Theorem grpinvval 13419

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 12935	. . . 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 13418	. . . 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 5585	. 2 |- ( X e. B -> ( N ` X ) = ( ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) ` X ) )
9		eqid 2206	. . 3 |- ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) ) = ( x e. B |-> ( iota_ y e. B ( y .+ x ) = .0. ) )
10		oveq2 5959	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
11	10	eqeq1d 2215	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
12	11	riotabidv 5908	. . 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 12934	. . . . . 6 |- Base Fn _V
15		funfvex 5600	. . . . . . 7 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
16	15	funfni 5381	. . . . . 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 2293	. . . 4 |- ( X e. B -> B e. _V )
19		riotaexg 5910	. . . 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 5680	. 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 2239	1 |- ( X e. B -> ( N ` X ) = ( iota_ y e. B ( y .+ X ) = .0. ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   |-> cmpt 4109   Fn wfn 5271   ` cfv 5276   iota_crio 5905  (class class class)co 5951   Basecbs 12876   +g cplusg 12953   0gc0g 13132   invgcminusg 13377

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-ndx 12879  df-slot 12880  df-base 12882  df-minusg 13380

This theorem is referenced by:  grplinv  13426  isgrpinv  13430