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Theorem ishlo 28083
 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.)
Assertion
Ref Expression
ishlo (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))

Proof of Theorem ishlo
StepHypRef Expression
1 df-hlo 28082 . 2 CHilOLD = (CBan ∩ CPreHilOLD)
21elin2 3952 1 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 382   ∈ wcel 2145  CPreHilOLDccphlo 28007  CBanccbn 28058  CHilOLDchlo 28081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-hlo 28082 This theorem is referenced by:  hlobn  28084  hlph  28085  cnchl  28112  ssphl  28113  hhhl  28401  hhsshl  28478
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