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Theorem nvcli 24201
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 24200 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
61, 2, 5mp2an 672 1  |-  ( N `
 A )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   ` cfv 5527   RRcr 9393   NrmCVeccnv 24115   BaseSetcba 24117   normCVcnmcv 24121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-1st 6688  df-2nd 6689  df-vc 24077  df-nv 24123  df-va 24126  df-ba 24127  df-sm 24128  df-0v 24129  df-nmcv 24131
This theorem is referenced by:  ip0i  24378  ip1ilem  24379  ipasslem10  24392  siilem1  24404  siii  24406
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