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Theorem nmfval 18629
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 5721 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 nmfval.x . . . . . 6  |-  X  =  ( Base `  W
)
42, 3syl6eqr 2486 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  X )
5 fveq2 5721 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
6 nmfval.d . . . . . . 7  |-  D  =  ( dist `  W
)
75, 6syl6eqr 2486 . . . . . 6  |-  ( w  =  W  ->  ( dist `  w )  =  D )
8 eqidd 2437 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5721 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 nmfval.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2486 . . . . . 6  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
127, 8, 11oveq123d 6095 . . . . 5  |-  ( w  =  W  ->  (
x ( dist `  w
) ( 0g `  w ) )  =  ( x D  .0.  ) )
134, 12mpteq12dv 4280 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
14 df-nm 18623 . . . 4  |-  norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) ) )
15 eqid 2436 . . . . . 6  |-  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  X  |->  ( x D  .0.  ) )
16 df-ov 6077 . . . . . . . 8  |-  ( x D  .0.  )  =  ( D `  <. x ,  .0.  >. )
17 fvrn0 5746 . . . . . . . 8  |-  ( D `
 <. x ,  .0.  >.
)  e.  ( ran 
D  u.  { (/) } )
1816, 17eqeltri 2506 . . . . . . 7  |-  ( x D  .0.  )  e.  ( ran  D  u.  {
(/) } )
1918a1i 11 . . . . . 6  |-  ( x  e.  X  ->  (
x D  .0.  )  e.  ( ran  D  u.  {
(/) } ) )
2015, 19fmpti 5885 . . . . 5  |-  ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran 
D  u.  { (/) } )
21 fvex 5735 . . . . . 6  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2506 . . . . 5  |-  X  e. 
_V
23 fvex 5735 . . . . . . . 8  |-  ( dist `  W )  e.  _V
246, 23eqeltri 2506 . . . . . . 7  |-  D  e. 
_V
2524rnex 5126 . . . . . 6  |-  ran  D  e.  _V
26 p0ex 4379 . . . . . 6  |-  { (/) }  e.  _V
2725, 26unex 4700 . . . . 5  |-  ( ran 
D  u.  { (/) } )  e.  _V
28 fex2 5596 . . . . 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 1279 . . . 4  |-  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V
3013, 14, 29fvmpt 5799 . . 3  |-  ( W  e.  _V  ->  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
31 fvprc 5715 . . . . 5  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  (/) )
32 mpt0 5565 . . . . 5  |-  ( x  e.  (/)  |->  ( x D  .0.  ) )  =  (/)
3331, 32syl6eqr 2486 . . . 4  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
34 fvprc 5715 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
353, 34syl5eq 2480 . . . . 5  |-  ( -.  W  e.  _V  ->  X  =  (/) )
3635mpteq1d 4283 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3733, 36eqtr4d 2471 . . 3  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
3830, 37pm2.61i 158 . 2  |-  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  )
)
391, 38eqtri 2456 1  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2949    u. cun 3311   (/)c0 3621   {csn 3807   <.cop 3810    e. cmpt 4259   ran crn 4872   -->wf 5443   ` cfv 5447  (class class class)co 6074   Basecbs 13462   distcds 13531   0gc0g 13716   normcnm 18617
This theorem is referenced by:  nmval  18630  nmfval2  18631  nmpropd  18634  subgnm  18667  tngnm  18685  cnfldnm  18806  nmcn  18868  ressnm  24177
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 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-nm 18623
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