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Theorem hlnv 28595
Description: Every complex Hilbert space is a normed complex vector space. (Contributed by NM, 17-Mar-2007.) (New usage is discouraged.)
Assertion
Ref Expression
hlnv (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)

Proof of Theorem hlnv
StepHypRef Expression
1 hlobn 28592 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
2 bnnv 28570 . 2 (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
31, 2syl 17 1 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  NrmCVeccnv 28288  CBanccbn 28566  CHilOLDchlo 28589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-cbn 28567  df-hlo 28590
This theorem is referenced by:  hlnvi  28596  hlvc  28597  hladdf  28603  hlcom  28604  hlass  28605  hl0cl  28606  hladdid  28607  hlmulf  28608  hlmulid  28609  hlmulass  28610  hldi  28611  hldir  28612  hlmul0  28613  hlipf  28614  hlipcj  28615  hlipgt0  28618
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