Theorem bncms

Description: A Banach space is a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Assertion	bncms	⊢ ( 𝑊 ∈ Ban → 𝑊 ∈ CMetSp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
2	1	isbn	⊢ ( 𝑊 ∈ Ban ↔ ( 𝑊 ∈ NrmVec ∧ 𝑊 ∈ CMetSp ∧ ( Scalar ‘ 𝑊 ) ∈ CMetSp ) )
3	2	simp2bi	⊢ ( 𝑊 ∈ Ban → 𝑊 ∈ CMetSp )