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Metamath Proof Explorer

Theorem slmd0vrid

Description: Right identity law for the zero vector. ( ax-hvaddid analog.) (Contributed by NM, 10-Jan-2014) (Revised by Mario Carneiro, 19-Jun-2014) (Revised by Thierry Arnoux, 1-Apr-2018)

Ref Expression
Hypotheses slmd0vlid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
slmd0vlid.a ⊢ + = ( +g ‘ 𝑊 )
slmd0vlid.z ⊢ 0 = ( 0g ‘ 𝑊 )
Assertion slmd0vrid ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 slmd0vlid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 slmd0vlid.a ⊢ + = ( +g ‘ 𝑊 )
3 slmd0vlid.z ⊢ 0 = ( 0g ‘ 𝑊 )
4 slmdmnd ⊢ ( 𝑊 ∈ SLMod → 𝑊 ∈ Mnd )
5 1 2 3 mndrid ⊢ ( ( 𝑊 ∈ Mnd ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )
6 4 5 sylan ⊢ ( ( 𝑊 ∈ SLMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )