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Theorem nmval 18128
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 5881 . 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 18127 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
7 ovex 5899 . 2  |-  ( A D  .0.  )  e. 
_V
81, 6, 7fvmpt 5618 1  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   distcds 13233   0gc0g 13416   normcnm 18115
This theorem is referenced by:  nmval2  18130  ngpds2  18143  isngp4  18149  nmge0  18154  nmeq0  18155  nminv  18158  nmmtri  18159  nmrtri  18161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-nm 18121
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