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 ∈ CBan → 𝐷 ∈ ( CMet ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		iscbn.x	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		iscbn.8	⊢ 𝐷 = ( IndMet ‘ 𝑈 )
3	1 2	iscbn	⊢ ( 𝑈 ∈ CBan ↔ ( 𝑈 ∈ NrmCVec ∧ 𝐷 ∈ ( CMet ‘ 𝑋 ) ) )
4	3	simprbi	⊢ ( 𝑈 ∈ CBan → 𝐷 ∈ ( CMet ‘ 𝑋 ) )