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| Mirrors > Home > MPE Home > Th. List > ishl | Structured version Visualization version GIF version | ||
| Description: The predicate "is a subcomplex Hilbert space". A Hilbert space is a Banach space which is also an inner product space, i.e. whose norm satisfies the parallelogram law. (Contributed by Steve Rodriguez, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| ishl | ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-hl 25371 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
| 2 | 1 | elin2 4203 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ℂPreHilccph 25200 Bancbn 25367 ℂHilchl 25368 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-hl 25371 | 
| This theorem is referenced by: hlbn 25397 hlcph 25398 ishl2 25404 cphssphl 25405 cmslsschl 25411 chlcsschl 25412 | 
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