Theorem slmd0vlid

Description: Left identity law for the zero vector. ( hvaddlid analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmd0vlid.v	|- V = ( Base ` W )
	slmd0vlid.a	|- .+ = ( +g ` W )
	slmd0vlid.z	|- .0. = ( 0g ` W )
Assertion	slmd0vlid	|- ( ( W e. SLMod /\ X e. V ) -> ( .0. .+ X ) = X )

Proof

Step	Hyp	Ref	Expression
1		slmd0vlid.v	|- V = ( Base ` W )
2		slmd0vlid.a	|- .+ = ( +g ` W )
3		slmd0vlid.z	|- .0. = ( 0g ` W )
4		slmdmnd	|- ( W e. SLMod -> W e. Mnd )
5	1 2 3	mndlid	|- ( ( W e. Mnd /\ X e. V ) -> ( .0. .+ X ) = X )
6	4 5	sylan	|- ( ( W e. SLMod /\ X e. V ) -> ( .0. .+ X ) = X )