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Theorem nmfval 18127
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 5541 . . . . . 6  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
3 nmfval.x . . . . . 6  |-  X  =  ( Base `  W
)
42, 3syl6eqr 2346 . . . . 5  |-  ( w  =  W  ->  ( Base `  w )  =  X )
5 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( dist `  w )  =  ( dist `  W
) )
6 nmfval.d . . . . . . 7  |-  D  =  ( dist `  W
)
75, 6syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( dist `  w )  =  D )
8 eqidd 2297 . . . . . 6  |-  ( w  =  W  ->  x  =  x )
9 fveq2 5541 . . . . . . 7  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
10 nmfval.z . . . . . . 7  |-  .0.  =  ( 0g `  W )
119, 10syl6eqr 2346 . . . . . 6  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
127, 8, 11oveq123d 5895 . . . . 5  |-  ( w  =  W  ->  (
x ( dist `  w
) ( 0g `  w ) )  =  ( x D  .0.  ) )
134, 12mpteq12dv 4114 . . . 4  |-  ( w  =  W  ->  (
x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
14 df-nm 18121 . . . 4  |-  norm  =  ( w  e.  _V  |->  ( x  e.  ( Base `  w )  |->  ( x ( dist `  w
) ( 0g `  w ) ) ) )
15 eqid 2296 . . . . . 6  |-  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  X  |->  ( x D  .0.  ) )
16 df-ov 5877 . . . . . . . 8  |-  ( x D  .0.  )  =  ( D `  <. x ,  .0.  >. )
17 fvrn0 5566 . . . . . . . 8  |-  ( D `
 <. x ,  .0.  >.
)  e.  ( ran 
D  u.  { (/) } )
1816, 17eqeltri 2366 . . . . . . 7  |-  ( x D  .0.  )  e.  ( ran  D  u.  {
(/) } )
1918a1i 10 . . . . . 6  |-  ( x  e.  X  ->  (
x D  .0.  )  e.  ( ran  D  u.  {
(/) } ) )
2015, 19fmpti 5699 . . . . 5  |-  ( x  e.  X  |->  ( x D  .0.  ) ) : X --> ( ran 
D  u.  { (/) } )
21 fvex 5555 . . . . . 6  |-  ( Base `  W )  e.  _V
223, 21eqeltri 2366 . . . . 5  |-  X  e. 
_V
23 fvex 5555 . . . . . . . 8  |-  ( dist `  W )  e.  _V
246, 23eqeltri 2366 . . . . . . 7  |-  D  e. 
_V
2524rnex 4958 . . . . . 6  |-  ran  D  e.  _V
26 p0ex 4213 . . . . . 6  |-  { (/) }  e.  _V
2725, 26unex 4534 . . . . 5  |-  ( ran 
D  u.  { (/) } )  e.  _V
28 fex2 5417 . . . . 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 1277 . . . 4  |-  ( x  e.  X  |->  ( x D  .0.  ) )  e.  _V
3013, 14, 29fvmpt 5618 . . 3  |-  ( W  e.  _V  ->  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
31 fvprc 5535 . . . . 5  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  (/) )
32 mpt0 5387 . . . . 5  |-  ( x  e.  (/)  |->  ( x D  .0.  ) )  =  (/)
3331, 32syl6eqr 2346 . . . 4  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
34 fvprc 5535 . . . . . 6  |-  ( -.  W  e.  _V  ->  (
Base `  W )  =  (/) )
353, 34syl5eq 2340 . . . . 5  |-  ( -.  W  e.  _V  ->  X  =  (/) )
36 mpteq1 4116 . . . . 5  |-  ( X  =  (/)  ->  ( x  e.  X  |->  ( x D  .0.  ) )  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3735, 36syl 15 . . . 4  |-  ( -.  W  e.  _V  ->  ( x  e.  X  |->  ( x D  .0.  )
)  =  ( x  e.  (/)  |->  ( x D  .0.  ) ) )
3833, 37eqtr4d 2331 . . 3  |-  ( -.  W  e.  _V  ->  (
norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  ) ) )
3930, 38pm2.61i 156 . 2  |-  ( norm `  W )  =  ( x  e.  X  |->  ( x D  .0.  )
)
401, 39eqtri 2316 1  |-  N  =  ( x  e.  X  |->  ( x D  .0.  ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   distcds 13233   0gc0g 13416   normcnm 18115
This theorem is referenced by:  nmval  18128  nmfval2  18129  nmpropd  18132  subgnm  18165  tngnm  18183  cnfldnm  18304  nmcn  18365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-nm 18121
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