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Metamath Proof Explorer

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 = Base W
slmd0vcl.z ⊢ 0 ˙ = 0 W
Assertion slmd0vcl ⊢ W ∈ SLMod → 0 ˙ ∈ V

Proof

Step Hyp Ref Expression
1 slmd0vcl.v ⊢ V = Base W
2 slmd0vcl.z ⊢ 0 ˙ = 0 W
3 slmdmnd ⊢ W ∈ SLMod → W ∈ Mnd
4 1 2 mndidcl ⊢ W ∈ Mnd → 0 ˙ ∈ V
5 3 4 syl ⊢ W ∈ SLMod → 0 ˙ ∈ V