Theorem vrgpinv 18103

Description: The inverse of a generating element is represented by ⟨𝐴, 1⟩ instead of ⟨𝐴, 0⟩. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
vrgpfval.r	⊢ ∼ = ( ~FG ‘𝐼)
vrgpfval.u	⊢ 𝑈 = (varFGrp‘𝐼)
vrgpf.m	⊢ 𝐺 = (freeGrp‘𝐼)
vrgpinv.n	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
vrgpinv	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] ∼ )

Proof of Theorem vrgpinv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vrgpfval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
2		vrgpfval.u	. . . 4 ⊢ 𝑈 = (varFGrp‘𝐼)
3	1, 2	vrgpval 18101	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
4	3	fveq2d 6152	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ))
5		simpr 477	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼)
6		0ex 4750	. . . . . . . 8 ⊢ ∅ ∈ V
7	6	prid1 4267	. . . . . . 7 ⊢ ∅ ∈ {∅, 1𝑜}
8		df2o3 7518	. . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜}
9	7, 8	eleqtrri 2697	. . . . . 6 ⊢ ∅ ∈ 2𝑜
10		opelxpi 5108	. . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
11	5, 9, 10	sylancl 693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
12	11	s1cld 13322	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
13		simpl 473	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉)
14		2on 7513	. . . . . 6 ⊢ 2𝑜 ∈ On
15		xpexg 6913	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
16	13, 14, 15	sylancl 693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2𝑜) ∈ V)
17		wrdexg 13254	. . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
18		fvi 6212	. . . . 5 ⊢ (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
19	16, 17, 18	3syl 18	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
20	12, 19	eleqtrrd 2701	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“⟨𝐴, ∅⟩”⟩ ∈ ( I ‘Word (𝐼 × 2𝑜)))
21		eqid 2621	. . . 4 ⊢ ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
22		vrgpf.m	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
23		vrgpinv.n	. . . 4 ⊢ 𝑁 = (invg‘𝐺)
24		eqid 2621	. . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)
25	21, 22, 1, 23, 24	frgpinv 18098	. . 3 ⊢ (⟨“⟨𝐴, ∅⟩”⟩ ∈ ( I ‘Word (𝐼 × 2𝑜)) → (𝑁‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩))] ∼ )
26	20, 25	syl 17	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = [((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩))] ∼ )
27		revs1 13451	. . . . . 6 ⊢ (reverse‘⟨“⟨𝐴, ∅⟩”⟩) = ⟨“⟨𝐴, ∅⟩”⟩
28	27	a1i 11	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘⟨“⟨𝐴, ∅⟩”⟩) = ⟨“⟨𝐴, ∅⟩”⟩)
29	28	coeq2d 5244	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩)) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ ⟨“⟨𝐴, ∅⟩”⟩))
30	24	efgmf 18047	. . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
31		s1co 13516	. . . . 5 ⊢ ((⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜) ∧ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)”⟩)
32	11, 30, 31	sylancl 693	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)”⟩)
33	24	efgmval 18046	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)∅) = ⟨𝐴, (1𝑜 ∖ ∅)⟩)
34	5, 9, 33	sylancl 693	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)∅) = ⟨𝐴, (1𝑜 ∖ ∅)⟩)
35		df-ov 6607	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)
36		dif0 3924	. . . . . . 7 ⊢ (1𝑜 ∖ ∅) = 1𝑜
37	36	opeq2i 4374	. . . . . 6 ⊢ ⟨𝐴, (1𝑜 ∖ ∅)⟩ = ⟨𝐴, 1𝑜⟩
38	34, 35, 37	3eqtr3g 2678	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩) = ⟨𝐴, 1𝑜⟩)
39	38	s1eqd 13320	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)”⟩ = ⟨“⟨𝐴, 1𝑜⟩”⟩)
40	29, 32, 39	3eqtrd 2659	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩)) = ⟨“⟨𝐴, 1𝑜⟩”⟩)
41	40	eceq1d 7728	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩))] ∼ = [⟨“⟨𝐴, 1𝑜⟩”⟩] ∼ )
42	4, 26, 41	3eqtrd 2659	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ∖ cdif 3552  ∅c0 3891  {cpr 4150  ⟨cop 4154   I cid 4984   × cxp 5072   ∘ ccom 5078  Oncon0 5682  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↦ cmpt2 6606  1_𝑜c1o 7498  2_𝑜c2o 7499  [cec 7685  Word cword 13230  ⟨“cs1 13233  reversecreverse 13236  inv_gcminusg 17344   ~_FG cefg 18040  freeGrpcfrgp 18041  var_FGrpcvrgp 18042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-qs 7693  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-lsw 13239  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-reverse 13244  df-s2 13530  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-0g 16023  df-imas 16089  df-qus 16090  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-frmd 17307  df-grp 17346  df-minusg 17347  df-efg 18043  df-frgp 18044  df-vrgp 18045

This theorem is referenced by:  frgpup3lem  18111