Theorem grpinvval 13118

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		grpinvval.b	. . . . 5 |- B = ( Base ` G )
2	1	basmex 12680	. . . 4 |- ( X e. B -> G e. _V )
3		grpinvval.p	. . . . 5 |- .+ = ( +g ` G )
4		grpinvval.o	. . . . 5 |- .0. = ( 0g ` G )
5		grpinvval.n	. . . . 5 |- N = ( invg ` G )
6	1, 3, 4, 5	grpinvfvalg 13117	. . . 4 |- ( G e. _V -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
7	2, 6	syl 14	. . 3 |- ( X e. B -> N = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) )
8	7	fveq1d 5557	. 2 |- ( X e. B -> ( N ` X ) = ( ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) ` X ) )
9		eqid 2193	. . 3 |- ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
10		oveq2 5927	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
11	10	eqeq1d 2202	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
12	11	riotabidv 5876	. . 3 |- ( x = X -> ( iota_ y e. B ( y .+ x ) = .0. ) = ( iota_ y e. B ( y .+ X ) = .0. ) )
13		id 19	. . 3 |- ( X e. B -> X e. B )
14		basfn 12679	. . . . . 6 |- Base Fn _V
15		funfvex 5572	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
16	15	funfni 5355	. . . . . 6 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
17	14, 2, 16	sylancr 414	. . . . 5 |- ( X e. B -> ( Base ` G ) e. _V )
18	1, 17	eqeltrid 2280	. . . 4 |- ( X e. B -> B e. _V )
19		riotaexg 5878	. . . 4 |- ( B e. _V -> ( iota_ y e. B ( y .+ X ) = .0. ) e. _V )
20	18, 19	syl 14	. . 3 |- ( X e. B -> ( iota_ y e. B ( y .+ X ) = .0. ) e. _V )
21	9, 12, 13, 20	fvmptd3 5652	. 2 |- ( X e. B -> ( ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )
22	8, 21	eqtrd 2226	1 |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   _Vcvv 2760   |-> cmpt 4091   Fn wfn 5250   ` cfv 5255   iota_crio 5873  (class class class)co 5919   Basecbs 12621   +g cplusg 12698   0gc0g 12870   invgcminusg 13076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-ndx 12624  df-slot 12625  df-base 12627  df-minusg 13079

This theorem is referenced by:  grplinv  13125  isgrpinv  13129