Theorem grpinvex

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

	Ref	Expression
Hypotheses	grpcl.b	|- B = ( Base ` G )
	grpcl.p	|- .+ = ( +g ` G )
	grpinvex.p	|- .0. = ( 0g ` G )
Assertion	grpinvex	|- ( ( G e. Grp /\ X e. B ) -> E. y e. B ( y .+ X ) = .0. )

Proof

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