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Theorem hlph 21468
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 21466 . 2  |-  ( U  e.  CHil OLD  <->  ( U  e. 
CBan  /\  U  e.  CPreHil OLD ) )
21simprbi 450 1  |-  ( U  e.  CHil OLD  ->  U  e.  CPreHil
OLD )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   CPreHil OLDccphlo 21390   CBanccbn 21441   CHil
OLDchlo 21464
This theorem is referenced by:  hlpar2  21475  hlpar  21476  hlipdir  21491  hlipass  21492  htthlem  21497
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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