Theorem hlbn

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

	Ref	Expression
Assertion	hlbn	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ Ban )

Proof

Step	Hyp	Ref	Expression
1		ishl	⊢ ( 𝑊 ∈ ℂHil ↔ ( 𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil ) )
2	1	simplbi	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ Ban )