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Mirrors > Home > MPE Home > Th. List > ishlo | Structured version Visualization version GIF version |
Description: The predicate "is a complex 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.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ishlo | ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hlo 27126 | . 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD) | |
2 | 1 | elin2 3763 | 1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 CPreHilOLDccphlo 27051 CBanccbn 27102 CHilOLDchlo 27125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-in 3547 df-hlo 27126 |
This theorem is referenced by: hlobn 27128 hlph 27129 cnchl 27156 ssphl 27157 hhhl 27445 hhsshl 27522 |
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