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Theorem ishl 19347
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.) (Revised by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
ishl  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )

Proof of Theorem ishl
StepHypRef Expression
1 df-hl 19321 . 2  |-  CHil  =  (Ban  i^i  CPreHil )
21elin2 3517 1  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1727   CPreHilccph 19160  Bancbn 19317   CHilchl 19318
This theorem is referenced by:  hlbn  19348  hlcph  19349  ishl2  19355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-in 3313  df-hl 19321
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