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Theorem hlnvi 28661
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 28660 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  NrmCVeccnv 28353  CHilOLDchlo 28654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-cbn 28632  df-hlo 28655
This theorem is referenced by:  htthlem  28686  axhfvadd-zf  28751  axhvcom-zf  28752  axhvass-zf  28753  axhvaddid-zf  28755  axhfvmul-zf  28756  axhvmulid-zf  28757  axhvmulass-zf  28758  axhvdistr1-zf  28759  axhvdistr2-zf  28760  axhvmul0-zf  28761  axhis2-zf  28764  axhis3-zf  28765  axhcompl-zf  28767  hilcompl  28970
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