MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cphnvc Structured version   Unicode version

Theorem cphnvc 19139
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 19135 . 2  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
2 cphlvec 19138 . 2  |-  ( W  e.  CPreHil  ->  W  e.  LVec )
3 isnvc 18730 . 2  |-  ( W  e. NrmVec 
<->  ( W  e. NrmMod  /\  W  e.  LVec ) )
41, 2, 3sylanbrc 646 1  |-  ( W  e.  CPreHil  ->  W  e. NrmVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   LVecclvec 16174  NrmModcnlm 18628  NrmVeccnvc 18629   CPreHilccph 19129
This theorem is referenced by:  ishl2  19324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-ov 6084  df-phl 16857  df-nvc 18635  df-cph 19131
  Copyright terms: Public domain W3C validator