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Theorem ishlo 25495
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.  CHilOLD  <->  ( U  e.  CBan  /\  U  e.  CPreHil OLD ) )

Proof of Theorem ishlo
StepHypRef Expression
1 df-hlo 25494 . 2  |-  CHilOLD  =  ( CBan  i^i  CPreHil OLD )
21elin2 3689 1  |-  ( U  e.  CHilOLD  <->  ( U  e.  CBan  /\  U  e.  CPreHil OLD ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1767   CPreHil OLDccphlo 25419   CBanccbn 25470   CHilOLDchlo 25493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-hlo 25494
This theorem is referenced by:  hlobn  25496  hlph  25497  cnchl  25524  ssphl  25525  hhhl  25813  hhsshl  25889
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