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Theorem nvcli 25764
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 25763 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( N `  A )  e.  RR )
61, 2, 5mp2an 670 1  |-  ( N `
 A )  e.  RR
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   ` cfv 5570   RRcr 9480   NrmCVeccnv 25678   BaseSetcba 25680   normCVcnmcv 25684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-1st 6773  df-2nd 6774  df-vc 25640  df-nv 25686  df-va 25689  df-ba 25690  df-sm 25691  df-0v 25692  df-nmcv 25694
This theorem is referenced by:  ip0i  25941  ip1ilem  25942  ipasslem10  25955  siilem1  25967  siii  25969
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