Theorem lmod0vcl

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

	Ref	Expression
Hypotheses	0vcl.v	$$⊢V={\mathrm{Base}}_W$$
	0vcl.z	$$⊢\underset{\dot{}}{0}=0_W$$
Assertion	lmod0vcl	$$⊢W\in \mathrm{LMod}\to \underset{\dot{}}{0}\in V$$

Proof

Step	Hyp	Ref	Expression
1		0vcl.v	$$⊢V={\mathrm{Base}}_W$$
2		0vcl.z	$$⊢\underset{\dot{}}{0}=0_W$$
3		lmodgrp	$$⊢W\in \mathrm{LMod}\to W\in \mathrm{Grp}$$
4	1 2	grpidcl	$$⊢W\in \mathrm{Grp}\to \underset{\dot{}}{0}\in V$$
5	3 4	syl	$$⊢W\in \mathrm{LMod}\to \underset{\dot{}}{0}\in V$$