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Metamath Proof Explorer

Theorem lmod0vrid

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

Ref Expression
Hypotheses 0vlid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
0vlid.a ⊢ + = ( +g ‘ 𝑊 )
0vlid.z ⊢ 0 = ( 0g ‘ 𝑊 )
Assertion lmod0vrid ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 0vlid.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 0vlid.a ⊢ + = ( +g ‘ 𝑊 )
3 0vlid.z ⊢ 0 = ( 0g ‘ 𝑊 )
4 lmodgrp ⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
5 1 2 3 grprid ⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )
6 4 5 sylan ⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 𝑋 + 0 ) = 𝑋 )