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Theorem gexval 15175
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 15131 . . . 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 9970 . . . . . 6  |-  NN  e.  _V
54rabex 4322 . . . . 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 448 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  g  =  G )
87fveq2d 5699 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
9 gexval.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
108, 9syl6eqr 2462 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  X )
117fveq2d 5699 . . . . . . . . . . . . . 14  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  =  (.g `  G ) )
12 gexval.2 . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
1311, 12syl6eqr 2462 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  = 
.x.  )
1413oveqd 6065 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( y (.g `  g
) x )  =  ( y  .x.  x
) )
157fveq2d 5699 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  ( 0g
`  G ) )
16 gexval.3 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  G )
1715, 16syl6eqr 2462 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  .0.  )
1814, 17eqeq12d 2426 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( ( y (.g `  g ) x )  =  ( 0g `  g )  <->  ( y  .x.  x )  =  .0.  ) )
1910, 18raleqbidv 2884 . . . . . . . . . 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 2916 . . . . . . . . 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 2462 . . . . . . . 8  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  I )
2322eqeq2d 2423 . . . . . . 7  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( i  =  {
y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  <-> 
i  =  I ) )
2423biimpa 471 . . . . . 6  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
i  =  I )
2524eqeq1d 2420 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
( i  =  (/)  <->  I  =  (/) ) )
2624supeq1d 7417 . . . . 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 3727 . . . 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 3261 . . 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 2932 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 c0ex 9049 . . . . 5  |-  0  e.  _V
31 ltso 9120 . . . . . . 7  |-  <  Or  RR
32 cnvso 5378 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
3331, 32mpbi 200 . . . . . 6  |-  `'  <  Or  RR
3433supex 7432 . . . . 5  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
3530, 34ifex 3765 . . . 4  |-  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V
3635a1i 11 . . 3  |-  ( G  e.  V  ->  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) )  e.  _V )
373, 28, 29, 36fvmptd 5777 . 2  |-  ( G  e.  V  ->  (gEx `  G )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) )
381, 37syl5eq 2456 1  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   {crab 2678   _Vcvv 2924   [_csb 3219   (/)c0 3596   ifcif 3707    e. cmpt 4234    Or wor 4470   `'ccnv 4844   ` cfv 5421  (class class class)co 6048   supcsup 7411   RRcr 8953   0cc0 8954    < clt 9084   NNcn 9964   Basecbs 13432   0gc0g 13686  .gcmg 14652  gExcgex 15127
This theorem is referenced by:  gexlem1  15176  gexlem2  15179
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-i2m1 9022  ax-1ne0 9023  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-sup 7412  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-nn 9965  df-gex 15131
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