Theorem hlcmet

Description: The induced metric on a complex Hilbert space is complete. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hlcmet.x	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
	hlcmet.8	⊢ 𝐷 = ( IndMet ‘ 𝑈 )
Assertion	hlcmet	⊢ ( 𝑈 ∈ CHilOLD → 𝐷 ∈ ( CMet ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		hlcmet.x	⊢ 𝑋 = ( BaseSet ‘ 𝑈 )
2		hlcmet.8	⊢ 𝐷 = ( IndMet ‘ 𝑈 )
3		hlobn	⊢ ( 𝑈 ∈ CHilOLD → 𝑈 ∈ CBan )
4	1 2	cbncms	⊢ ( 𝑈 ∈ CBan → 𝐷 ∈ ( CMet ‘ 𝑋 ) )
5	3 4	syl	⊢ ( 𝑈 ∈ CHilOLD → 𝐷 ∈ ( CMet ‘ 𝑋 ) )