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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 23966 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ Ban) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ℂPreHilccph 23771 Bancbn 23937 ℂHilchl 23938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-hl 23941 |
This theorem is referenced by: hlcms 23970 hlprlem 23971 cmslsschl 23981 chlcsschl 23982 |
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