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Theorem odfval 16430
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 5872 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 odval.1 . . . . . 6  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2526 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  X )
5 fveq2 5872 . . . . . . . . . 10  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
6 odval.2 . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
75, 6syl6eqr 2526 . . . . . . . . 9  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
87oveqd 6312 . . . . . . . 8  |-  ( g  =  G  ->  (
y (.g `  g ) x )  =  ( y 
.x.  x ) )
9 fveq2 5872 . . . . . . . . 9  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 odval.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2526 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2489 . . . . . . 7  |-  ( g  =  G  ->  (
( y (.g `  g
) x )  =  ( 0g `  g
)  <->  ( y  .x.  x )  =  .0.  ) )
1312rabbidv 3110 . . . . . 6  |-  ( g  =  G  ->  { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g
`  g ) }  =  { y  e.  NN  |  ( y 
.x.  x )  =  .0.  } )
1413csbeq1d 3447 . . . . 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 4531 . . . 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 16426 . . . 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 5882 . . . . . 6  |-  ( Base `  G )  e.  _V
183, 17eqeltri 2551 . . . . 5  |-  X  e. 
_V
1918mptex 6142 . . . 4  |-  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  e.  _V
2015, 16, 19fvmpt 5957 . . 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 5866 . . . 4  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  (/) )
22 fvprc 5866 . . . . . . 7  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
233, 22syl5eq 2520 . . . . . 6  |-  ( -.  G  e.  _V  ->  X  =  (/) )
2423mpteq1d 4534 . . . . 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 ,  `'  <  ) ) ) )
25 mpt0 5714 . . . . 5  |-  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  (/)
2624, 25syl6eq 2524 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  (/) )
2721, 26eqtr4d 2511 . . 3  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
2820, 27pm2.61i 164 . 2  |-  ( od
`  G )  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
291, 28eqtri 2496 1  |-  O  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1379    e. wcel 1767   {crab 2821   _Vcvv 3118   [_csb 3440   (/)c0 3790   ifcif 3945    |-> cmpt 4511   `'ccnv 5004   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   0cc0 9504    < clt 9640   NNcn 10548   Basecbs 14507   0gc0g 14712  .gcmg 15928   odcod 16422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-od 16426
This theorem is referenced by:  odval  16431  odf  16434
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