Theorem grpsubid

Description: Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014)

	Ref	Expression
Hypotheses	grpsubid.b	|- B = ( Base ` G )
	grpsubid.o	|- .0. = ( 0g ` G )
	grpsubid.m	|- .- = ( -g ` G )
Assertion	grpsubid	|- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		grpsubid.b	|- B = ( Base ` G )
2		grpsubid.o	|- .0. = ( 0g ` G )
3		grpsubid.m	|- .- = ( -g ` G )
4		eqid	|- ( +g ` G ) = ( +g ` G )
5		eqid	|- ( invg ` G ) = ( invg ` G )
6	1 4 5 3	grpsubval	|- ( ( X e. B /\ X e. B ) -> ( X .- X ) = ( X ( +g ` G ) ( ( invg ` G ) ` X ) ) )
7	6	anidms	|- ( X e. B -> ( X .- X ) = ( X ( +g ` G ) ( ( invg ` G ) ` X ) ) )
8	7	adantl	|- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = ( X ( +g ` G ) ( ( invg ` G ) ` X ) ) )
9	1 4 2 5	grprinv	|- ( ( G e. Grp /\ X e. B ) -> ( X ( +g ` G ) ( ( invg ` G ) ` X ) ) = .0. )
10	8 9	eqtrd	|- ( ( G e. Grp /\ X e. B ) -> ( X .- X ) = .0. )