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Theorem gexex 17478
Description: In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if  E  =  0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so  E is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
Hypotheses
Ref Expression
gexex.1  |-  X  =  ( Base `  G
)
gexex.2  |-  E  =  (gEx `  G )
gexex.3  |-  O  =  ( od `  G
)
Assertion
Ref Expression
gexex  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  E )
Distinct variable groups:    x, E    x, G    x, O    x, X

Proof of Theorem gexex
Dummy variables  y  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gexex.1 . . 3  |-  X  =  ( Base `  G
)
2 gexex.2 . . 3  |-  E  =  (gEx `  G )
3 gexex.3 . . 3  |-  O  =  ( od `  G
)
4 simpll 758 . . 3  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  G  e.  Abel )
5 simplr 760 . . 3  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  E  e.  NN )
6 simprl 762 . . 3  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  x  e.  X )
71, 3odf 17173 . . . . . . . 8  |-  O : X
--> NN0
8 frn 5748 . . . . . . . 8  |-  ( O : X --> NN0  ->  ran 
O  C_  NN0 )
97, 8ax-mp 5 . . . . . . 7  |-  ran  O  C_ 
NN0
10 nn0ssz 10958 . . . . . . 7  |-  NN0  C_  ZZ
119, 10sstri 3473 . . . . . 6  |-  ran  O  C_  ZZ
1211a1i 11 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ran  O 
C_  ZZ )
13 nnz 10959 . . . . . . . 8  |-  ( E  e.  NN  ->  E  e.  ZZ )
1413adantl 467 . . . . . . 7  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E  e.  ZZ )
15 ablgrp 17422 . . . . . . . . . . . 12  |-  ( G  e.  Abel  ->  G  e. 
Grp )
1615adantr 466 . . . . . . . . . . 11  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  G  e.  Grp )
171, 2, 3gexod 17225 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( O `  x
)  ||  E )
1816, 17sylan 473 . . . . . . . . . 10  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  ||  E
)
191, 3odcl 17172 . . . . . . . . . . . . 13  |-  ( x  e.  X  ->  ( O `  x )  e.  NN0 )
2019adantl 467 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  NN0 )
2120nn0zd 11038 . . . . . . . . . . 11  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  ZZ )
22 simplr 760 . . . . . . . . . . 11  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  E  e.  NN )
23 dvdsle 14337 . . . . . . . . . . 11  |-  ( ( ( O `  x
)  e.  ZZ  /\  E  e.  NN )  ->  ( ( O `  x )  ||  E  ->  ( O `  x
)  <_  E )
)
2421, 22, 23syl2anc 665 . . . . . . . . . 10  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( ( O `  x )  ||  E  ->  ( O `
 x )  <_  E ) )
2518, 24mpd 15 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  <_  E
)
2625ralrimiva 2839 . . . . . . . 8  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. x  e.  X  ( O `  x )  <_  E
)
27 ffn 5742 . . . . . . . . . 10  |-  ( O : X --> NN0  ->  O  Fn  X )
287, 27ax-mp 5 . . . . . . . . 9  |-  O  Fn  X
29 breq1 4423 . . . . . . . . . 10  |-  ( y  =  ( O `  x )  ->  (
y  <_  E  <->  ( O `  x )  <_  E
) )
3029ralrn 6036 . . . . . . . . 9  |-  ( O  Fn  X  ->  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
) )
3128, 30ax-mp 5 . . . . . . . 8  |-  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
)
3226, 31sylibr 215 . . . . . . 7  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. y  e.  ran  O  y  <_  E )
33 breq2 4424 . . . . . . . . 9  |-  ( n  =  E  ->  (
y  <_  n  <->  y  <_  E ) )
3433ralbidv 2864 . . . . . . . 8  |-  ( n  =  E  ->  ( A. y  e.  ran  O  y  <_  n  <->  A. y  e.  ran  O  y  <_  E ) )
3534rspcev 3182 . . . . . . 7  |-  ( ( E  e.  ZZ  /\  A. y  e.  ran  O  y  <_  E )  ->  E. n  e.  ZZ  A. y  e.  ran  O  y  <_  n )
3614, 32, 35syl2anc 665 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. n  e.  ZZ  A. y  e. 
ran  O  y  <_  n )
3736ad2antrr 730 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  E. n  e.  ZZ  A. y  e. 
ran  O  y  <_  n )
3828a1i 11 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  O  Fn  X )
39 fnfvelrn 6030 . . . . . 6  |-  ( ( O  Fn  X  /\  y  e.  X )  ->  ( O `  y
)  e.  ran  O
)
4038, 39sylan 473 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  e.  ran  O )
41 suprzub 11255 . . . . 5  |-  ( ( ran  O  C_  ZZ  /\ 
E. n  e.  ZZ  A. y  e.  ran  O  y  <_  n  /\  ( O `  y )  e.  ran  O )  -> 
( O `  y
)  <_  sup ( ran  O ,  RR ,  <  ) )
4212, 37, 40, 41syl3anc 1264 . . . 4  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  <_  sup ( ran  O ,  RR ,  <  )
)
43 simplrr 769 . . . 4  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  x )  =  sup ( ran  O ,  RR ,  <  )
)
4442, 43breqtrrd 4447 . . 3  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  <_  ( O `  x
) )
451, 2, 3, 4, 5, 6, 44gexexlem 17477 . 2  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  ( O `  x )  =  E )
4611a1i 11 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O 
C_  ZZ )
471grpbn0 16682 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
4816, 47syl 17 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  X  =/=  (/) )
497fdmi 5747 . . . . . . . 8  |-  dom  O  =  X
5049eqeq1i 2429 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  X  =  (/) )
51 dm0rn0 5066 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  ran  O  =  (/) )
5250, 51bitr3i 254 . . . . . 6  |-  ( X  =  (/)  <->  ran  O  =  (/) )
5352necon3bii 2692 . . . . 5  |-  ( X  =/=  (/)  <->  ran  O  =/=  (/) )
5448, 53sylib 199 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O  =/=  (/) )
55 suprzcl2 11254 . . . 4  |-  ( ( ran  O  C_  ZZ  /\ 
ran  O  =/=  (/)  /\  E. n  e.  ZZ  A. y  e.  ran  O  y  <_  n )  ->  sup ( ran  O ,  RR ,  <  )  e.  ran  O )
5646, 54, 36, 55syl3anc 1264 . . 3  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  sup ( ran  O ,  RR ,  <  )  e.  ran  O )
57 fvelrnb 5924 . . . 4  |-  ( O  Fn  X  ->  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) ) )
5828, 57ax-mp 5 . . 3  |-  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) )
5956, 58sylib 199 . 2  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  sup ( ran  O ,  RR ,  <  ) )
6045, 59reximddv 2901 1  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776    C_ wss 3436   (/)c0 3761   class class class wbr 4420   dom cdm 4849   ran crn 4850    Fn wfn 5592   -->wf 5593   ` cfv 5597   supcsup 7956   RRcr 9538    < clt 9675    <_ cle 9676   NNcn 10609   NN0cn0 10869   ZZcz 10937    || cdvds 14292   Basecbs 15108   Grpcgrp 16656   odcod 17152  gExcgex 17154   Abelcabl 17418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  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 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-sup 7958  df-inf 7959  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-dvds 14293  df-gcd 14456  df-prm 14610  df-pc 14774  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-grp 16660  df-minusg 16661  df-sbg 16662  df-mulg 16663  df-od 17159  df-gex 17161  df-cmn 17419  df-abl 17420
This theorem is referenced by:  cyggexb  17520  pgpfaclem3  17703
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