Theorem grpinvval 17461

Description: The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)

Hypotheses

Ref	Expression
grpinvval.b	|- B = ( Base ` G )
grpinvval.p	|- .+ = ( +g ` G )
grpinvval.o	|- .0. = ( 0g ` G )
grpinvval.n	|- N = ( invg ` G )

Assertion

Ref	Expression
grpinvval	|- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Distinct variable groups:   y, B   y, G   y, X

Allowed substitution hints:   .+ ( y)   N( y)   .0. ( y)

Proof of Theorem grpinvval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
2	1	eqeq1d 2624	. . 3 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
3	2	riotabidv 6613	. 2 |- ( x = X -> ( iota_ y e. B ( y .+ x ) = .0. ) = ( iota_ y e. B ( y .+ X ) = .0. ) )
4		grpinvval.b	. . 3 |- B = ( Base ` G )
5		grpinvval.p	. . 3 |- .+ = ( +g ` G )
6		grpinvval.o	. . 3 |- .0. = ( 0g ` G )
7		grpinvval.n	. . 3 |- N = ( invg ` G )
8	4, 5, 6, 7	grpinvfval 17460	. 2 |- N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
9		riotaex 6615	. 2 |- ( iota_ y e. B ( y .+ X ) = .0. ) e. _V
10	3, 8, 9	fvmpt 6282	1 |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   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-minusg 17426

This theorem is referenced by:  grplinv  17468  isgrpinv  17472  xrsinvgval  29677  ringinvval  29792