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Theorem odfval 14848
Description: Value of the order function. (Contributed by Mario Carneiro, 13-Jul-2014.)
Hypotheses
Ref Expression
odval.1  |-  X  =  ( Base `  G
)
odval.2  |-  .x.  =  (.g
`  G )
odval.3  |-  .0.  =  ( 0g `  G )
odval.4  |-  O  =  ( od `  G
)
Assertion
Ref Expression
odfval  |-  O  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
Distinct variable groups:    y, i, x    x, G, y    x,  .x. , y    x,  .0. , y    x, i    x, X
Allowed substitution hints:    .x. ( i)    G( i)    O( x, y, i)    X( y, i)    .0. ( i)

Proof of Theorem odfval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 odval.4 . 2  |-  O  =  ( od `  G
)
2 fveq2 5525 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 odval.1 . . . . . 6  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2333 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  X )
5 fveq2 5525 . . . . . . . . . 10  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
6 odval.2 . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
75, 6syl6eqr 2333 . . . . . . . . 9  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
87oveqd 5875 . . . . . . . 8  |-  ( g  =  G  ->  (
y (.g `  g ) x )  =  ( y 
.x.  x ) )
9 fveq2 5525 . . . . . . . . 9  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 odval.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2333 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2297 . . . . . . 7  |-  ( g  =  G  ->  (
( y (.g `  g
) x )  =  ( 0g `  g
)  <->  ( y  .x.  x )  =  .0.  ) )
1312rabbidv 2780 . . . . . 6  |-  ( g  =  G  ->  { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g
`  g ) }  =  { y  e.  NN  |  ( y 
.x.  x )  =  .0.  } )
1413csbeq1d 3087 . . . . 5  |-  ( g  =  G  ->  [_ {
y  e.  NN  | 
( y (.g `  g
) x )  =  ( 0g `  g
) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) )  =  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0.  }  / 
i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
154, 14mpteq12dv 4098 . . . 4  |-  ( g  =  G  ->  (
x  e.  ( Base `  g )  |->  [_ {
y  e.  NN  | 
( y (.g `  g
) x )  =  ( 0g `  g
) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) ) )
16 df-od 14844 . . . 4  |-  od  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g )  |->  [_ { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g `  g ) }  / 
i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
17 fvex 5539 . . . . . 6  |-  ( Base `  G )  e.  _V
183, 17eqeltri 2353 . . . . 5  |-  X  e. 
_V
1918mptex 5746 . . . 4  |-  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  e.  _V
2015, 16, 19fvmpt 5602 . . 3  |-  ( G  e.  _V  ->  ( od `  G )  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
21 fvprc 5519 . . . 4  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  (/) )
22 fvprc 5519 . . . . . . 7  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
233, 22syl5eq 2327 . . . . . 6  |-  ( -.  G  e.  _V  ->  X  =  (/) )
24 eqidd 2284 . . . . . 6  |-  ( -.  G  e.  _V  ->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  = 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
2523, 24mpteq12dv 4098 . . . . 5  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
26 mpt0 5371 . . . . 5  |-  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  (/)
2725, 26syl6eq 2331 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  (/) )
2821, 27eqtr4d 2318 . . 3  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
2920, 28pm2.61i 156 . 2  |-  ( od
`  G )  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
301, 29eqtri 2303 1  |-  O  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081   (/)c0 3455   ifcif 3565    e. cmpt 4077   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867   NNcn 9746   Basecbs 13148   0gc0g 13400  .gcmg 14366   odcod 14840
This theorem is referenced by:  odval  14849  odf  14852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-od 14844
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