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Theorem hlnv 21470
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  e.  CHil OLD  ->  U  e.  NrmCVec )

Proof of Theorem hlnv
StepHypRef Expression
1 hlobn 21467 . 2  |-  ( U  e.  CHil OLD  ->  U  e. 
CBan )
2 bnnv 21445 . 2  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
31, 2syl 15 1  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   NrmCVeccnv 21140   CBanccbn 21441   CHil
OLDchlo 21464
This theorem is referenced by:  hlnvi  21471  hlvc  21472  hladdf  21478  hlcom  21479  hlass  21480  hl0cl  21481  hladdid  21482  hlmulf  21483  hlmulid  21484  hlmulass  21485  hldi  21486  hldir  21487  hlmul0  21488  hlipf  21489  hlipcj  21490  hlipgt0  21493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-cbn 21442  df-hlo 21465
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