MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  odfval Structured version   Unicode version

Theorem odfval 15171
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 5728 . . . . . 6  |-  ( g  =  G  ->  ( Base `  g )  =  ( Base `  G
) )
3 odval.1 . . . . . 6  |-  X  =  ( Base `  G
)
42, 3syl6eqr 2486 . . . . 5  |-  ( g  =  G  ->  ( Base `  g )  =  X )
5 fveq2 5728 . . . . . . . . . 10  |-  ( g  =  G  ->  (.g `  g )  =  (.g `  G ) )
6 odval.2 . . . . . . . . . 10  |-  .x.  =  (.g
`  G )
75, 6syl6eqr 2486 . . . . . . . . 9  |-  ( g  =  G  ->  (.g `  g )  =  .x.  )
87oveqd 6098 . . . . . . . 8  |-  ( g  =  G  ->  (
y (.g `  g ) x )  =  ( y 
.x.  x ) )
9 fveq2 5728 . . . . . . . . 9  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
10 odval.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
119, 10syl6eqr 2486 . . . . . . . 8  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
128, 11eqeq12d 2450 . . . . . . 7  |-  ( g  =  G  ->  (
( y (.g `  g
) x )  =  ( 0g `  g
)  <->  ( y  .x.  x )  =  .0.  ) )
1312rabbidv 2948 . . . . . 6  |-  ( g  =  G  ->  { y  e.  NN  |  ( y (.g `  g ) x )  =  ( 0g
`  g ) }  =  { y  e.  NN  |  ( y 
.x.  x )  =  .0.  } )
1413csbeq1d 3257 . . . . 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 4287 . . . 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 15167 . . . 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 5742 . . . . . 6  |-  ( Base `  G )  e.  _V
183, 17eqeltri 2506 . . . . 5  |-  X  e. 
_V
1918mptex 5966 . . . 4  |-  ( x  e.  X  |->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  e.  _V
2015, 16, 19fvmpt 5806 . . 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 5722 . . . 4  |-  ( -.  G  e.  _V  ->  ( od `  G )  =  (/) )
22 fvprc 5722 . . . . . . 7  |-  ( -.  G  e.  _V  ->  (
Base `  G )  =  (/) )
233, 22syl5eq 2480 . . . . . 6  |-  ( -.  G  e.  _V  ->  X  =  (/) )
2423mpteq1d 4290 . . . . 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 5572 . . . . 5  |-  ( x  e.  (/)  |->  [_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )  =  (/)
2624, 25syl6eq 2484 . . . 4  |-  ( -.  G  e.  _V  ->  ( x  e.  X  |->  [_ { y  e.  NN  |  ( y  .x.  x )  =  .0. 
}  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) ) )  =  (/) )
2721, 26eqtr4d 2471 . . 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 158 . 2  |-  ( od
`  G )  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
291, 28eqtri 2456 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 1652    e. wcel 1725   {crab 2709   _Vcvv 2956   [_csb 3251   (/)c0 3628   ifcif 3739    e. cmpt 4266   `'ccnv 4877   ` cfv 5454  (class class class)co 6081   supcsup 7445   RRcr 8989   0cc0 8990    < clt 9120   NNcn 10000   Basecbs 13469   0gc0g 13723  .gcmg 14689   odcod 15163
This theorem is referenced by:  odval  15172  odf  15175
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-od 15167
  Copyright terms: Public domain W3C validator