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Theorem ishlo 21466
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  e.  CHil OLD  <->  ( U  e. 
CBan  /\  U  e.  CPreHil OLD ) )

Proof of Theorem ishlo
StepHypRef Expression
1 df-hlo 21465 . 2  |-  CHil OLD  =  ( CBan  i^i  CPreHil OLD )
21elin2 3359 1  |-  ( U  e.  CHil OLD  <->  ( U  e. 
CBan  /\  U  e.  CPreHil OLD ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    e. wcel 1684   CPreHil OLDccphlo 21390   CBanccbn 21441   CHil
OLDchlo 21464
This theorem is referenced by:  hlobn  21467  hlph  21468  cnchl  21495  ssphl  21496  hhhl  21783  hhsshl  21858
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-hlo 21465
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