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Mirrors > Home > MPE Home > Th. List > hlcph | Structured version Visualization version GIF version |
Description: Every subcomplex Hilbert space is a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlcph | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishl 25212 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | 1 | simprbi 496 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ℂPreHilccph 25016 Bancbn 25183 ℂHilchl 25184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3947 df-hl 25187 |
This theorem is referenced by: hlphl 25215 hlprlem 25217 cmslsschl 25227 chlcsschl 25228 pjthlem1 25287 pjthlem2 25288 cldcss 25291 |
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