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Theorem hlnvi 28997
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 6-Jun-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
hlnvi.1 𝑈 ∈ CHilOLD
Assertion
Ref Expression
hlnvi 𝑈 ∈ NrmCVec

Proof of Theorem hlnvi
StepHypRef Expression
1 hlnvi.1 . 2 𝑈 ∈ CHilOLD
2 hlnv 28996 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2111  NrmCVeccnv 28689  CHilOLDchlo 28990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-ext 2709
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-iota 6355  df-fv 6405  df-cbn 28968  df-hlo 28991
This theorem is referenced by:  htthlem  29022  axhfvadd-zf  29087  axhvcom-zf  29088  axhvass-zf  29089  axhvaddid-zf  29091  axhfvmul-zf  29092  axhvmulid-zf  29093  axhvmulass-zf  29094  axhvdistr1-zf  29095  axhvdistr2-zf  29096  axhvmul0-zf  29097  axhis2-zf  29100  axhis3-zf  29101  axhcompl-zf  29103  hilcompl  29306
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