Theorem grpinvex 13082

Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpcl.b	|- B = ( Base ` G )
grpcl.p	|- .+ = ( +g ` G )
grpinvex.p	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
grpinvex	|- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )

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

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

Proof of Theorem grpinvex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpcl.b	. . . 4 |- B = ( Base ` G )
2		grpcl.p	. . . 4 |- .+ = ( +g ` G )
3		grpinvex.p	. . . 4 |- .0. = ( 0g ` G )
4	1, 2, 3	isgrp 13078	. . 3 |- ( G e. Grp <-> ( G e. Mnd /\ A. x e. B E. y e. B ( y .+ x ) = .0. ) )
5	4	simprbi 275	. 2 |- ( G e. Grp -> A. x e. B E. y e. B ( y .+ x ) = .0. )
6		oveq2 5926	. . . . 5 |- ( x = X -> ( y .+ x ) = ( y .+ X ) )
7	6	eqeq1d 2202	. . . 4 |- ( x = X -> ( ( y .+ x ) = .0. <-> ( y .+ X ) = .0. ) )
8	7	rexbidv 2495	. . 3 |- ( x = X -> ( E. y e. B ( y .+ x ) = .0. <-> E. y e. B ( y .+ X ) = .0. ) )
9	8	rspccva 2863	. 2 |- ( ( A. x e. B E. y e. B ( y .+ x ) = .0. /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )
10	5, 9	sylan 283	1 |- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   0gc0g 12867   Mndcmnd 12997   Grpcgrp 13072

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-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921  df-grp 13075

This theorem is referenced by:  dfgrp2  13099  grprcan  13109  grpinveu  13110  grprinv  13123