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Theorem hlnv 30144
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 30141 . 2 (𝑈 ∈ CHilOLD𝑈 ∈ CBan)
2 bnnv 30119 . 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 29837  CBanccbn 30115  CHilOLDchlo 30138
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 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-cbn 30116  df-hlo 30139
This theorem is referenced by:  hlnvi  30145  hlvc  30146  hladdf  30152  hlcom  30153  hlass  30154  hl0cl  30155  hladdid  30156  hlmulf  30157  hlmulid  30158  hlmulass  30159  hldi  30160  hldir  30161  hlmul0  30162  hlipf  30163  hlipcj  30164  hlipgt0  30167
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