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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 23648 | . 2 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | |
2 | 1 | simprbi 497 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2081 ℂPreHilccph 23453 Bancbn 23619 ℂHilchl 23620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-v 3439 df-in 3866 df-hl 23623 |
This theorem is referenced by: hlphl 23651 hlprlem 23653 cmslsschl 23663 chlcsschl 23664 pjthlem1 23723 pjthlem2 23724 cldcss 23727 |
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