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Mirrors > Home > MPE Home > Th. List > hlobn | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is a complex Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlobn | ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishlo 30140 | . 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | |
2 | 1 | simplbi 499 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 CPreHilOLDccphlo 30065 CBanccbn 30115 CHilOLDchlo 30138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3956 df-hlo 30139 |
This theorem is referenced by: hlrel 30143 hlnv 30144 hlcmet 30147 htthlem 30170 |
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