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Theorem hlnvi 30820
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 30819 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
31, 2ax-mp 5 1 𝑈 ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  NrmCVeccnv 30512  CHilOLDchlo 30813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-br 5145  df-iota 6496  df-fv 6552  df-cbn 30791  df-hlo 30814
This theorem is referenced by:  htthlem  30845  axhfvadd-zf  30910  axhvcom-zf  30911  axhvass-zf  30912  axhvaddid-zf  30914  axhfvmul-zf  30915  axhvmulid-zf  30916  axhvmulass-zf  30917  axhvdistr1-zf  30918  axhvdistr2-zf  30919  axhvmul0-zf  30920  axhis2-zf  30923  axhis3-zf  30924  axhcompl-zf  30926  hilcompl  31129
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