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Theorem nmfval 20184
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 5694 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 nmfval.x . . . . . 6  |-  X  =  ( Base `  W
)
42, 3syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  X )
5 fveq2 5694 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
6 nmfval.d . . . . . . 7  |-  D  =  ( dist `  W
)
75, 6syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( dist `  w )  =  D )
8 eqidd 2444 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5694 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 nmfval.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2493 . . . . . 6  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
127, 8, 11oveq123d 6115 . . . . 5  |-  ( w  =  W  ->  (
x ( dist `  w
) ( 0g `  w ) )  =  ( x D  .0.  ) )
134, 12mpteq12dv 4373 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
14 df-nm 20178 . . . 4  |-  norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) ) )
15 eqid 2443 . . . . . 6  |-  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  X  |->  ( x D  .0.  ) )
16 df-ov 6097 . . . . . . . 8  |-  ( x D  .0.  )  =  ( D `  <. x ,  .0.  >. )
17 fvrn0 5715 . . . . . . . 8  |-  ( D `
 <. x ,  .0.  >.
)  e.  ( ran 
D  u.  { (/) } )
1816, 17eqeltri 2513 . . . . . . 7  |-  ( x D  .0.  )  e.  ( ran  D  u.  {
(/) } )
1918a1i 11 . . . . . 6  |-  ( x  e.  X  ->  (
x D  .0.  )  e.  ( ran  D  u.  {
(/) } ) )
2015, 19fmpti 5869 . . . . 5  |-  ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran 
D  u.  { (/) } )
21 fvex 5704 . . . . . 6  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2513 . . . . 5  |-  X  e. 
_V
23 fvex 5704 . . . . . . . 8  |-  ( dist `  W )  e.  _V
246, 23eqeltri 2513 . . . . . . 7  |-  D  e. 
_V
2524rnex 6515 . . . . . 6  |-  ran  D  e.  _V
26 p0ex 4482 . . . . . 6  |-  { (/) }  e.  _V
2725, 26unex 6381 . . . . 5  |-  ( ran 
D  u.  { (/) } )  e.  _V
28 fex2 6535 . . . . 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 1314 . . . 4  |-  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V
3013, 14, 29fvmpt 5777 . . 3  |-  ( W  e.  _V  ->  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
31 fvprc 5688 . . . . 5  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  (/) )
32 mpt0 5541 . . . . 5  |-  ( x  e.  (/)  |->  ( x D  .0.  ) )  =  (/)
3331, 32syl6eqr 2493 . . . 4  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
34 fvprc 5688 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
353, 34syl5eq 2487 . . . . 5  |-  ( -.  W  e.  _V  ->  X  =  (/) )
3635mpteq1d 4376 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3733, 36eqtr4d 2478 . . 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 2463 1  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2975    u. cun 3329   (/)c0 3640   {csn 3880   <.cop 3886    e. cmpt 4353   ran crn 4844   -->wf 5417   ` cfv 5421  (class class class)co 6094   Basecbs 14177   distcds 14250   0gc0g 14381   normcnm 20172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6097  df-nm 20178
This theorem is referenced by:  nmval  20185  nmfval2  20186  nmpropd  20189  subgnm  20222  tngnm  20240  cnfldnm  20361  nmcn  20424  ressnm  26115
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