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

Theorem phnvi 28011
Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
phnvi.1 𝑈 ∈ CPreHilOLD
Assertion
Ref Expression
phnvi 𝑈 ∈ NrmCVec

Proof of Theorem phnvi
StepHypRef Expression
1 phnvi.1 . 2 𝑈 ∈ CPreHilOLD
2 phnv 28009 . 2 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2145  NrmCVeccnv 27779  CPreHilOLDccphlo 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-ph 28008
This theorem is referenced by:  elimph  28015  ip0i  28020  ip1ilem  28021  ip2i  28023  ipdirilem  28024  ipasslem1  28026  ipasslem2  28027  ipasslem4  28029  ipasslem5  28030  ipasslem7  28031  ipasslem8  28032  ipasslem9  28033  ipasslem10  28034  ipasslem11  28035  ip2dii  28039  pythi  28045  siilem1  28046  siilem2  28047  siii  28048  ipblnfi  28051  ip2eqi  28052  ajfuni  28055
  Copyright terms: Public domain W3C validator