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Theorem nvf 22147
Description: Mapping for the norm function. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvf.1  |-  X  =  ( BaseSet `  U )
nvf.6  |-  N  =  ( normCV `  U )
Assertion
Ref Expression
nvf  |-  ( U  e.  NrmCVec  ->  N : X --> RR )

Proof of Theorem nvf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvf.1 . . 3  |-  X  =  ( BaseSet `  U )
2 eqid 2436 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2436 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2436 . . 3  |-  ( 0vec `  U )  =  (
0vec `  U )
5 nvf.6 . . 3  |-  N  =  ( normCV `  U )
61, 2, 3, 4, 5nvi 22093 . 2  |-  ( U  e.  NrmCVec  ->  ( <. ( +v `  U ) ,  ( .s OLD `  U
) >.  e.  CVec OLD  /\  N : X --> RR  /\  A. x  e.  X  ( ( ( N `  x )  =  0  ->  x  =  (
0vec `  U )
)  /\  A. y  e.  CC  ( N `  ( y ( .s
OLD `  U )
x ) )  =  ( ( abs `  y
)  x.  ( N `
 x ) )  /\  A. y  e.  X  ( N `  ( x ( +v
`  U ) y ) )  <_  (
( N `  x
)  +  ( N `
 y ) ) ) ) )
76simp2d 970 1  |-  ( U  e.  NrmCVec  ->  N : X --> RR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817   class class class wbr 4212   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   RRcr 8989   0cc0 8990    + caddc 8993    x. cmul 8995    <_ cle 9121   abscabs 12039   CVec
OLDcvc 22024   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s
OLDcns 22066   0veccn0v 22067   normCVcnmcv 22069
This theorem is referenced by:  nvcl  22148  nvdm  22150  imsdf  22181  nmcvcn  22191  sspn  22235  hilnormi  22665
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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