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Metamath Proof Explorer

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 )