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Theorem cphnvc 18665
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 18661 . 2  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
2 cphlvec 18664 . 2  |-  ( W  e.  CPreHil  ->  W  e.  LVec )
3 isnvc 18257 . 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 1701   LVecclvec 15904  NrmModcnlm 18155  NrmVeccnvc 18156   CPreHilccph 18655
This theorem is referenced by:  ishl2  18840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-xp 4732  df-cnv 4734  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fv 5300  df-ov 5903  df-phl 16586  df-nvc 18162  df-cph 18657
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