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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 30689 | . 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD) | |
2 | 1 | elin2 4193 | 1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∈ wcel 2099 CPreHilOLDccphlo 30615 CBanccbn 30665 CHilOLDchlo 30688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-in 3952 df-hlo 30689 |
This theorem is referenced by: hlobn 30691 hlph 30692 cnchl 30719 hhhl 31007 |
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