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 ∈ CHilOLD ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) ) |
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 ∈ CHilOLD → 𝐷 ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) ) |
5 | 2 | cmetcau | ⊢ ( ( 𝐷 ∈ ( CMet ‘ ( BaseSet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) ) |
6 | 4 5 | sylan | ⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) ) |