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Theorem phnvi 27559
 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 27557 . 2 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 1987  NrmCVeccnv 27327  CPreHilOLDccphlo 27555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3192  df-in 3567  df-ss 3574  df-ph 27556 This theorem is referenced by:  elimph  27563  ip0i  27568  ip1ilem  27569  ip2i  27571  ipdirilem  27572  ipasslem1  27574  ipasslem2  27575  ipasslem4  27577  ipasslem5  27578  ipasslem7  27579  ipasslem8  27580  ipasslem9  27581  ipasslem10  27582  ipasslem11  27583  ip2dii  27587  pythi  27593  siilem1  27594  siilem2  27595  siii  27596  ipblnfi  27599  ip2eqi  27600  ajfuni  27603
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