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Theorem ishl 18779
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 18759 . 2  |-  CHil  =  (Ban  i^i  CPreHil )
21elin2 3359 1  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   CPreHilccph 18602  Bancbn 18755   CHilchl 18756
This theorem is referenced by:  hlbn  18780  hlcph  18781  ishl2  18787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-hl 18759
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