Theorem vrgpinv 18005

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 18003	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
4	3	fveq2d 6107	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = (𝑁‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ))
5		simpr 476	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐴 ∈ 𝐼)
6		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
7	6	prid1 4241	. . . . . . 7 ⊢ ∅ ∈ {∅, 1𝑜}
8		df2o3 7460	. . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜}
9	7, 8	eleqtrri 2687	. . . . . 6 ⊢ ∅ ∈ 2𝑜
10		opelxpi 5072	. . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
11	5, 9, 10	sylancl 693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2𝑜))
12	11	s1cld 13236	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2𝑜))
13		simpl 472	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → 𝐼 ∈ 𝑉)
14		2on 7455	. . . . . 6 ⊢ 2𝑜 ∈ On
15		xpexg 6858	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
16	13, 14, 15	sylancl 693	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝐼 × 2𝑜) ∈ V)
17		wrdexg 13170	. . . . 5 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
18		fvi 6165	. . . . 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 2691	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“⟨𝐴, ∅⟩”⟩ ∈ ( I ‘Word (𝐼 × 2𝑜)))
21		eqid 2610	. . . 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 2610	. . . 4 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) = (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)
25	21, 22, 1, 23, 24	frgpinv 18000	. . 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 13365	. . . . . 6 ⊢ (reverse‘⟨“⟨𝐴, ∅⟩”⟩) = ⟨“⟨𝐴, ∅⟩”⟩
28	27	a1i 11	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (reverse‘⟨“⟨𝐴, ∅⟩”⟩) = ⟨“⟨𝐴, ∅⟩”⟩)
29	28	coeq2d 5206	. . . 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 17949	. . . . 5 ⊢ (𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
31		s1co 13430	. . . . 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 17948	. . . . . . 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 6552	. . . . . 6 ⊢ (𝐴(𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)∅) = ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)
36		dif0 3904	. . . . . . 7 ⊢ (1𝑜 ∖ ∅) = 1𝑜
37	36	opeq2i 4344	. . . . . 6 ⊢ ⟨𝐴, (1𝑜 ∖ ∅)⟩ = ⟨𝐴, 1𝑜⟩
38	34, 35, 37	3eqtr3g 2667	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩) = ⟨𝐴, 1𝑜⟩)
39	38	s1eqd 13234	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ⟨“((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩)‘⟨𝐴, ∅⟩)”⟩ = ⟨“⟨𝐴, 1𝑜⟩”⟩)
40	29, 32, 39	3eqtrd 2648	. . 3 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → ((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩)) = ⟨“⟨𝐴, 1𝑜⟩”⟩)
41	40	eceq1d 7670	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → [((𝑥 ∈ 𝐼, 𝑦 ∈ 2𝑜 ↦ ⟨𝑥, (1𝑜 ∖ 𝑦)⟩) ∘ (reverse‘⟨“⟨𝐴, ∅⟩”⟩))] ∼ = [⟨“⟨𝐴, 1𝑜⟩”⟩] ∼ )
42	4, 26, 41	3eqtrd 2648	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑁‘(𝑈‘𝐴)) = [⟨“⟨𝐴, 1𝑜⟩”⟩] ∼ )

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {cpr 4127  ⟨cop 4131   I cid 4948   × cxp 5036   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441  [cec 7627  Word cword 13146  ⟨“cs1 13149  reversecreverse 13152  inv_gcminusg 17246   ~_FG cefg 17942  freeGrpcfrgp 17943  var_FGrpcvrgp 17944

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  df-vrgp 17947

This theorem is referenced by:  frgpup3lem  18013