Theorem lmodsubid

Description: Subtraction of a vector from itself. ( hvsubid analog.) (Contributed by NM, 16-Apr-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	lmodsubeq0.v	|- V = ( Base ` W )
	lmodsubeq0.o	|- .0. = ( 0g ` W )
	lmodsubeq0.m	|- .- = ( -g ` W )
Assertion	lmodsubid	|- ( ( W e. LMod /\ A e. V ) -> ( A .- A ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		lmodsubeq0.v	|- V = ( Base ` W )
2		lmodsubeq0.o	|- .0. = ( 0g ` W )
3		lmodsubeq0.m	|- .- = ( -g ` W )
4		lmodgrp	|- ( W e. LMod -> W e. Grp )
5	1 2 3	grpsubid	|- ( ( W e. Grp /\ A e. V ) -> ( A .- A ) = .0. )
6	4 5	sylan	|- ( ( W e. LMod /\ A e. V ) -> ( A .- A ) = .0. )