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Theorem cphnvc 22202
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 22198 . 2  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
2 cphlvec 22201 . 2  |-  ( W  e.  CPreHil  ->  W  e.  LVec )
3 isnvc 21745 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
41, 2, 3sylanbrc 675 1  |-  ( W  e.  CPreHil  ->  W  e. NrmVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1897   LVecclvec 18373  NrmModcnlm 21643  NrmVeccnvc 21644   CPreHilccph 22192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-nul 4547
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-xp 4858  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fv 5608  df-ov 6317  df-phl 19241  df-nvc 21650  df-cph 22194
This theorem is referenced by:  ishl2  22385
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