Theorem invrfval 19158

Description: Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)

Hypotheses

Ref	Expression
invrfval.u	⊢ 𝑈 = (Unit‘𝑅)
invrfval.g	⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
invrfval.i	⊢ 𝐼 = (invr‘𝑅)

Assertion

Ref	Expression
invrfval	⊢ 𝐼 = (invg‘𝐺)

Proof of Theorem invrfval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		invrfval.i	. 2 ⊢ 𝐼 = (invr‘𝑅)
2		fveq2 6496	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
3		fveq2 6496	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
4		invrfval.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
5	3, 4	syl6eqr 2825	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
6	2, 5	oveq12d 6992	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈))
7		invrfval.g	. . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)
8	6, 7	syl6eqr 2825	. . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺)
9	8	fveq2d 6500	. . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺))
10		df-invr 19157	. . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))
11		fvex 6509	. . . 4 ⊢ (invg‘𝐺) ∈ V
12	9, 10, 11	fvmpt 6593	. . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺))
13		fvprc 6489	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅)
14		base0 16390	. . . . . . 7 ⊢ ∅ = (Base‘∅)
15		eqid 2771	. . . . . . 7 ⊢ (invg‘∅) = (invg‘∅)
16	14, 15	grpinvfn 17944	. . . . . 6 ⊢ (invg‘∅) Fn ∅
17		fn0 6306	. . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅)
18	16, 17	mpbi 222	. . . . 5 ⊢ (invg‘∅) = ∅
19	13, 18	syl6eqr 2825	. . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅))
20		fvprc 6489	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅)
21	20	oveq1d 6989	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈))
22	7, 21	syl5eq 2819	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈))
23		ress0 16412	. . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅
24	22, 23	syl6eq 2823	. . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅)
25	24	fveq2d 6500	. . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅))
26	19, 25	eqtr4d 2810	. . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺))
27	12, 26	pm2.61i 177	. 2 ⊢ (invr‘𝑅) = (invg‘𝐺)
28	1, 27	eqtri 2795	1 ⊢ 𝐼 = (invg‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1508   ∈ wcel 2051  Vcvv 3408  ∅c0 4172   Fn wfn 6180  ‘cfv 6185  (class class class)co 6974   ↾_s cress 16338  inv_gcminusg 17904  mulGrpcmgp 18974  Unitcui 19124  inv_rcinvr 19156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-slot 16341  df-base 16343  df-ress 16345  df-minusg 17907  df-invr 19157

This theorem is referenced by:  unitinvcl  19159  unitinvinv  19160  unitlinv  19162  unitrinv  19163  invrpropd  19183  subrgugrp  19289  cntzsdrg  19315  cnmsubglem  20325  psgninv  20443  invrvald  21004  invrcn2  22506  nrginvrcn  23019  nrgtdrg  23020  sum2dchr  25567  rdivmuldivd  30573  ringinvval  30574  dvrcan5  30575