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	$$⊢V={\mathrm{Base}}_W$$
	slmd0vcl.z	$$⊢\underset{\dot{}}{0}=0_W$$
Assertion	slmd0vcl	$$⊢W\in \mathrm{SLMod}\to \underset{\dot{}}{0}\in V$$

Proof

Step	Hyp	Ref	Expression
1		slmd0vcl.v	$$⊢V={\mathrm{Base}}_W$$
2		slmd0vcl.z	$$⊢\underset{\dot{}}{0}=0_W$$
3		slmdmnd	$$⊢W\in \mathrm{SLMod}\to W\in \mathrm{Mnd}$$
4	1 2	mndidcl	$$⊢W\in \mathrm{Mnd}\to \underset{\dot{}}{0}\in V$$
5	3 4	syl	$$⊢W\in \mathrm{SLMod}\to \underset{\dot{}}{0}\in V$$