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Theorem hlph 22232
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 22230 . 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 1717   CPreHil OLDccphlo 22154   CBanccbn 22205   CHil
OLDchlo 22228
This theorem is referenced by:  hlpar2  22239  hlpar  22240  hlipdir  22255  hlipass  22256  htthlem  22261
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-in 3263  df-hlo 22229
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