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Theorem hlph 22383
Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hlph  |-  ( U  e.  CHil OLD  ->  U  e.  CPreHil
OLD )

Proof of Theorem hlph
StepHypRef Expression
1 ishlo 22381 . 2  |-  ( U  e.  CHil OLD  <->  ( U  e. 
CBan  /\  U  e.  CPreHil OLD ) )
21simprbi 451 1  |-  ( U  e.  CHil OLD  ->  U  e.  CPreHil
OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   CPreHil OLDccphlo 22305   CBanccbn 22356   CHil
OLDchlo 22379
This theorem is referenced by:  hlpar2  22390  hlpar  22391  hlipdir  22406  hlipass  22407  htthlem  22412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-hlo 22380
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