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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 30633 | . 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD) | |
2 | 1 | elin2 4190 | 1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2098 CPreHilOLDccphlo 30559 CBanccbn 30609 CHilOLDchlo 30632 |
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-hlo 30633 |
This theorem is referenced by: hlobn 30635 hlph 30636 cnchl 30663 hhhl 30951 |
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