Theorem bnsca

Description: The scalar field of a Banach space is complete. (Contributed by NM, 8-Sep-2007) (Revised by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Hypothesis	isbn.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
Assertion	bnsca	$$⊢W\in \mathrm{Ban}\to F\in \mathrm{CMetSp}$$

Proof

Step	Hyp	Ref	Expression
1		isbn.1	$$⊢F=\mathrm{Scalar}\left(W\right)$$
2	1	isbn	$$⊢W\in \mathrm{Ban}\leftrightarrow \left(W\in \mathrm{NrmVec}\wedge W\in \mathrm{CMetSp}\wedge F\in \mathrm{CMetSp}\right)$$
3	2	simp3bi	$$⊢W\in \mathrm{Ban}\to F\in \mathrm{CMetSp}$$