Metamath Proof Explorer

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