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 = ( Base ‘ 𝑊 )
	0vcl.z	⊢ 0 = ( 0g ‘ 𝑊 )
Assertion	lmod0vcl	⊢ ( 𝑊 ∈ LMod → 0 ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		0vcl.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		0vcl.z	⊢ 0 = ( 0g ‘ 𝑊 )
3		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
4	1 2	grpidcl	⊢ ( 𝑊 ∈ Grp → 0 ∈ 𝑉 )
5	3 4	syl	⊢ ( 𝑊 ∈ LMod → 0 ∈ 𝑉 )