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Theorem gexval 14889
Description: Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.)
Hypotheses
Ref Expression
gexval.1  |-  X  =  ( Base `  G
)
gexval.2  |-  .x.  =  (.g
`  G )
gexval.3  |-  .0.  =  ( 0g `  G )
gexval.4  |-  E  =  (gEx `  G )
gexval.i  |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
Assertion
Ref Expression
gexval  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Distinct variable groups:    x, y,  .0.    x, G, y    x, V, y    x,  .x. , y    x, X
Allowed substitution hints:    E( x, y)    I( x, y)    X( y)

Proof of Theorem gexval
Dummy variables  g 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gexval.4 . 2  |-  E  =  (gEx `  G )
2 df-gex 14845 . . . 4  |- gEx  =  ( g  e.  _V  |->  [_ { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) )
32a1i 10 . . 3  |-  ( G  e.  V  -> gEx  =  ( g  e.  _V  |->  [_ { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  /  i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) ) ) )
4 nnex 9752 . . . . . 6  |-  NN  e.  _V
54rabex 4165 . . . . 5  |-  { y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  e.  _V
65a1i 10 . . . 4  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  e.  _V )
7 simpr 447 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  g  =  G )
87fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
9 gexval.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
108, 9syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  X )
117fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  =  (.g `  G ) )
12 gexval.2 . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
1311, 12syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  = 
.x.  )
1413oveqd 5875 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( y (.g `  g
) x )  =  ( y  .x.  x
) )
157fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  ( 0g
`  G ) )
16 gexval.3 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  G )
1715, 16syl6eqr 2333 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  .0.  )
1814, 17eqeq12d 2297 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( ( y (.g `  g ) x )  =  ( 0g `  g )  <->  ( y  .x.  x )  =  .0.  ) )
1910, 18raleqbidv 2748 . . . . . . . . . 10  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( A. x  e.  ( Base `  g
) ( y (.g `  g ) x )  =  ( 0g `  g )  <->  A. x  e.  X  ( y  .x.  x )  =  .0.  ) )
2019rabbidv 2780 . . . . . . . . 9  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  {
y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
)
21 gexval.i . . . . . . . . 9  |-  I  =  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
2220, 21syl6eqr 2333 . . . . . . . 8  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  I )
2322eqeq2d 2294 . . . . . . 7  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( i  =  {
y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  <-> 
i  =  I ) )
2423biimpa 470 . . . . . 6  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
i  =  I )
2524eqeq1d 2291 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
( i  =  (/)  <->  I  =  (/) ) )
2624supeq1d 7199 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  ->  sup ( i ,  RR ,  `'  <  )  =  sup ( I ,  RR ,  `'  <  ) )
2725, 26ifbieq2d 3585 . . . 4  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  ->  if ( i  =  (/) ,  0 ,  sup (
i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
286, 27csbied 3123 . . 3  |-  ( ( G  e.  V  /\  g  =  G )  ->  [_ { y  e.  NN  |  A. x  e.  ( Base `  g
) ( y (.g `  g ) x )  =  ( 0g `  g ) }  / 
i ]_ if ( i  =  (/) ,  0 ,  sup ( i ,  RR ,  `'  <  ) )  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
29 elex 2796 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 c0ex 8832 . . . . 5  |-  0  e.  _V
31 ltso 8903 . . . . . . 7  |-  <  Or  RR
32 cnvso 5214 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
3331, 32mpbi 199 . . . . . 6  |-  `'  <  Or  RR
3433supex 7214 . . . . 5  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
3530, 34ifex 3623 . . . 4  |-  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V
3635a1i 10 . . 3  |-  ( G  e.  V  ->  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V )
373, 28, 29, 36fvmptd 5606 . 2  |-  ( G  e.  V  ->  (gEx `  G )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) )
381, 37syl5eq 2327 1  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   [_csb 3081   (/)c0 3455   ifcif 3565    e. cmpt 4077    Or wor 4313   `'ccnv 4688   ` cfv 5255  (class class class)co 5858   supcsup 7193   RRcr 8736   0cc0 8737    < clt 8867   NNcn 9746   Basecbs 13148   0gc0g 13400  .gcmg 14366  gExcgex 14841
This theorem is referenced by:  gexlem1  14890  gexlem2  14893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-i2m1 8805  ax-1ne0 8806  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-nn 9747  df-gex 14845
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