Theorem slmd0vcl

Description: The zero vector is a vector. ( ax-hv0cl analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

	Ref	Expression
Hypotheses	slmd0vcl.v	|- V = ( Base ` W )
	slmd0vcl.z	|- .0. = ( 0g ` W )
Assertion	slmd0vcl	|- ( W e. SLMod -> .0. e. V )

Proof

Step	Hyp	Ref	Expression
1		slmd0vcl.v	|- V = ( Base ` W )
2		slmd0vcl.z	|- .0. = ( 0g ` W )
3		slmdmnd	|- ( W e. SLMod -> W e. Mnd )
4	1 2	mndidcl	|- ( W e. Mnd -> .0. e. V )
5	3 4	syl	|- ( W e. SLMod -> .0. e. V )