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Theorem hlphl 18798
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 18797 . 2  |-  ( W  e.  CHil  ->  W  e.  CPreHil )
2 cphphl 18623 . 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 1696   PreHilcphl 16544   CPreHilccph 18618   CHilchl 18772
This theorem is referenced by:  pjthlem1  18817  pjth  18819  pjth2  18820  cldcss  18821  hlhil  18823
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  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-cph 18620  df-hl 18775
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