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Theorem odfval 17128
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 5881 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 odval.1 . . . . . 6  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2488 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  X )
5 fveq2 5881 . . . . . . . . . 10  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
6 odval.2 . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
75, 6syl6eqr 2488 . . . . . . . . 9  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
87oveqd 6322 . . . . . . . 8  |-  ( g  =  G  ->  (
y (.g `  g ) x )  =  ( y 
.x.  x ) )
9 fveq2 5881 . . . . . . . . 9  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 odval.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2488 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2451 . . . . . . 7  |-  ( g  =  G  ->  (
( y (.g `  g
) x )  =  ( 0g `  g
)  <->  ( y  .x.  x )  =  .0.  ) )
1312rabbidv 3079 . . . . . 6  |-  ( g  =  G  ->  { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g
`  g ) }  =  { y  e.  NN  |  ( y 
.x.  x )  =  .0.  } )
1413csbeq1d 3408 . . . . 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 4504 . . . 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 17124 . . . 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 5891 . . . . . 6  |-  ( Base `  G )  e.  _V
183, 17eqeltri 2513 . . . . 5  |-  X  e. 
_V
1918mptex 6151 . . . 4  |-  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  e.  _V
2015, 16, 19fvmpt 5964 . . 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 5875 . . . 4  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  (/) )
22 fvprc 5875 . . . . . . 7  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
233, 22syl5eq 2482 . . . . . 6  |-  ( -.  G  e.  _V  ->  X  =  (/) )
2423mpteq1d 4507 . . . . 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 5723 . . . . 5  |-  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  (/)
2624, 25syl6eq 2486 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  (/) )
2721, 26eqtr4d 2473 . . 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 167 . 2  |-  ( od
`  G )  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
291, 28eqtri 2458 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 1437    e. wcel 1870   {crab 2786   _Vcvv 3087   [_csb 3401   (/)c0 3767   ifcif 3915    |-> cmpt 4484   `'ccnv 4853   ` cfv 5601  (class class class)co 6305   supcsup 7960   RRcr 9537   0cc0 9538    < clt 9674   NNcn 10609   Basecbs 15084   0gc0g 15301  .gcmg 16627   odcod 17120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-od 17124
This theorem is referenced by:  odval  17129  odf  17132
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