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Theorem odfval 16055
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 5710 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 odval.1 . . . . . 6  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2493 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  X )
5 fveq2 5710 . . . . . . . . . 10  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
6 odval.2 . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
75, 6syl6eqr 2493 . . . . . . . . 9  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
87oveqd 6127 . . . . . . . 8  |-  ( g  =  G  ->  (
y (.g `  g ) x )  =  ( y 
.x.  x ) )
9 fveq2 5710 . . . . . . . . 9  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 odval.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2493 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2457 . . . . . . 7  |-  ( g  =  G  ->  (
( y (.g `  g
) x )  =  ( 0g `  g
)  <->  ( y  .x.  x )  =  .0.  ) )
1312rabbidv 2983 . . . . . 6  |-  ( g  =  G  ->  { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g
`  g ) }  =  { y  e.  NN  |  ( y 
.x.  x )  =  .0.  } )
1413csbeq1d 3314 . . . . 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 4389 . . . 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 16051 . . . 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 5720 . . . . . 6  |-  ( Base `  G )  e.  _V
183, 17eqeltri 2513 . . . . 5  |-  X  e. 
_V
1918mptex 5967 . . . 4  |-  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  e.  _V
2015, 16, 19fvmpt 5793 . . 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 5704 . . . 4  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  (/) )
22 fvprc 5704 . . . . . . 7  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
233, 22syl5eq 2487 . . . . . 6  |-  ( -.  G  e.  _V  ->  X  =  (/) )
2423mpteq1d 4392 . . . . 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 5557 . . . . 5  |-  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  (/)
2624, 25syl6eq 2491 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  (/) )
2721, 26eqtr4d 2478 . . 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 2463 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 1369    e. wcel 1756   {crab 2738   _Vcvv 2991   [_csb 3307   (/)c0 3656   ifcif 3810    e. cmpt 4369   `'ccnv 4858   ` cfv 5437  (class class class)co 6110   supcsup 7709   RRcr 9300   0cc0 9301    < clt 9437   NNcn 10341   Basecbs 14193   0gc0g 14397  .gcmg 15433   odcod 16047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-ov 6113  df-od 16051
This theorem is referenced by:  odval  16056  odf  16059
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