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Theorem ishl 19277
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 19251 . 2  |-  CHil  =  (Ban  i^i  CPreHil )
21elin2 3499 1  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    e. wcel 1721   CPreHilccph 19090  Bancbn 19247   CHilchl 19248
This theorem is referenced by:  hlbn  19278  hlcph  19279  ishl2  19285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-in 3295  df-hl 19251
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