Theorem grpinvval 13619

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 13135	. . . 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 13618	. . . 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 5637	. 2 |- ( X e. B -> ( N ` X ) = ( ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) ` X ) )
9		eqid 2229	. . 3 |- ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
10		oveq2 6021	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
11	10	eqeq1d 2238	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
12	11	riotabidv 5968	. . 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 13134	. . . . . 6 |- Base Fn _V
15		funfvex 5652	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
16	15	funfni 5429	. . . . . 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 2316	. . . 4 |- ( X e. B -> B e. _V )
19		riotaexg 5970	. . . 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 5736	. 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 2262	1 |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2800   |-> cmpt 4148   Fn wfn 5319   ` cfv 5324   iota_crio 5965  (class class class)co 6013   Basecbs 13075   +g cplusg 13153   0gc0g 13332   invgcminusg 13577

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8116  ax-resscn 8117  ax-1re 8119  ax-addrcl 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-inn 9137  df-ndx 13078  df-slot 13079  df-base 13081  df-minusg 13580

This theorem is referenced by:  grplinv  13626  isgrpinv  13630