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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 28658 | . 2 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ CBan) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 CPreHilOLDccphlo 28583 CBanccbn 28633 CHilOLDchlo 28656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3497 df-in 3943 df-hlo 28657 |
This theorem is referenced by: hlrel 28661 hlnv 28662 hlcmet 28665 htthlem 28688 |
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