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Theorem nmval 20873
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 6291 . 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 20872 . 2  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
7 ovex 6309 . 2  |-  ( A D  .0.  )  e. 
_V
81, 6, 7fvmpt 5950 1  |-  ( A  e.  X  ->  ( N `  A )  =  ( A D  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   Basecbs 14490   distcds 14564   0gc0g 14695   normcnm 20860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-nm 20866
This theorem is referenced by:  nmval2  20875  ngpds2  20888  isngp4  20894  nmge0  20899  nmeq0  20900  nminv  20903  nmmtri  20904  nmrtri  20906
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