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Theorem nmfval 21615
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 5870 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 nmfval.x . . . . . 6  |-  X  =  ( Base `  W
)
42, 3syl6eqr 2505 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  X )
5 fveq2 5870 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
6 nmfval.d . . . . . . 7  |-  D  =  ( dist `  W
)
75, 6syl6eqr 2505 . . . . . 6  |-  ( w  =  W  ->  ( dist `  w )  =  D )
8 eqidd 2454 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5870 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 nmfval.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2505 . . . . . 6  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
127, 8, 11oveq123d 6316 . . . . 5  |-  ( w  =  W  ->  (
x ( dist `  w
) ( 0g `  w ) )  =  ( x D  .0.  ) )
134, 12mpteq12dv 4484 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
14 df-nm 21609 . . . 4  |-  norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) ) )
15 eqid 2453 . . . . . 6  |-  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  X  |->  ( x D  .0.  ) )
16 df-ov 6298 . . . . . . . 8  |-  ( x D  .0.  )  =  ( D `  <. x ,  .0.  >. )
17 fvrn0 5892 . . . . . . . 8  |-  ( D `
 <. x ,  .0.  >.
)  e.  ( ran 
D  u.  { (/) } )
1816, 17eqeltri 2527 . . . . . . 7  |-  ( x D  .0.  )  e.  ( ran  D  u.  {
(/) } )
1918a1i 11 . . . . . 6  |-  ( x  e.  X  ->  (
x D  .0.  )  e.  ( ran  D  u.  {
(/) } ) )
2015, 19fmpti 6050 . . . . 5  |-  ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran 
D  u.  { (/) } )
21 fvex 5880 . . . . . 6  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2527 . . . . 5  |-  X  e. 
_V
23 fvex 5880 . . . . . . . 8  |-  ( dist `  W )  e.  _V
246, 23eqeltri 2527 . . . . . . 7  |-  D  e. 
_V
2524rnex 6732 . . . . . 6  |-  ran  D  e.  _V
26 p0ex 4593 . . . . . 6  |-  { (/) }  e.  _V
2725, 26unex 6594 . . . . 5  |-  ( ran 
D  u.  { (/) } )  e.  _V
28 fex2 6753 . . . . 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 1366 . . . 4  |-  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V
3013, 14, 29fvmpt 5953 . . 3  |-  ( W  e.  _V  ->  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
31 fvprc 5864 . . . . 5  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  (/) )
32 mpt0 5710 . . . . 5  |-  ( x  e.  (/)  |->  ( x D  .0.  ) )  =  (/)
3331, 32syl6eqr 2505 . . . 4  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
34 fvprc 5864 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
353, 34syl5eq 2499 . . . . 5  |-  ( -.  W  e.  _V  ->  X  =  (/) )
3635mpteq1d 4487 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3733, 36eqtr4d 2490 . . 3  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
3830, 37pm2.61i 168 . 2  |-  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  )
)
391, 38eqtri 2475 1  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1446    e. wcel 1889   _Vcvv 3047    u. cun 3404   (/)c0 3733   {csn 3970   <.cop 3976    |-> cmpt 4464   ran crn 4838   -->wf 5581   ` cfv 5585  (class class class)co 6295   Basecbs 15133   distcds 15211   0gc0g 15350   normcnm 21603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-nm 21609
This theorem is referenced by:  nmval  21616  nmfval2  21617  nmpropd  21620  subgnm  21653  tngnm  21671  cnfldnm  21811  nmcn  21874  ressnm  28424
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