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Mirrors > Home > MPE Home > Th. List > hlbn | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a Banach space. (Contributed by Steve Rodriguez, 28-Apr-2007.) |
Ref | Expression |
---|---|
hlbn | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 25410 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ℂPreHilccph 25214 Bancbn 25381 ℂHilchl 25382 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-in 3970 df-hl 25385 |
This theorem is referenced by: hlcms 25414 hlprlem 25415 cmslsschl 25425 chlcsschl 25426 |
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