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Theorem gexex 15161
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 14868 . . . . . . 7  |-  O : X
--> NN0
4 frn 5411 . . . . . . 7  |-  ( O : X --> NN0  ->  ran 
O  C_  NN0 )
53, 4ax-mp 8 . . . . . 6  |-  ran  O  C_ 
NN0
6 nn0ssz 10060 . . . . . 6  |-  NN0  C_  ZZ
75, 6sstri 3201 . . . . 5  |-  ran  O  C_  ZZ
87a1i 10 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O 
C_  ZZ )
9 ablgrp 15110 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
109adantr 451 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  G  e.  Grp )
111grpbn0 14527 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
1210, 11syl 15 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  X  =/=  (/) )
133fdmi 5410 . . . . . . . 8  |-  dom  O  =  X
1413eqeq1i 2303 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  X  =  (/) )
15 dm0rn0 4911 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  ran  O  =  (/) )
1614, 15bitr3i 242 . . . . . 6  |-  ( X  =  (/)  <->  ran  O  =  (/) )
1716necon3bii 2491 . . . . 5  |-  ( X  =/=  (/)  <->  ran  O  =/=  (/) )
1812, 17sylib 188 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O  =/=  (/) )
19 nnz 10061 . . . . . 6  |-  ( E  e.  NN  ->  E  e.  ZZ )
2019adantl 452 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E  e.  ZZ )
21 gexex.2 . . . . . . . . . 10  |-  E  =  (gEx `  G )
221, 21, 2gexod 14913 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  X )  ->  ( O `  x
)  ||  E )
2310, 22sylan 457 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  ||  E
)
241, 2odcl 14867 . . . . . . . . . . 11  |-  ( x  e.  X  ->  ( O `  x )  e.  NN0 )
2524adantl 452 . . . . . . . . . 10  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  NN0 )
2625nn0zd 10131 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  e.  ZZ )
27 simplr 731 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  E  e.  NN )
28 dvdsle 12590 . . . . . . . . 9  |-  ( ( ( O `  x
)  e.  ZZ  /\  E  e.  NN )  ->  ( ( O `  x )  ||  E  ->  ( O `  x
)  <_  E )
)
2926, 27, 28syl2anc 642 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( ( O `  x )  ||  E  ->  ( O `
 x )  <_  E ) )
3023, 29mpd 14 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( O `  x )  <_  E
)
3130ralrimiva 2639 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. x  e.  X  ( O `  x )  <_  E
)
32 ffn 5405 . . . . . . . 8  |-  ( O : X --> NN0  ->  O  Fn  X )
333, 32ax-mp 8 . . . . . . 7  |-  O  Fn  X
34 breq1 4042 . . . . . . . 8  |-  ( y  =  ( O `  x )  ->  (
y  <_  E  <->  ( O `  x )  <_  E
) )
3534ralrn 5684 . . . . . . 7  |-  ( O  Fn  X  ->  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
) )
3633, 35ax-mp 8 . . . . . 6  |-  ( A. y  e.  ran  O  y  <_  E  <->  A. x  e.  X  ( O `  x )  <_  E
)
3731, 36sylibr 203 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. y  e.  ran  O  y  <_  E )
38 breq2 4043 . . . . . . 7  |-  ( n  =  E  ->  (
y  <_  n  <->  y  <_  E ) )
3938ralbidv 2576 . . . . . 6  |-  ( n  =  E  ->  ( A. y  e.  ran  O  y  <_  n  <->  A. y  e.  ran  O  y  <_  E ) )
4039rspcev 2897 . . . . 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 642 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. n  e.  ZZ  A. y  e. 
ran  O  y  <_  n )
42 suprzcl2 10324 . . . 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 1182 . . 3  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  sup ( ran  O ,  RR ,  <  )  e.  ran  O )
44 fvelrnb 5586 . . . 4  |-  ( O  Fn  X  ->  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) ) )
4533, 44ax-mp 8 . . 3  |-  ( sup ( ran  O ,  RR ,  <  )  e. 
ran  O  <->  E. x  e.  X  ( O `  x )  =  sup ( ran 
O ,  RR ,  <  ) )
4643, 45sylib 188 . 2  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  sup ( ran  O ,  RR ,  <  ) )
47 simpll 730 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  G  e.  Abel )
48 simplr 731 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  E  e.  NN )
49 simprl 732 . . . . 5  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  x  e.  X )
507a1i 10 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  /\  y  e.  X )  ->  ran  O 
C_  ZZ )
5141ad2antrr 706 . . . . . . 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 10 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  O  Fn  X )
53 fnfvelrn 5678 . . . . . . . 8  |-  ( ( O  Fn  X  /\  y  e.  X )  ->  ( O `  y
)  e.  ran  O
)
5452, 53sylan 457 . . . . . . 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 10325 . . . . . . 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 1182 . . . . . 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 737 . . . . . 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 4065 . . . . 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 15160 . . . 4  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  ( x  e.  X  /\  ( O `
 x )  =  sup ( ran  O ,  RR ,  <  )
) )  ->  ( O `  x )  =  E )
6059expr 598 . . 3  |-  ( ( ( G  e.  Abel  /\  E  e.  NN )  /\  x  e.  X
)  ->  ( ( O `  x )  =  sup ( ran  O ,  RR ,  <  )  ->  ( O `  x
)  =  E ) )
6160reximdva 2668 . 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 14 1  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271   supcsup 7209   RRcr 8752    < clt 8883    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040    || cdivides 12547   Basecbs 13164   Grpcgrp 14378   odcod 14856  gExcgex 14857   Abelcabel 15106
This theorem is referenced by:  cyggexb  15201  pgpfaclem3  15334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-od 14860  df-gex 14861  df-cmn 15107  df-abl 15108
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