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Theorem gexval 16471
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 16427 . . . 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 11 . . 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 10554 . . . . . 6  |-  NN  e.  _V
54rabex 4604 . . . . 5  |-  { y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  e.  _V
65a1i 11 . . . 4  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  e.  _V )
7 simpr 461 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  g  =  G )
87fveq2d 5876 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
9 gexval.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
108, 9syl6eqr 2526 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  X )
117fveq2d 5876 . . . . . . . . . . . . . 14  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  =  (.g `  G ) )
12 gexval.2 . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
1311, 12syl6eqr 2526 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  = 
.x.  )
1413oveqd 6312 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( y (.g `  g
) x )  =  ( y  .x.  x
) )
157fveq2d 5876 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  ( 0g
`  G ) )
16 gexval.3 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  G )
1715, 16syl6eqr 2526 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  .0.  )
1814, 17eqeq12d 2489 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( ( y (.g `  g ) x )  =  ( 0g `  g )  <->  ( y  .x.  x )  =  .0.  ) )
1910, 18raleqbidv 3077 . . . . . . . . . 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 3110 . . . . . . . . 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 2526 . . . . . . . 8  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  I )
2322eqeq2d 2481 . . . . . . 7  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( i  =  {
y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  <-> 
i  =  I ) )
2423biimpa 484 . . . . . 6  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
i  =  I )
2524eqeq1d 2469 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
( i  =  (/)  <->  I  =  (/) ) )
2624supeq1d 7918 . . . . 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 3970 . . . 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 3467 . . 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 3127 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 c0ex 9602 . . . . 5  |-  0  e.  _V
31 gtso 9678 . . . . . 6  |-  `'  <  Or  RR
3231supex 7935 . . . . 5  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
3330, 32ifex 4014 . . . 4  |-  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V
3433a1i 11 . . 3  |-  ( G  e.  V  ->  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V )
353, 28, 29, 34fvmptd 5962 . 2  |-  ( G  e.  V  ->  (gEx `  G )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) )
361, 35syl5eq 2520 1  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   _Vcvv 3118   [_csb 3440   (/)c0 3790   ifcif 3945    |-> cmpt 4511   `'ccnv 5004   ` cfv 5594  (class class class)co 6295   supcsup 7912   RRcr 9503   0cc0 9504    < clt 9640   NNcn 10548   Basecbs 14507   0gc0g 14712  .gcmg 15928  gExcgex 16423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-i2m1 9572  ax-1ne0 9573  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-om 6696  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-ltxr 9645  df-nn 10549  df-gex 16427
This theorem is referenced by:  gexlem1  16472  gexlem2  16475
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