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Theorem nvcli 22149
Description: The norm of a normed complex vector space is a real number. (Contributed by NM, 20-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1  |-  X  =  ( BaseSet `  U )
nvf.6  |-  N  =  ( normCV `  U )
nvcli.9  |-  U  e.  NrmCVec
nvcli.7  |-  A  e.  X
Assertion
Ref Expression
nvcli  |-  ( N `
 A )  e.  RR

Proof of Theorem nvcli
StepHypRef Expression
1 nvcli.9 . 2  |-  U  e.  NrmCVec
2 nvcli.7 . 2  |-  A  e.  X
3 nvf.1 . . 3  |-  X  =  ( BaseSet `  U )
4 nvf.6 . . 3  |-  N  =  ( normCV `  U )
53, 4nvcl 22148 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
61, 2, 5mp2an 654 1  |-  ( N `
 A )  e.  RR
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   ` cfv 5454   RRcr 8989   NrmCVeccnv 22063   BaseSetcba 22065   normCVcnmcv 22069
This theorem is referenced by:  ip0i  22326  ip1ilem  22327  ipasslem10  22340  siilem1  22352  siii  22354
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  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-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-1st 6349  df-2nd 6350  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079
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