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Theorem cphnvc 18612
Description: A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
Assertion
Ref Expression
cphnvc  |-  ( W  e.  CPreHil  ->  W  e. NrmVec )

Proof of Theorem cphnvc
StepHypRef Expression
1 cphnlm 18608 . 2  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
2 cphlvec 18611 . 2  |-  ( W  e.  CPreHil  ->  W  e.  LVec )
3 isnvc 18205 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
41, 2, 3sylanbrc 645 1  |-  ( W  e.  CPreHil  ->  W  e. NrmVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   LVecclvec 15855  NrmModcnlm 18103  NrmVeccnvc 18104   CPreHilccph 18602
This theorem is referenced by:  ishl2  18787
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-phl 16530  df-nvc 18110  df-cph 18604
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