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Theorem nmval 18637
Description: The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
nmfval.n  |-  N  =  ( norm `  W
)
nmfval.x  |-  X  =  ( Base `  W
)
nmfval.z  |-  .0.  =  ( 0g `  W )
nmfval.d  |-  D  =  ( dist `  W
)
Assertion
Ref Expression
nmval  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )

Proof of Theorem nmval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 6088 . 2  |-  ( x  =  A  ->  (
x D  .0.  )  =  ( A D  .0.  ) )
2 nmfval.n . . 3  |-  N  =  ( norm `  W
)
3 nmfval.x . . 3  |-  X  =  ( Base `  W
)
4 nmfval.z . . 3  |-  .0.  =  ( 0g `  W )
5 nmfval.d . . 3  |-  D  =  ( dist `  W
)
62, 3, 4, 5nmfval 18636 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
7 ovex 6106 . 2  |-  ( A D  .0.  )  e. 
_V
81, 6, 7fvmpt 5806 1  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   distcds 13538   0gc0g 13723   normcnm 18624
This theorem is referenced by:  nmval2  18639  ngpds2  18652  isngp4  18658  nmge0  18663  nmeq0  18664  nminv  18667  nmmtri  18668  nmrtri  18670
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-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-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  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-fv 5462  df-ov 6084  df-nm 18630
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