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Theorem ishlo 30690
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 30689 . 2 CHilOLD = (CBan ∩ CPreHilOLD)
21elin2 4193 1 (𝑈 ∈ CHilOLD ↔ (𝑈 ∈ CBan ∧ 𝑈 ∈ CPreHilOLD))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  CPreHilOLDccphlo 30615  CBanccbn 30665  CHilOLDchlo 30688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-in 3952  df-hlo 30689
This theorem is referenced by:  hlobn  30691  hlph  30692  cnchl  30719  hhhl  31007
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