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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 25385 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
2 | 1 | elin2 4213 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∈ 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: hlbn 25411 hlcph 25412 ishl2 25418 cphssphl 25419 cmslsschl 25425 chlcsschl 25426 |
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