Theorem grpinvval 12959

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 12545	. . . 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 12958	. . . 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 5532	. 2 |- ( X e. B -> ( N ` X ) = ( ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) ` X ) )
9		eqid 2189	. . 3 |- ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
10		oveq2 5899	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
11	10	eqeq1d 2198	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
12	11	riotabidv 5849	. . 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 12544	. . . . . 6 |- Base Fn _V
15		funfvex 5547	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
16	15	funfni 5331	. . . . . 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 2276	. . . 4 |- ( X e. B -> B e. _V )
19		riotaexg 5851	. . . 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 5625	. 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 2222	1 |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   _Vcvv 2752   |-> cmpt 4079   Fn wfn 5226   ` cfv 5231   iota_crio 5846  (class class class)co 5891   Basecbs 12486   +g cplusg 12561   0gc0g 12733   invgcminusg 12918

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-cnex 7921  ax-resscn 7922  ax-1re 7924  ax-addrcl 7927

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-inn 8939  df-ndx 12489  df-slot 12490  df-base 12492  df-minusg 12921

This theorem is referenced by:  grplinv  12966  isgrpinv  12970