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Theorem gexlem2OLD 17235
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 17232 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 6312 . . . . . 6  |-  ( y  =  N  ->  (
y  .x.  x )  =  ( N  .x.  x ) )
21eqeq1d 2424 . . . . 5  |-  ( y  =  N  ->  (
( y  .x.  x
)  =  .0.  <->  ( N  .x.  x )  =  .0.  ) )
32ralbidv 2861 . . . 4  |-  ( y  =  N  ->  ( A. x  e.  X  ( y  .x.  x
)  =  .0.  <->  A. x  e.  X  ( N  .x.  x )  =  .0.  ) )
43elrab 3228 . . 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 2422 . . . . . 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 17228 . . . . 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 3767 . . . . . 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 3923 . . . . . 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 2483 . . . 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 3546 . . . . . 6  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  NN
16 nnuz 11201 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
1715, 16sseqtri 3496 . . . . . . 7  |-  { y  e.  NN  |  A. x  e.  X  (
y  .x.  x )  =  .0.  }  C_  ( ZZ>=
`  1 )
1811adantl 467 . . . . . . 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 11254 . . . . . . 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 667 . . . . . 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 3462 . . . . 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 11253 . . . . . . 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 674 . . . . . 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 467 . . . . 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 3225 . . . . . . . 8  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  N  e.  NN )
2625nnzd 11046 . . . . . . 7  |-  ( N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
}  ->  N  e.  ZZ )
27 fznn 11870 . . . . . . 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 467 . . . . 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 930 . . . 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 2507 . . 3  |-  ( ( G  e.  V  /\  N  e.  { y  e.  NN  |  A. x  e.  X  ( y  .x.  x )  =  .0. 
} )  ->  E  e.  ( 1 ... N
) )
324, 31sylan2br 478 . 2  |-  ( ( G  e.  V  /\  ( N  e.  NN  /\ 
A. x  e.  X  ( N  .x.  x )  =  .0.  ) )  ->  E  e.  ( 1 ... N ) )
33323impb 1201 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   {crab 2775    C_ wss 3436   (/)c0 3761   ifcif 3911   class class class wbr 4423   `'ccnv 4852   ` cfv 5601  (class class class)co 6305   supcsup 7963   RRcr 9545   0cc0 9546   1c1 9547    < clt 9682    <_ cle 9683   NNcn 10616   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791   Basecbs 15120   0gc0g 15337  .gcmg 16671  gExcgexold 17167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-gexOLD 17174
This theorem is referenced by: (None)
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