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Theorem gexex 16732
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 . . . . . . . 8  |-  X  =  ( Base `  G
)
2 gexex.3 . . . . . . . 8  |-  O  =  ( od `  G
)
31, 2odf 16434 . . . . . . 7  |-  O : X
--> NN0
4 frn 5743 . . . . . . 7  |-  ( O : X --> NN0  ->  ran 
O  C_  NN0 )
53, 4ax-mp 5 . . . . . 6  |-  ran  O  C_ 
NN0
6 nn0ssz 10897 . . . . . 6  |-  NN0  C_  ZZ
75, 6sstri 3518 . . . . 5  |-  ran  O  C_  ZZ
87a1i 11 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O 
C_  ZZ )
9 ablgrp 16676 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
109adantr 465 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  G  e.  Grp )
111grpbn0 15951 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
1210, 11syl 16 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  X  =/=  (/) )
133fdmi 5742 . . . . . . . 8  |-  dom  O  =  X
1413eqeq1i 2474 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  X  =  (/) )
15 dm0rn0 5225 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  ran  O  =  (/) )
1614, 15bitr3i 251 . . . . . 6  |-  ( X  =  (/)  <->  ran  O  =  (/) )
1716necon3bii 2735 . . . . 5  |-  ( X  =/=  (/)  <->  ran  O  =/=  (/) )
1812, 17sylib 196 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O  =/=  (/) )
19 nnz 10898 . . . . . 6  |-  ( E  e.  NN  ->  E  e.  ZZ )
2019adantl 466 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E  e.  ZZ )
21 gexex.2 . . . . . . . . . 10  |-  E  =  (gEx `  G )
221, 21, 2gexod 16479 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( O `  x
)  ||  E )
2310, 22sylan 471 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  ||  E
)
241, 2odcl 16433 . . . . . . . . . . 11  |-  ( x  e.  X  ->  ( O `  x )  e.  NN0 )
2524adantl 466 . . . . . . . . . 10  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  NN0 )
2625nn0zd 10976 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  ZZ )
27 simplr 754 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  E  e.  NN )
28 dvdsle 13907 . . . . . . . . 9  |-  ( ( ( O `  x
)  e.  ZZ  /\  E  e.  NN )  ->  ( ( O `  x )  ||  E  ->  ( O `  x
)  <_  E )
)
2926, 27, 28syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( ( O `  x )  ||  E  ->  ( O `
 x )  <_  E ) )
3023, 29mpd 15 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  <_  E
)
3130ralrimiva 2881 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. x  e.  X  ( O `  x )  <_  E
)
32 ffn 5737 . . . . . . . 8  |-  ( O : X --> NN0  ->  O  Fn  X )
333, 32ax-mp 5 . . . . . . 7  |-  O  Fn  X
34 breq1 4456 . . . . . . . 8  |-  ( y  =  ( O `  x )  ->  (
y  <_  E  <->  ( O `  x )  <_  E
) )
3534ralrn 6035 . . . . . . 7  |-  ( O  Fn  X  ->  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
) )
3633, 35ax-mp 5 . . . . . 6  |-  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
)
3731, 36sylibr 212 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. y  e.  ran  O  y  <_  E )
38 breq2 4457 . . . . . . 7  |-  ( n  =  E  ->  (
y  <_  n  <->  y  <_  E ) )
3938ralbidv 2906 . . . . . 6  |-  ( n  =  E  ->  ( A. y  e.  ran  O  y  <_  n  <->  A. y  e.  ran  O  y  <_  E ) )
4039rspcev 3219 . . . . 5  |-  ( ( E  e.  ZZ  /\  A. y  e.  ran  O  y  <_  E )  ->  E. n  e.  ZZ  A. y  e.  ran  O  y  <_  n )
4120, 37, 40syl2anc 661 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. n  e.  ZZ  A. y  e. 
ran  O  y  <_  n )
42 suprzcl2 11184 . . . 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 )
438, 18, 41, 42syl3anc 1228 . . 3  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  sup ( ran  O ,  RR ,  <  )  e.  ran  O )
44 fvelrnb 5921 . . . 4  |-  ( O  Fn  X  ->  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) ) )
4533, 44ax-mp 5 . . 3  |-  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) )
4643, 45sylib 196 . 2  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  sup ( ran  O ,  RR ,  <  ) )
47 simpll 753 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  G  e.  Abel )
48 simplr 754 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  E  e.  NN )
49 simprl 755 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  x  e.  X )
507a1i 11 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ran  O 
C_  ZZ )
5141ad2antrr 725 . . . . . . 7  |-  ( ( ( ( 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 )
5233a1i 11 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  O  Fn  X )
53 fnfvelrn 6029 . . . . . . . 8  |-  ( ( O  Fn  X  /\  y  e.  X )  ->  ( O `  y
)  e.  ran  O
)
5452, 53sylan 471 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  e.  ran  O )
55 suprzub 11185 . . . . . . 7  |-  ( ( 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 ,  <  ) )
5650, 51, 54, 55syl3anc 1228 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  <_  sup ( ran  O ,  RR ,  <  )
)
57 simplrr 760 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  x )  =  sup ( ran  O ,  RR ,  <  )
)
5856, 57breqtrrd 4479 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ( O `  y )  <_  ( O `  x
) )
591, 21, 2, 47, 48, 49, 58gexexlem 16731 . . . 4  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  ( O `  x )  =  E )
6059expr 615 . . 3  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( ( O `  x )  =  sup ( ran  O ,  RR ,  <  )  ->  ( O `  x
)  =  E ) )
6160reximdva 2942 . 2  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ( E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  )  ->  E. x  e.  X  ( O `  x )  =  E ) )
6246, 61mpd 15 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 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   (/)c0 3790   class class class wbr 4453   dom cdm 5005   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594   supcsup 7912   RRcr 9503    < clt 9640    <_ cle 9641   NNcn 10548   NN0cn0 10807   ZZcz 10876    || cdivides 13864   Basecbs 14507   Grpcgrp 15925   odcod 16422  gExcgex 16423   Abelcabl 16672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-q 11195  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-prm 14094  df-pc 14237  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-od 16426  df-gex 16427  df-cmn 16673  df-abl 16674
This theorem is referenced by:  cyggexb  16774  pgpfaclem3  17006
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