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Theorem odval 15100
Description: Second substitution for the group order definition. (Contributed by Mario Carneiro, 13-Jul-2014.) (Revised by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
odval.1  |-  X  =  ( Base `  G
)
odval.2  |-  .x.  =  (.g
`  G )
odval.3  |-  .0.  =  ( 0g `  G )
odval.4  |-  O  =  ( od `  G
)
odval.i  |-  I  =  { y  e.  NN  |  ( y  .x.  A )  =  .0. 
}
Assertion
Ref Expression
odval  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Distinct variable groups:    y, A    y, G    y,  .x.    y,  .0.
Allowed substitution hints:    I( y)    O( y)    X( y)

Proof of Theorem odval
Dummy variables  i  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6029 . . . . . . 7  |-  ( x  =  A  ->  (
y  .x.  x )  =  ( y  .x.  A ) )
21eqeq1d 2396 . . . . . 6  |-  ( x  =  A  ->  (
( y  .x.  x
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
32rabbidv 2892 . . . . 5  |-  ( x  =  A  ->  { y  e.  NN  |  ( y  .x.  x )  =  .0.  }  =  { y  e.  NN  |  ( y  .x.  A )  =  .0. 
} )
4 odval.i . . . . 5  |-  I  =  { y  e.  NN  |  ( y  .x.  A )  =  .0. 
}
53, 4syl6eqr 2438 . . . 4  |-  ( x  =  A  ->  { y  e.  NN  |  ( y  .x.  x )  =  .0.  }  =  I )
65csbeq1d 3201 . . 3  |-  ( x  =  A  ->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  = 
[_ I  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
7 nnex 9939 . . . . . 6  |-  NN  e.  _V
87rabex 4296 . . . . 5  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  e.  _V
94, 8eqeltri 2458 . . . 4  |-  I  e. 
_V
10 nfcv 2524 . . . 4  |-  F/_ i if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) )
11 eqeq1 2394 . . . . 5  |-  ( i  =  I  ->  (
i  =  (/)  <->  I  =  (/) ) )
12 supeq1 7386 . . . . 5  |-  ( i  =  I  ->  sup ( i ,  RR ,  `'  <  )  =  sup ( I ,  RR ,  `'  <  ) )
1311, 12ifbieq2d 3703 . . . 4  |-  ( i  =  I  ->  if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
149, 10, 13csbief 3236 . . 3  |-  [_ I  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )
156, 14syl6eq 2436 . 2  |-  ( x  =  A  ->  [_ {
y  e.  NN  | 
( y  .x.  x
)  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
16 odval.1 . . 3  |-  X  =  ( Base `  G
)
17 odval.2 . . 3  |-  .x.  =  (.g
`  G )
18 odval.3 . . 3  |-  .0.  =  ( 0g `  G )
19 odval.4 . . 3  |-  O  =  ( od `  G
)
2016, 17, 18, 19odfval 15099 . 2  |-  O  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
21 c0ex 9019 . . 3  |-  0  e.  _V
22 ltso 9090 . . . . 5  |-  <  Or  RR
23 cnvso 5352 . . . . 5  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
2422, 23mpbi 200 . . . 4  |-  `'  <  Or  RR
2524supex 7402 . . 3  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
2621, 25ifex 3741 . 2  |-  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V
2715, 20, 26fvmpt 5746 1  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900   [_csb 3195   (/)c0 3572   ifcif 3683    Or wor 4444   `'ccnv 4818   ` cfv 5395  (class class class)co 6021   supcsup 7381   RRcr 8923   0cc0 8924    < clt 9054   NNcn 9933   Basecbs 13397   0gc0g 13651  .gcmg 14617   odcod 15091
This theorem is referenced by:  odlem1  15101  odlem2  15105  submod  15131  ofldchr  24071
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-i2m1 8992  ax-1ne0 8993  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-sup 7382  df-pnf 9056  df-mnf 9057  df-ltxr 9059  df-nn 9934  df-od 15095
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