Theorem hlobn

Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007) (New usage is discouraged.)

	Ref	Expression
Assertion	hlobn	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CBan )

Proof

Step	Hyp	Ref	Expression
1		ishlo	⊢ ( 𝑈 ∈ CHilOLD ↔ ( 𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD ) )
2	1	simplbi	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CBan )