Theorem bncms

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

	Ref	Expression
Assertion	bncms	$$⊢W\in \mathrm{Ban}\to W\in \mathrm{CMetSp}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\mathrm{Scalar}\left(W\right)=\mathrm{Scalar}\left(W\right)$$
2	1	isbn	$$⊢W\in \mathrm{Ban}\leftrightarrow \left(W\in \mathrm{NrmVec}\wedge W\in \mathrm{CMetSp}\wedge \mathrm{Scalar}\left(W\right)\in \mathrm{CMetSp}\right)$$
3	2	simp2bi	$$⊢W\in \mathrm{Ban}\to W\in \mathrm{CMetSp}$$