Theorem invrfval 18673

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 6191	. . . . . . 7 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
3		fveq2 6191	. . . . . . . 8 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
4		invrfval.u	. . . . . . . 8 |- U = (Unit ` R )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( r = R -> (Unit ` r ) = U )
6	2, 5	oveq12d 6668	. . . . . 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 2674	. . . . 5 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = G )
9	8	fveq2d 6195	. . . 4 |- ( r = R -> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) = ( invg ` G ) )
10		df-invr 18672	. . . 4 |- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
11		fvex 6201	. . . 4 |- ( invg ` G ) e. _V
12	9, 10, 11	fvmpt 6282	. . 3 |- ( R e. _V -> ( invr ` R ) = ( invg ` G ) )
13		fvprc 6185	. . . . 5 |- ( -. R e. _V -> ( invr ` R ) = (/) )
14		base0 15912	. . . . . . 7 |- (/) = ( Base ` (/) )
15		eqid 2622	. . . . . . 7 |- ( invg ` (/) ) = ( invg ` (/) )
16	14, 15	grpinvfn 17462	. . . . . 6 |- ( invg ` (/) ) Fn (/)
17		fn0 6011	. . . . . 6 |- ( ( invg ` (/) ) Fn (/) <-> ( invg ` (/) ) = (/) )
18	16, 17	mpbi 220	. . . . 5 |- ( invg ` (/) ) = (/)
19	13, 18	syl6eqr 2674	. . . 4 |- ( -. R e. _V -> ( invr ` R ) = ( invg ` (/) ) )
20		fvprc 6185	. . . . . . . 8 |- ( -. R e. _V -> (mulGrp ` R ) = (/) )
21	20	oveq1d 6665	. . . . . . 7 |- ( -. R e. _V -> ( (mulGrp ` R ) ↾s U ) = ( (/) ↾s U ) )
22	7, 21	syl5eq 2668	. . . . . 6 |- ( -. R e. _V -> G = ( (/) ↾s U ) )
23		ress0 15934	. . . . . 6 |- ( (/) ↾s U ) = (/)
24	22, 23	syl6eq 2672	. . . . 5 |- ( -. R e. _V -> G = (/) )
25	24	fveq2d 6195	. . . 4 |- ( -. R e. _V -> ( invg ` G ) = ( invg ` (/) ) )
26	19, 25	eqtr4d 2659	. . 3 |- ( -. R e. _V -> ( invr ` R ) = ( invg ` G ) )
27	12, 26	pm2.61i 176	. 2 |- ( invr ` R ) = ( invg ` G )
28	1, 27	eqtri 2644	1 |- I = ( invg ` G )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858   invgcminusg 17423  mulGrpcmgp 18489  Unitcui 18639   invrcinvr 18671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-slot 15861  df-base 15863  df-ress 15865  df-minusg 17426  df-invr 18672

This theorem is referenced by:  unitinvcl  18674  unitinvinv  18675  unitlinv  18677  unitrinv  18678  invrpropd  18698  subrgugrp  18799  cnmsubglem  19809  psgninv  19928  invrvald  20482  invrcn2  21983  nrginvrcn  22496  nrgtdrg  22497  sum2dchr  24999  rdivmuldivd  29791  ringinvval  29792  dvrcan5  29793  cntzsdrg  37772