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Theorem hlphl 18782
Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
hlphl  |-  ( W  e.  CHil  ->  W  e. 
PreHil )

Proof of Theorem hlphl
StepHypRef Expression
1 hlcph 18781 . 2  |-  ( W  e.  CHil  ->  W  e.  CPreHil )
2 cphphl 18607 . 2  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
31, 2syl 15 1  |-  ( W  e.  CHil  ->  W  e. 
PreHil )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   PreHilcphl 16528   CPreHilccph 18602   CHilchl 18756
This theorem is referenced by:  pjthlem1  18801  pjth  18803  pjth2  18804  cldcss  18805  hlhil  18807
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  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-cph 18604  df-hl 18759
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