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Theorem ishlo 27127
 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 27126 . 2 CHilOLD = (CBan ∩ CPreHilOLD)
21elin2 3763 1 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  CPreHilOLDccphlo 27051  CBanccbn 27102  CHilOLDchlo 27125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-hlo 27126 This theorem is referenced by:  hlobn  27128  hlph  27129  cnchl  27156  ssphl  27157  hhhl  27445  hhsshl  27522
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