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Theorem gexcl3 14898
Description: If the order of every group element is bounded by  N, the group has finite exponent. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gexod.1  |-  X  =  ( Base `  G
)
gexod.2  |-  E  =  (gEx `  G )
gexod.3  |-  O  =  ( od `  G
)
Assertion
Ref Expression
gexcl3  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  E  e.  NN )
Distinct variable groups:    x, E    x, G    x, N    x, X
Allowed substitution hint:    O( x)

Proof of Theorem gexcl3
StepHypRef Expression
1 simpl 443 . . 3  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  G  e.  Grp )
2 gexod.1 . . . . . . . 8  |-  X  =  ( Base `  G
)
32grpbn0 14511 . . . . . . 7  |-  ( G  e.  Grp  ->  X  =/=  (/) )
4 r19.2z 3543 . . . . . . 7  |-  ( ( X  =/=  (/)  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N
) )  ->  E. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )
53, 4sylan 457 . . . . . 6  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  E. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )
6 elfzuz2 10801 . . . . . . . 8  |-  ( ( O `  x )  e.  ( 1 ... N )  ->  N  e.  ( ZZ>= `  1 )
)
7 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
86, 7syl6eleqr 2374 . . . . . . 7  |-  ( ( O `  x )  e.  ( 1 ... N )  ->  N  e.  NN )
98rexlimivw 2663 . . . . . 6  |-  ( E. x  e.  X  ( O `  x )  e.  ( 1 ... N )  ->  N  e.  NN )
105, 9syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  N  e.  NN )
1110nnnn0d 10018 . . . 4  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  N  e.  NN0 )
12 faccl 11298 . . . 4  |-  ( N  e.  NN0  ->  ( ! `
 N )  e.  NN )
1311, 12syl 15 . . 3  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  -> 
( ! `  N
)  e.  NN )
14 elfzuzb 10792 . . . . . . . . 9  |-  ( ( O `  x )  e.  ( 1 ... N )  <->  ( ( O `  x )  e.  ( ZZ>= `  1 )  /\  N  e.  ( ZZ>=
`  ( O `  x ) ) ) )
15 elnnuz 10264 . . . . . . . . . 10  |-  ( ( O `  x )  e.  NN  <->  ( O `  x )  e.  (
ZZ>= `  1 ) )
16 dvdsfac 12583 . . . . . . . . . 10  |-  ( ( ( O `  x
)  e.  NN  /\  N  e.  ( ZZ>= `  ( O `  x ) ) )  ->  ( O `  x )  ||  ( ! `  N
) )
1715, 16sylanbr 459 . . . . . . . . 9  |-  ( ( ( O `  x
)  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( O `  x ) ) )  ->  ( O `  x )  ||  ( ! `  N )
)
1814, 17sylbi 187 . . . . . . . 8  |-  ( ( O `  x )  e.  ( 1 ... N )  ->  ( O `  x )  ||  ( ! `  N
) )
1918adantl 452 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  ( O `  x )  ||  ( ! `  N )
)
20 simpll 730 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  G  e.  Grp )
21 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  x  e.  X )
228adantl 452 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  N  e.  NN )
2322nnnn0d 10018 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  N  e.  NN0 )
2423, 12syl 15 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  ( ! `  N )  e.  NN )
2524nnzd 10116 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  ( ! `  N )  e.  ZZ )
26 gexod.3 . . . . . . . . 9  |-  O  =  ( od `  G
)
27 eqid 2283 . . . . . . . . 9  |-  (.g `  G
)  =  (.g `  G
)
28 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
292, 26, 27, 28oddvds 14862 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  ( ! `  N )  e.  ZZ )  -> 
( ( O `  x )  ||  ( ! `  N )  <->  ( ( ! `  N
) (.g `  G ) x )  =  ( 0g
`  G ) ) )
3020, 21, 25, 29syl3anc 1182 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  ( ( O `  x )  ||  ( ! `  N
)  <->  ( ( ! `
 N ) (.g `  G ) x )  =  ( 0g `  G ) ) )
3119, 30mpbid 201 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  X )  /\  ( O `  x )  e.  ( 1 ... N ) )  ->  ( ( ! `  N )
(.g `  G ) x )  =  ( 0g
`  G ) )
3231ex 423 . . . . 5  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( ( O `  x )  e.  ( 1 ... N )  ->  ( ( ! `
 N ) (.g `  G ) x )  =  ( 0g `  G ) ) )
3332ralimdva 2621 . . . 4  |-  ( G  e.  Grp  ->  ( A. x  e.  X  ( O `  x )  e.  ( 1 ... N )  ->  A. x  e.  X  ( ( ! `  N )
(.g `  G ) x )  =  ( 0g
`  G ) ) )
3433imp 418 . . 3  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  A. x  e.  X  ( ( ! `  N ) (.g `  G
) x )  =  ( 0g `  G
) )
35 gexod.2 . . . 4  |-  E  =  (gEx `  G )
362, 35, 27, 28gexlem2 14893 . . 3  |-  ( ( G  e.  Grp  /\  ( ! `  N )  e.  NN  /\  A. x  e.  X  (
( ! `  N
) (.g `  G ) x )  =  ( 0g
`  G ) )  ->  E  e.  ( 1 ... ( ! `
 N ) ) )
371, 13, 34, 36syl3anc 1182 . 2  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  E  e.  ( 1 ... ( ! `  N ) ) )
38 elfznn 10819 . 2  |-  ( E  e.  ( 1 ... ( ! `  N
) )  ->  E  e.  NN )
3937, 38syl 15 1  |-  ( ( G  e.  Grp  /\  A. x  e.  X  ( O `  x )  e.  ( 1 ... N ) )  ->  E  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   1c1 8738   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   !cfa 11288    || cdivides 12531   Basecbs 13148   0gc0g 13400   Grpcgrp 14362  .gcmg 14366   odcod 14840  gExcgex 14841
This theorem is referenced by:  gexcl2  14900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-od 14844  df-gex 14845
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