Theorem bnnv

Description: Every complex Banach space is a normed complex vector space. (Contributed by NM, 17-Mar-2007) Use bnnvc instead. (New usage is discouraged.)

	Ref	Expression
Assertion	bnnv	⊢ ( 𝑈 ∈ CBan → 𝑈 ∈ NrmCVec )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 )
2		eqid	⊢ ( IndMet ‘ 𝑈 ) = ( IndMet ‘ 𝑈 )
3	1 2	iscbn	⊢ ( 𝑈 ∈ CBan ↔ ( 𝑈 ∈ NrmCVec ∧ ( IndMet ‘ 𝑈 ) ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) ) )
4	3	simplbi	⊢ ( 𝑈 ∈ CBan → 𝑈 ∈ NrmCVec )