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Theorem phnvi 24151
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  e.  CPreHil
OLD
Assertion
Ref Expression
phnvi  |-  U  e.  NrmCVec

Proof of Theorem phnvi
StepHypRef Expression
1 phnvi.1 . 2  |-  U  e.  CPreHil
OLD
2 phnv 24149 . 2  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
31, 2ax-mp 5 1  |-  U  e.  NrmCVec
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1761   NrmCVeccnv 23897   CPreHil OLDccphlo 24147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-in 3332  df-ss 3339  df-ph 24148
This theorem is referenced by:  elimph  24155  ip0i  24160  ip1ilem  24161  ip2i  24163  ipdirilem  24164  ipasslem1  24166  ipasslem2  24167  ipasslem4  24169  ipasslem5  24170  ipasslem7  24171  ipasslem8  24172  ipasslem9  24173  ipasslem10  24174  ipasslem11  24175  ip2dii  24179  pythi  24185  siilem1  24186  siilem2  24187  siii  24188  ipblnfi  24191  ip2eqi  24192  ajfuni  24195
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