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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 25189 | . 2 ⊢ ℂHil = (Ban ∩ ℂPreHil) | |
2 | 1 | elin2 4190 | 1 ⊢ (𝑊 ∈ ℂHil ↔ (𝑊 ∈ Ban ∧ 𝑊 ∈ ℂPreHil)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2098 ℂPreHilccph 25018 Bancbn 25185 ℂHilchl 25186 |
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 3948 df-hl 25189 |
This theorem is referenced by: hlbn 25215 hlcph 25216 ishl2 25222 cphssphl 25223 cmslsschl 25229 chlcsschl 25230 |
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