Theorem hlcms

Description: Every subcomplex Hilbert space is a complete metric space. (Contributed by Mario Carneiro, 17-Oct-2015)

	Ref	Expression
Assertion	hlcms	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp )

Proof

Step	Hyp	Ref	Expression
1		hlbn	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ Ban )
2		bncms	⊢ ( 𝑊 ∈ Ban → 𝑊 ∈ CMetSp )
3	1 2	syl	⊢ ( 𝑊 ∈ ℂHil → 𝑊 ∈ CMetSp )