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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 30905 | . 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD) | |
| 2 | 1 | elin2 4203 | 1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 CPreHilOLDccphlo 30831 CBanccbn 30881 CHilOLDchlo 30904 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-in 3958 df-hlo 30905 |
| This theorem is referenced by: hlobn 30907 hlph 30908 cnchl 30935 hhhl 31223 |
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