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Theorem hlcph 19186
Description: Every complex Hilbert space is a complex pre-Hilbert space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
hlcph  |-  ( W  e.  CHil  ->  W  e.  CPreHil )

Proof of Theorem hlcph
StepHypRef Expression
1 ishl 19184 . 2  |-  ( W  e.  CHil  <->  ( W  e. Ban  /\  W  e.  CPreHil ) )
21simprbi 451 1  |-  ( W  e.  CHil  ->  W  e.  CPreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   CPreHilccph 19001  Bancbn 19156   CHilchl 19157
This theorem is referenced by:  hlphl  19187  hlprlem  19189  pjthlem1  19206  pjthlem2  19207  cldcss  19210
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 2369
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 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-v 2902  df-in 3271  df-hl 19160
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