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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 28297 | . 2 ⊢ CHilOLD = (CBan ∩ CPreHilOLD) | |
2 | 1 | elin2 4028 | 1 ⊢ (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 ∈ wcel 2166 CPreHilOLDccphlo 28222 CBanccbn 28273 CHilOLDchlo 28296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-in 3805 df-hlo 28297 |
This theorem is referenced by: hlobn 28299 hlph 28300 cnchl 28327 ssphlOLD 28328 hhhl 28616 hhsshlOLD 28693 |
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