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