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Theorem ishlo 27613
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 27612 . 2 CHilOLD = (CBan ∩ CPreHilOLD)
21elin2 3784 1 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1987  CPreHilOLDccphlo 27537  CBanccbn 27588  CHilOLDchlo 27611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3191  df-in 3566  df-hlo 27612
This theorem is referenced by:  hlobn  27614  hlph  27615  cnchl  27642  ssphl  27643  hhhl  27931  hhsshl  28008
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