Theorem isnvc

Description: A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015)

	Ref	Expression
Assertion	isnvc	⊢ ( 𝑊 ∈ NrmVec ↔ ( 𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec ) )

Proof

Step	Hyp	Ref	Expression
1		df-nvc	⊢ NrmVec = ( NrmMod ∩ LVec )
2	1	elin2	⊢ ( 𝑊 ∈ NrmVec ↔ ( 𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec ) )