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Theorem hlnv 22393
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 22390 . 2  |-  ( U  e.  CHil OLD  ->  U  e. 
CBan )
2 bnnv 22368 . 2  |-  ( U  e.  CBan  ->  U  e.  NrmCVec )
31, 2syl 16 1  |-  ( U  e.  CHil OLD  ->  U  e.  NrmCVec )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   NrmCVeccnv 22063   CBanccbn 22364   CHil
OLDchlo 22387
This theorem is referenced by:  hlnvi  22394  hlvc  22395  hladdf  22401  hlcom  22402  hlass  22403  hl0cl  22404  hladdid  22405  hlmulf  22406  hlmulid  22407  hlmulass  22408  hldi  22409  hldir  22410  hlmul0  22411  hlipf  22412  hlipcj  22413  hlipgt0  22416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-cbn 22365  df-hlo 22388
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