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Theorem nmfval 21235
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
nmfval  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Distinct variable groups:    x, D    x, W    x, X    x,  .0.
Allowed substitution hint:    N( x)

Proof of Theorem nmfval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nmfval.n . 2  |-  N  =  ( norm `  W
)
2 fveq2 5872 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 nmfval.x . . . . . 6  |-  X  =  ( Base `  W
)
42, 3syl6eqr 2516 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  X )
5 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
6 nmfval.d . . . . . . 7  |-  D  =  ( dist `  W
)
75, 6syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( dist `  w )  =  D )
8 eqidd 2458 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 nmfval.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
127, 8, 11oveq123d 6317 . . . . 5  |-  ( w  =  W  ->  (
x ( dist `  w
) ( 0g `  w ) )  =  ( x D  .0.  ) )
134, 12mpteq12dv 4535 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
14 df-nm 21229 . . . 4  |-  norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) ) )
15 eqid 2457 . . . . . 6  |-  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  X  |->  ( x D  .0.  ) )
16 df-ov 6299 . . . . . . . 8  |-  ( x D  .0.  )  =  ( D `  <. x ,  .0.  >. )
17 fvrn0 5894 . . . . . . . 8  |-  ( D `
 <. x ,  .0.  >.
)  e.  ( ran 
D  u.  { (/) } )
1816, 17eqeltri 2541 . . . . . . 7  |-  ( x D  .0.  )  e.  ( ran  D  u.  {
(/) } )
1918a1i 11 . . . . . 6  |-  ( x  e.  X  ->  (
x D  .0.  )  e.  ( ran  D  u.  {
(/) } ) )
2015, 19fmpti 6055 . . . . 5  |-  ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran 
D  u.  { (/) } )
21 fvex 5882 . . . . . 6  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2541 . . . . 5  |-  X  e. 
_V
23 fvex 5882 . . . . . . . 8  |-  ( dist `  W )  e.  _V
246, 23eqeltri 2541 . . . . . . 7  |-  D  e. 
_V
2524rnex 6733 . . . . . 6  |-  ran  D  e.  _V
26 p0ex 4643 . . . . . 6  |-  { (/) }  e.  _V
2725, 26unex 6597 . . . . 5  |-  ( ran 
D  u.  { (/) } )  e.  _V
28 fex2 6754 . . . . 5  |-  ( ( ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran  D  u.  { (/)
} )  /\  X  e.  _V  /\  ( ran 
D  u.  { (/) } )  e.  _V )  ->  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V )
2920, 22, 27, 28mp3an 1324 . . . 4  |-  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V
3013, 14, 29fvmpt 5956 . . 3  |-  ( W  e.  _V  ->  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
31 fvprc 5866 . . . . 5  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  (/) )
32 mpt0 5714 . . . . 5  |-  ( x  e.  (/)  |->  ( x D  .0.  ) )  =  (/)
3331, 32syl6eqr 2516 . . . 4  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
34 fvprc 5866 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
353, 34syl5eq 2510 . . . . 5  |-  ( -.  W  e.  _V  ->  X  =  (/) )
3635mpteq1d 4538 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3733, 36eqtr4d 2501 . . 3  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
3830, 37pm2.61i 164 . 2  |-  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  )
)
391, 38eqtri 2486 1  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038    |-> cmpt 4515   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296   Basecbs 14644   distcds 14721   0gc0g 14857   normcnm 21223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-nm 21229
This theorem is referenced by:  nmval  21236  nmfval2  21237  nmpropd  21240  subgnm  21273  tngnm  21291  cnfldnm  21412  nmcn  21475  ressnm  27799
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