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Theorem ishlo 26004
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 26003 . 2  |-  CHilOLD  =  ( CBan  i^i  CPreHil OLD )
21elin2 3675 1  |-  ( U  e.  CHilOLD  <->  ( U  e.  CBan  /\  U  e.  CPreHil OLD ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    e. wcel 1823   CPreHil OLDccphlo 25928   CBanccbn 25979   CHilOLDchlo 26002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-in 3468  df-hlo 26003
This theorem is referenced by:  hlobn  26005  hlph  26006  cnchl  26033  ssphl  26034  hhhl  26322  hhsshl  26398
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