Theorem lmod0vlid

Description: Left identity law for the zero vector. ( hvaddid2 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	lmod0vlid	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 0 + 𝑋 ) = 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	grplid	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( 0 + 𝑋 ) = 𝑋 )
6	4 5	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( 0 + 𝑋 ) = 𝑋 )