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Theorem hlnvi 28273
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 28272 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  NrmCVeccnv 27964  CHilOLDchlo 28266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-cbn 28244  df-hlo 28267
This theorem is referenced by:  htthlem  28299  axhfvadd-zf  28364  axhvcom-zf  28365  axhvass-zf  28366  axhvaddid-zf  28368  axhfvmul-zf  28369  axhvmulid-zf  28370  axhvmulass-zf  28371  axhvdistr1-zf  28372  axhvdistr2-zf  28373  axhvmul0-zf  28374  axhis2-zf  28377  axhis3-zf  28378  axhcompl-zf  28380  hilcompl  28583
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