Theorem lmod0vid

Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014) (Revised by Mario Carneiro, 19-Jun-2014)

	Ref	Expression
Hypotheses	0vlid.v	|- V = ( Base ` W )
	0vlid.a	|- .+ = ( +g ` W )
	0vlid.z	|- .0. = ( 0g ` W )
Assertion	lmod0vid	|- ( ( W e. LMod /\ X e. V ) -> ( ( X .+ X ) = X <-> .0. = X ) )

Proof

Step	Hyp	Ref	Expression
1		0vlid.v	|- V = ( Base ` W )
2		0vlid.a	|- .+ = ( +g ` W )
3		0vlid.z	|- .0. = ( 0g ` W )
4		lmodgrp	|- ( W e. LMod -> W e. Grp )
5	1 2 3	grpid	|- ( ( W e. Grp /\ X e. V ) -> ( ( X .+ X ) = X <-> .0. = X ) )
6	4 5	sylan	|- ( ( W e. LMod /\ X e. V ) -> ( ( X .+ X ) = X <-> .0. = X ) )