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Theorem phnvi 26443
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 26441 . 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 1868   NrmCVeccnv 26189   CPreHil OLDccphlo 26439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-in 3443  df-ss 3450  df-ph 26440
This theorem is referenced by:  elimph  26447  ip0i  26452  ip1ilem  26453  ip2i  26455  ipdirilem  26456  ipasslem1  26458  ipasslem2  26459  ipasslem4  26461  ipasslem5  26462  ipasslem7  26463  ipasslem8  26464  ipasslem9  26465  ipasslem10  26466  ipasslem11  26467  ip2dii  26471  pythi  26477  siilem1  26478  siilem2  26479  siii  26480  ipblnfi  26483  ip2eqi  26484  ajfuni  26487
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