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Theorem hlnv 29875
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 29872 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
2 bnnv 29850 . 2 (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
31, 2syl 17 1 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  NrmCVeccnv 29568  CBanccbn 29846  CHilOLDchlo 29869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-cbn 29847  df-hlo 29870
This theorem is referenced by:  hlnvi  29876  hlvc  29877  hladdf  29883  hlcom  29884  hlass  29885  hl0cl  29886  hladdid  29887  hlmulf  29888  hlmulid  29889  hlmulass  29890  hldi  29891  hldir  29892  hlmul0  29893  hlipf  29894  hlipcj  29895  hlipgt0  29898
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