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Theorem gexval 15217
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 15173 . . . 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 10011 . . . . . 6  |-  NN  e.  _V
54rabex 4357 . . . . 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 449 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  g  =  G )
87fveq2d 5735 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  ( Base `  G ) )
9 gexval.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
108, 9syl6eqr 2488 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( Base `  g
)  =  X )
117fveq2d 5735 . . . . . . . . . . . . . 14  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  =  (.g `  G ) )
12 gexval.2 . . . . . . . . . . . . . 14  |-  .x.  =  (.g
`  G )
1311, 12syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  (.g `  g )  = 
.x.  )
1413oveqd 6101 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( y (.g `  g
) x )  =  ( y  .x.  x
) )
157fveq2d 5735 . . . . . . . . . . . . 13  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  ( 0g
`  G ) )
16 gexval.3 . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  G )
1715, 16syl6eqr 2488 . . . . . . . . . . . 12  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( 0g `  g
)  =  .0.  )
1814, 17eqeq12d 2452 . . . . . . . . . . 11  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( ( y (.g `  g ) x )  =  ( 0g `  g )  <->  ( y  .x.  x )  =  .0.  ) )
1910, 18raleqbidv 2918 . . . . . . . . . 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 2950 . . . . . . . . 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 2488 . . . . . . . 8  |-  ( ( G  e.  V  /\  g  =  G )  ->  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) }  =  I )
2322eqeq2d 2449 . . . . . . 7  |-  ( ( G  e.  V  /\  g  =  G )  ->  ( i  =  {
y  e.  NN  |  A. x  e.  ( Base `  g ) ( y (.g `  g ) x )  =  ( 0g
`  g ) }  <-> 
i  =  I ) )
2423biimpa 472 . . . . . 6  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
i  =  I )
2524eqeq1d 2446 . . . . 5  |-  ( ( ( G  e.  V  /\  g  =  G
)  /\  i  =  { y  e.  NN  |  A. x  e.  (
Base `  g )
( y (.g `  g
) x )  =  ( 0g `  g
) } )  -> 
( i  =  (/)  <->  I  =  (/) ) )
2624supeq1d 7454 . . . . 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 3761 . . . 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 3295 . . 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 2966 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
30 c0ex 9090 . . . . 5  |-  0  e.  _V
31 ltso 9161 . . . . . . 7  |-  <  Or  RR
32 cnvso 5414 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
3331, 32mpbi 201 . . . . . 6  |-  `'  <  Or  RR
3433supex 7471 . . . . 5  |-  sup (
I ,  RR ,  `'  <  )  e.  _V
3530, 34ifex 3799 . . . 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 5813 . 2  |-  ( G  e.  V  ->  (gEx `  G )  =  if ( I  =  (/) ,  0 ,  sup (
I ,  RR ,  `'  <  ) ) )
381, 37syl5eq 2482 1  |-  ( G  e.  V  ->  E  =  if ( I  =  (/) ,  0 ,  sup ( I ,  RR ,  `'  <  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   _Vcvv 2958   [_csb 3253   (/)c0 3630   ifcif 3741    e. cmpt 4269    Or wor 4505   `'ccnv 4880   ` cfv 5457  (class class class)co 6084   supcsup 7448   RRcr 8994   0cc0 8995    < clt 9125   NNcn 10005   Basecbs 13474   0gc0g 13728  .gcmg 14694  gExcgex 15169
This theorem is referenced by:  gexlem1  15218  gexlem2  15221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-i2m1 9063  ax-1ne0 9064  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-sup 7449  df-pnf 9127  df-mnf 9128  df-ltxr 9130  df-nn 10006  df-gex 15173
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