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Metamath Proof Explorer

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 ) )