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Theorem gexval 14988
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 14944 . . . 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 9842 . . . . . 6  |-  NN  e.  _V
54rabex 4246 . . . . 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 5612 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
9 gexval.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
108, 9syl6eqr 2408 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  X )
117fveq2d 5612 . . . . . . . . . . . . . 14  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  =  (.g `  G ) )
12 gexval.2 . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
1311, 12syl6eqr 2408 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  = 
.x.  )
1413oveqd 5962 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( y (.g `  g
) x )  =  ( y  .x.  x
) )
157fveq2d 5612 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  ( 0g
`  G ) )
16 gexval.3 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  G )
1715, 16syl6eqr 2408 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  .0.  )
1814, 17eqeq12d 2372 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( ( y (.g `  g ) x )  =  ( 0g `  g )  <->  ( y  .x.  x )  =  .0.  ) )
1910, 18raleqbidv 2824 . . . . . . . . . 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 2856 . . . . . . . . 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 2408 . . . . . . . 8  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  I )
2322eqeq2d 2369 . . . . . . 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 2366 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
( i  =  (/)  <->  I  =  (/) ) )
2624supeq1d 7289 . . . . 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 3661 . . . 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 3199 . . 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 2872 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 c0ex 8922 . . . . 5  |-  0  e.  _V
31 ltso 8993 . . . . . . 7  |-  <  Or  RR
32 cnvso 5296 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
3331, 32mpbi 199 . . . . . 6  |-  `'  <  Or  RR
3433supex 7304 . . . . 5  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
3530, 34ifex 3699 . . . 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 5689 . 2  |-  ( G  e.  V  ->  (gEx `  G )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) )
381, 37syl5eq 2402 1  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864   [_csb 3157   (/)c0 3531   ifcif 3641    e. cmpt 4158    Or wor 4395   `'ccnv 4770   ` cfv 5337  (class class class)co 5945   supcsup 7283   RRcr 8826   0cc0 8827    < clt 8957   NNcn 9836   Basecbs 13245   0gc0g 13499  .gcmg 14465  gExcgex 14940
This theorem is referenced by:  gexlem1  14989  gexlem2  14992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-i2m1 8895  ax-1ne0 8896  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-sup 7284  df-pnf 8959  df-mnf 8960  df-ltxr 8962  df-nn 9837  df-gex 14944
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