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 + = ( +g𝑊 )
slmd0vlid.z 0 = ( 0g𝑊 )
Assertion slmd0vrid ( ( 𝑊 ∈ SLMod ∧ 𝑋𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )

Proof

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