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Theorem gexlem2OLD 17291
Description: Any positive annihilator of all the group elements is an upper bound on the group exponent. (Contributed by Mario Carneiro, 24-Apr-2016.) Obsolete version of gexlem2 17288 as of 26-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gexlem2OLD.1  |-  X  =  ( Base `  G
)
gexlem2OLD.2  |-  E  =  (gEx `  G )
gexlem2OLD.3  |-  .x.  =  (.g
`  G )
gexlem2OLD.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
gexlem2OLD  |-  ( ( G  e.  V  /\  N  e.  NN  /\  A. x  e.  X  ( N  .x.  x )  =  .0.  )  ->  E  e.  ( 1 ... N
) )
Distinct variable groups:    x, E    x, G    x, N    x, V    x, X    x,  .0.    x, 
.x.

Proof of Theorem gexlem2OLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 oveq1 6327 . . . . . 6  |-  ( y  =  N  ->  (
y  .x.  x )  =  ( N  .x.  x ) )
21eqeq1d 2464 . . . . 5  |-  ( y  =  N  ->  (
( y  .x.  x
)  =  .0.  <->  ( N  .x.  x )  =  .0.  ) )
32ralbidv 2839 . . . 4  |-  ( y  =  N  ->  ( A. x  e.  X  ( y  .x.  x
)  =  .0.  <->  A. x  e.  X  ( N  .x.  x )  =  .0.  ) )
43elrab 3208 . . 3  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  <->  ( N  e.  NN  /\  A. x  e.  X  ( N  .x.  x )  =  .0.  ) )
5 gexlem2OLD.1 . . . . . 6  |-  X  =  ( Base `  G
)
6 gexlem2OLD.3 . . . . . 6  |-  .x.  =  (.g
`  G )
7 gexlem2OLD.4 . . . . . 6  |-  .0.  =  ( 0g `  G )
8 gexlem2OLD.2 . . . . . 6  |-  E  =  (gEx `  G )
9 eqid 2462 . . . . . 6  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  =  {
y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }
105, 6, 7, 8, 9gexvalOLD 17284 . . . . 5  |-  ( G  e.  V  ->  E  =  if ( { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  =  (/) ,  0 ,  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  ) ) )
11 ne0i 3749 . . . . . 6  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  =/=  (/) )
12 ifnefalse 3905 . . . . . 6  |-  ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  =/=  (/)  ->  if ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  =  (/) ,  0 ,  sup ( { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  } ,  RR ,  `'  <  ) )  =  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  ) )
1311, 12syl 17 . . . . 5  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  if ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  =  (/) ,  0 ,  sup ( { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  } ,  RR ,  `'  <  ) )  =  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  ) )
1410, 13sylan9eq 2516 . . . 4  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  E  =  sup ( { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  } ,  RR ,  `'  <  ) )
15 ssrab2 3526 . . . . . 6  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
16 nnuz 11228 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1715, 16sseqtri 3476 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  ( ZZ>=
`  1 )
1811adantl 472 . . . . . . 7  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  =/=  (/) )
19 infmssuzclOLD 11281 . . . . . . 7  |-  ( ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  C_  ( ZZ>= `  1 )  /\  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  =/=  (/) )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )
2017, 18, 19sylancr 674 . . . . . 6  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )
2115, 20sseldi 3442 . . . . 5  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  NN )
22 infmssuzleOLD 11280 . . . . . . 7  |-  ( ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  }  C_  ( ZZ>= `  1 )  /\  N  e.  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  } )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} ,  RR ,  `'  <  )  <_  N
)
2317, 22mpan 681 . . . . . 6  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  <_  N )
2423adantl 472 . . . . 5  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  <_  N )
25 elrabi 3205 . . . . . . . 8  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  N  e.  NN )
2625nnzd 11073 . . . . . . 7  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  N  e.  ZZ )
27 fznn 11898 . . . . . . 7  |-  ( N  e.  ZZ  ->  ( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} ,  RR ,  `'  <  )  e.  ( 1 ... N )  <-> 
( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  NN  /\  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  <_  N ) ) )
2826, 27syl 17 . . . . . 6  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  ( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  ( 1 ... N )  <->  ( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  NN  /\  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  <_  N ) ) )
2928adantl 472 . . . . 5  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  ( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} ,  RR ,  `'  <  )  e.  ( 1 ... N )  <-> 
( sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  NN  /\  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  <_  N ) ) )
3021, 24, 29mpbir2and 938 . . . 4  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  sup ( { y  e.  NN  |  A. x  e.  X  ( y  .x.  x
)  =  .0.  } ,  RR ,  `'  <  )  e.  ( 1 ... N ) )
3114, 30eqeltrd 2540 . . 3  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  E  e.  ( 1 ... N
) )
324, 31sylan2br 483 . 2  |-  ( ( G  e.  V  /\  ( N  e.  NN  /\ 
A. x  e.  X  ( N  .x.  x )  =  .0.  ) )  ->  E  e.  ( 1 ... N ) )
33323impb 1211 1  |-  ( ( G  e.  V  /\  N  e.  NN  /\  A. x  e.  X  ( N  .x.  x )  =  .0.  )  ->  E  e.  ( 1 ... N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   A.wral 2749   {crab 2753    C_ wss 3416   (/)c0 3743   ifcif 3893   class class class wbr 4418   `'ccnv 4855   ` cfv 5605  (class class class)co 6320   supcsup 7985   RRcr 9569   0cc0 9570   1c1 9571    < clt 9706    <_ cle 9707   NNcn 10642   ZZcz 10971   ZZ>=cuz 11193   ...cfz 11819   Basecbs 15176   0gc0g 15393  .gcmg 16727  gExcgexold 17223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615  ax-cnex 9626  ax-resscn 9627  ax-1cn 9628  ax-icn 9629  ax-addcl 9630  ax-addrcl 9631  ax-mulcl 9632  ax-mulrcl 9633  ax-mulcom 9634  ax-addass 9635  ax-mulass 9636  ax-distr 9637  ax-i2m1 9638  ax-1ne0 9639  ax-1rid 9640  ax-rnegex 9641  ax-rrecex 9642  ax-cnre 9643  ax-pre-lttri 9644  ax-pre-lttrn 9645  ax-pre-ltadd 9646  ax-pre-mulgt0 9647
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-tr 4514  df-eprel 4767  df-id 4771  df-po 4777  df-so 4778  df-fr 4815  df-we 4817  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-pred 5403  df-ord 5449  df-on 5450  df-lim 5451  df-suc 5452  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-riota 6282  df-ov 6323  df-oprab 6324  df-mpt2 6325  df-om 6725  df-1st 6825  df-2nd 6826  df-wrecs 7059  df-recs 7121  df-rdg 7159  df-er 7394  df-en 7601  df-dom 7602  df-sdom 7603  df-sup 7987  df-pnf 9708  df-mnf 9709  df-xr 9710  df-ltxr 9711  df-le 9712  df-sub 9893  df-neg 9894  df-nn 10643  df-n0 10904  df-z 10972  df-uz 11194  df-fz 11820  df-gexOLD 17230
This theorem is referenced by: (None)
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