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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 ˙ = 0 W
Assertion lmod0vcl ⊢ W ∈ LMod → 0 ˙ ∈ V

Proof

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