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Theorem hlnv 28665
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 28662 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
2 bnnv 28640 . 2 (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
31, 2syl 17 1 (𝑈 ∈ CHilOLD𝑈 ∈ NrmCVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  NrmCVeccnv 28358  CBanccbn 28636  CHilOLDchlo 28659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-iota 6297  df-fv 6346  df-cbn 28637  df-hlo 28660
This theorem is referenced by:  hlnvi  28666  hlvc  28667  hladdf  28673  hlcom  28674  hlass  28675  hl0cl  28676  hladdid  28677  hlmulf  28678  hlmulid  28679  hlmulass  28680  hldi  28681  hldir  28682  hlmul0  28683  hlipf  28684  hlipcj  28685  hlipgt0  28688
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