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Metamath Proof Explorer

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	$⊢ X = BaseSet (U)$
	iscbn.8	$⊢ D = IndMet (U)$
Assertion	cbncms	$⊢ U \in CBan \to D \in CMet (X)$

Proof

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