Theorem lmod0vid

Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-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	lmod0vid	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑋 + 𝑋 ) = 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	grpid	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑋 + 𝑋 ) = 𝑋 ↔ 0 = 𝑋 ) )
6	4 5	sylan	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ) → ( ( 𝑋 + 𝑋 ) = 𝑋 ↔ 0 = 𝑋 ) )