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Theorem phnvi 28587
 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 28585 . 2 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2110  NrmCVeccnv 28355  CPreHilOLDccphlo 28583 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3942  df-ss 3951  df-ph 28584 This theorem is referenced by:  elimph  28591  ip0i  28596  ip1ilem  28597  ip2i  28599  ipdirilem  28600  ipasslem1  28602  ipasslem2  28603  ipasslem4  28605  ipasslem5  28606  ipasslem7  28607  ipasslem8  28608  ipasslem9  28609  ipasslem10  28610  ipasslem11  28611  ip2dii  28615  pythi  28621  siilem1  28622  siilem2  28623  siii  28624  ipblnfi  28626  ip2eqi  28627  ajfuni  28630
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