Metamath Proof Explorer


Theorem hlcompl

Description: Completeness of a Hilbert space. (Contributed by NM, 8-Sep-2007) (Revised by Mario Carneiro, 9-May-2014) (New usage is discouraged.)

Ref Expression
Hypotheses hlcompl.1 𝐷 = ( IndMet ‘ 𝑈 )
hlcompl.2 𝐽 = ( MetOpen ‘ 𝐷 )
Assertion hlcompl ( ( 𝑈 ∈ CHilOLD𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡𝐽 ) )

Proof

Step Hyp Ref Expression
1 hlcompl.1 𝐷 = ( IndMet ‘ 𝑈 )
2 hlcompl.2 𝐽 = ( MetOpen ‘ 𝐷 )
3 eqid ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 𝑈 )
4 3 1 hlcmet ( 𝑈 ∈ CHilOLD𝐷 ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) )
5 2 cmetcau ( ( 𝐷 ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡𝐽 ) )
6 4 5 sylan ( ( 𝑈 ∈ CHilOLD𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡𝐽 ) )