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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 = ( 0g ‘ 𝑊 )
Assertion slmd0vcl ( 𝑊 ∈ SLMod → 0 ∈ 𝑉 )

Proof

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