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Theorem odval 14865
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 5882 . . . . . . 7  |-  ( x  =  A  ->  (
y  .x.  x )  =  ( y  .x.  A ) )
21eqeq1d 2304 . . . . . 6  |-  ( x  =  A  ->  (
( y  .x.  x
)  =  .0.  <->  ( y  .x.  A )  =  .0.  ) )
32rabbidv 2793 . . . . 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 2346 . . . 4  |-  ( x  =  A  ->  { y  e.  NN  |  ( y  .x.  x )  =  .0.  }  =  I )
65csbeq1d 3100 . . 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 9768 . . . . . 6  |-  NN  e.  _V
87rabex 4181 . . . . 5  |-  { y  e.  NN  |  ( y  .x.  A )  =  .0.  }  e.  _V
94, 8eqeltri 2366 . . . 4  |-  I  e. 
_V
10 nfcv 2432 . . . 4  |-  F/_ i if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) )
11 eqeq1 2302 . . . . 5  |-  ( i  =  I  ->  (
i  =  (/)  <->  I  =  (/) ) )
12 supeq1 7214 . . . . 5  |-  ( i  =  I  ->  sup ( i ,  RR ,  `'  <  )  =  sup ( I ,  RR ,  `'  <  ) )
1311, 12ifbieq2d 3598 . . . 4  |-  ( i  =  I  ->  if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
149, 10, 13csbief 3135 . . 3  |-  [_ I  /  i ]_ if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )
156, 14syl6eq 2344 . 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 14864 . 2  |-  O  =  ( x  e.  X  |-> 
[_ { y  e.  NN  |  ( y 
.x.  x )  =  .0.  }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
21 c0ex 8848 . . 3  |-  0  e.  _V
22 ltso 8919 . . . . 5  |-  <  Or  RR
23 cnvso 5230 . . . . 5  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
2422, 23mpbi 199 . . . 4  |-  `'  <  Or  RR
2524supex 7230 . . 3  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
2621, 25ifex 3636 . 2  |-  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V
2715, 20, 26fvmpt 5618 1  |-  ( A  e.  X  ->  ( O `  A )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   [_csb 3094   (/)c0 3468   ifcif 3578    Or wor 4329   `'ccnv 4704   ` cfv 5271  (class class class)co 5874   supcsup 7209   RRcr 8752   0cc0 8753    < clt 8883   NNcn 9762   Basecbs 13164   0gc0g 13416  .gcmg 14382   odcod 14856
This theorem is referenced by:  odlem1  14866  odlem2  14870  submod  14896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-i2m1 8821  ax-1ne0 8822  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-nn 9763  df-od 14860
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