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

Theorem hlnvi 28596
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 ∈ CHilOLD
Assertion
Ref Expression
hlnvi 𝑈 ∈ NrmCVec

Proof of Theorem hlnvi
StepHypRef Expression
1 hlnvi.1 . 2 𝑈 ∈ CHilOLD
2 hlnv 28595 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  NrmCVeccnv 28288  CHilOLDchlo 28589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-cbn 28567  df-hlo 28590
This theorem is referenced by:  htthlem  28621  axhfvadd-zf  28686  axhvcom-zf  28687  axhvass-zf  28688  axhvaddid-zf  28690  axhfvmul-zf  28691  axhvmulid-zf  28692  axhvmulass-zf  28693  axhvdistr1-zf  28694  axhvdistr2-zf  28695  axhvmul0-zf  28696  axhis2-zf  28699  axhis3-zf  28700  axhcompl-zf  28702  hilcompl  28905
  Copyright terms: Public domain W3C validator