Theorem cbncms

Description: The induced metric on complex Banach space is complete. (Contributed by NM, 8-Sep-2007) Use bncmet (or preferably bncms ) instead. (New usage is discouraged.)

	Ref	Expression
Hypotheses	iscbn.x	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
	iscbn.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
Assertion	cbncms	$$⊢U\in \mathrm{CBan}\to D\in \mathrm{CMet}\left(X\right)$$

Proof

Step	Hyp	Ref	Expression
1		iscbn.x	$$⊢X=\mathrm{BaseSet}\left(U\right)$$
2		iscbn.8	$$⊢D=\mathrm{IndMet}\left(U\right)$$
3	1 2	iscbn	$$⊢U\in \mathrm{CBan}\leftrightarrow \left(U\in \mathrm{NrmCVec}\wedge D\in \mathrm{CMet}\left(X\right)\right)$$
4	3	simprbi	$$⊢U\in \mathrm{CBan}\to D\in \mathrm{CMet}\left(X\right)$$