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Theorem phnvi 21394
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 21392 . 2  |-  ( U  e.  CPreHil OLD  ->  U  e.  NrmCVec )
31, 2ax-mp 8 1  |-  U  e.  NrmCVec
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   NrmCVeccnv 21140   CPreHil OLDccphlo 21390
This theorem is referenced by:  elimph  21398  ip0i  21403  ip1ilem  21404  ip2i  21406  ipdirilem  21407  ipasslem1  21409  ipasslem2  21410  ipasslem4  21412  ipasslem5  21413  ipasslem7  21414  ipasslem8  21415  ipasslem9  21416  ipasslem10  21417  ipasslem11  21418  ip2dii  21422  pythi  21428  siilem1  21429  siilem2  21430  siii  21431  ipblnfi  21434  ip2eqi  21435  ajfuni  21438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-ph 21391
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