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Theorem phnv 27539
Description: Every complex inner product space is a normed complex vector space. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
phnv (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)

Proof of Theorem phnv
Dummy variables 𝑔 𝑛 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ph 27538 . . 3 CPreHilOLD = (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))})
2 inss1 3816 . . 3 (NrmCVec ∩ {⟨⟨𝑔, 𝑠⟩, 𝑛⟩ ∣ ∀𝑥 ∈ ran 𝑔𝑦 ∈ ran 𝑔(((𝑛‘(𝑥𝑔𝑦))↑2) + ((𝑛‘(𝑥𝑔(-1𝑠𝑦)))↑2)) = (2 · (((𝑛𝑥)↑2) + ((𝑛𝑦)↑2)))}) ⊆ NrmCVec
31, 2eqsstri 3619 . 2 CPreHilOLD ⊆ NrmCVec
43sseli 3583 1 (𝑈 ∈ CPreHilOLD𝑈 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  cin 3558  ran crn 5080  cfv 5852  (class class class)co 6610  {coprab 6611  1c1 9889   + caddc 9891   · cmul 9893  -cneg 10219  2c2 11022  cexp 12808  NrmCVeccnv 27309  CPreHilOLDccphlo 27537
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 3191  df-in 3566  df-ss 3573  df-ph 27538
This theorem is referenced by:  phrel  27540  phnvi  27541  phop  27543  isph  27547  dipdi  27568  dipassr  27571  dipsubdir  27573  dipsubdi  27574  sspph  27580  ajval  27587  minvecolem1  27600  minvecolem2  27601  minvecolem3  27602  minvecolem4a  27603  minvecolem4b  27604  minvecolem4  27606  minvecolem5  27607  minvecolem6  27608  minvecolem7  27609
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