑊 → [(𝐴 ++ (𝑀 ∘ (reverse‘𝐴)))] ∼ = [∅] ∼ )
24	14, 15	frgp0 18380	. . . . . 6 ⊢ (𝐼 ∈ V → (𝐺 ∈ Grp ∧ [∅] ∼ = (0_g‘𝐺)))
25	24	adantr 466	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2_𝑜)) → (𝐺 ∈ Grp ∧ [∅] ∼ = (0_g‘𝐺)))
26	11, 25	syl 17	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐺 ∈ Grp ∧ [∅] ∼ = (0_g‘𝐺)))
27	26	simprd 483	. . 3 ⊢ (𝐴 ∈ 𝑊 → [∅] ∼ = (0_g‘𝐺))
28	18, 23, 27	3eqtrd 2809	. 2 ⊢ (𝐴 ∈ 𝑊 → ([𝐴] ∼ (+_g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0_g‘𝐺))
29	26	simpld 482	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐺 ∈ Grp)
30		eqid 2771	. . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺)
31	14, 15, 1, 30	frgpeccl 18381	. . 3 ⊢ (𝐴 ∈ 𝑊 → [𝐴] ∼ ∈ (Base‘𝐺))
32	14, 15, 1, 30	frgpeccl 18381	. . . 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 2771	. . . 4 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
35		frgpinv.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐺)
36	30, 16, 34, 35	grpinvid1 17678	. . 3 ⊢ ((𝐺 ∈ Grp ∧ [𝐴] ∼ ∈ (Base‘𝐺) ∧ [(𝑀 ∘ (reverse‘𝐴))] ∼ ∈ (Base‘𝐺)) → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+_g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0_g‘𝐺)))
37	29, 31, 33, 36	syl3anc 1476	. 2 ⊢ (𝐴 ∈ 𝑊 → ((𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ ↔ ([𝐴] ∼ (+_g‘𝐺)[(𝑀 ∘ (reverse‘𝐴))] ∼ ) = (0_g‘𝐺)))
38	28, 37	mpbird 247	1 ⊢ (𝐴 ∈ 𝑊 → (𝑁‘[𝐴] ∼ ) = [(𝑀 ∘ (reverse‘𝐴))] ∼ )

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∖ cdif 3720 ∅c0 4063 ⟨cop 4322 ⟨cotp 4324 ↦ cmpt 4863 I cid 5156 × cxp 5247 ∘ ccom 5253 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ↦ cmpt2 6795 1_𝑜c1o 7706 2_𝑜c2o 7707 Er wer 7893 [cec 7894 0cc0 10138 ...cfz 12533 ♯chash 13321 Word cword 13487 ++ cconcat 13489 splice csplice 13492 reversecreverse 13493 ⟨“cs2 13795 Basecbs 16064 +_gcplusg 16149 0_gc0g 16308 Grpcgrp 17630 inv_gcminusg 17631 ~_FG cefg 18326 freeGrpcfrgp 18327

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-ot 4325 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-2o 7714 df-oadd 7717 df-er 7896 df-ec 7898 df-qs 7902 df-map 8011 df-pm 8012 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-inf 8505 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-xnn0 11566 df-z 11580 df-dec 11696 df-uz 11889 df-fz 12534 df-fzo 12674 df-hash 13322 df-word 13495 df-lsw 13496 df-concat 13497 df-s1 13498 df-substr 13499 df-splice 13500 df-reverse 13501 df-s2 13802 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-ip 16167 df-tset 16168 df-ple 16169 df-ds 16172 df-0g 16310 df-imas 16376 df-qus 16377 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-frmd 17594 df-grp 17633 df-minusg 17634 df-efg 18329 df-frgp 18330

This theorem is referenced by: vrgpinv 18389

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