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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 24701 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
2 | 1 | elin2 4157 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ℂPreHilccph 24530 Bancbn 24697 ℂHilchl 24698 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-in 3917 df-hl 24701 |
This theorem is referenced by: hlbn 24727 hlcph 24728 ishl2 24734 cphssphl 24735 cmslsschl 24741 chlcsschl 24742 |
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