Theorem grpinvfvi 17463

Description: The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypothesis

Ref	Expression
grpinvfvi.t	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvfvi	|- N = ( invg ` ( _I ` G ) )

Proof of Theorem grpinvfvi

Step	Hyp	Ref	Expression
1		grpinvfvi.t	. 2 |- N = ( invg ` G )
2		fvi 6255	. . . 4 |- ( G e. _V -> ( _I ` G ) = G )
3	2	fveq2d 6195	. . 3 |- ( G e. _V -> ( invg ` ( _I ` G ) ) = ( invg ` G ) )
4		base0 15912	. . . . . 6 |- (/) = ( Base ` (/) )
5		eqid 2622	. . . . . 6 |- ( invg ` (/) ) = ( invg ` (/) )
6	4, 5	grpinvfn 17462	. . . . 5 |- ( invg ` (/) ) Fn (/)
7		fn0 6011	. . . . 5 |- ( ( invg ` (/) ) Fn (/) <-> ( invg ` (/) ) = (/) )
8	6, 7	mpbi 220	. . . 4 |- ( invg ` (/) ) = (/)
9		fvprc 6185	. . . . 5 |- ( -. G e. _V -> ( _I ` G ) = (/) )
10	9	fveq2d 6195	. . . 4 |- ( -. G e. _V -> ( invg ` ( _I ` G ) ) = ( invg ` (/) ) )
11		fvprc 6185	. . . 4 |- ( -. G e. _V -> ( invg ` G ) = (/) )
12	8, 10, 11	3eqtr4a 2682	. . 3 |- ( -. G e. _V -> ( invg ` ( _I ` G ) ) = ( invg ` G ) )
13	3, 12	pm2.61i 176	. 2 |- ( invg ` ( _I ` G ) ) = ( invg ` G )
14	1, 13	eqtr4i 2647	1 |- N = ( invg ` ( _I ` G ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   _I cid 5023   Fn wfn 5883   ` cfv 5888   invgcminusg 17423

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-slot 15861  df-base 15863  df-minusg 17426

This theorem is referenced by:  deg1invg  23866