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

Theorem hlnvi 26600
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlnvi.1  |-  U  e. 
CHilOLD
Assertion
Ref Expression
hlnvi  |-  U  e.  NrmCVec

Proof of Theorem hlnvi
StepHypRef Expression
1 hlnvi.1 . 2  |-  U  e. 
CHilOLD
2 hlnv 26599 . 2  |-  ( U  e.  CHilOLD  ->  U  e.  NrmCVec )
31, 2ax-mp 5 1  |-  U  e.  NrmCVec
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1898   NrmCVeccnv 26259   CHilOLDchlo 26593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-iota 5569  df-fv 5613  df-cbn 26561  df-hlo 26594
This theorem is referenced by:  htthlem  26626  axhfvadd-zf  26691  axhvcom-zf  26692  axhvass-zf  26693  axhvaddid-zf  26695  axhfvmul-zf  26696  axhvmulid-zf  26697  axhvmulass-zf  26698  axhvdistr1-zf  26699  axhvdistr2-zf  26700  axhvmul0-zf  26701  axhis2-zf  26704  axhis3-zf  26705  axhcompl-zf  26707  hilcompl  26910
  Copyright terms: Public domain W3C validator