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Theorem hlph 21484
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 21482 . 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 1696   CPreHil OLDccphlo 21406   CBanccbn 21457   CHil
OLDchlo 21480
This theorem is referenced by:  hlpar2  21491  hlpar  21492  hlipdir  21507  hlipass  21508  htthlem  21513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-hlo 21481
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