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Theorem hlnv 21525
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 21522 . 2  |-  ( U  e.  CHil OLD  ->  U  e. 
CBan )
2 bnnv 21500 . 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 1701   NrmCVeccnv 21195   CBanccbn 21496   CHil
OLDchlo 21519
This theorem is referenced by:  hlnvi  21526  hlvc  21527  hladdf  21533  hlcom  21534  hlass  21535  hl0cl  21536  hladdid  21537  hlmulf  21538  hlmulid  21539  hlmulass  21540  hldi  21541  hldir  21542  hlmul0  21543  hlipf  21544  hlipcj  21545  hlipgt0  21548
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-cbn 21497  df-hlo 21520
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