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Theorem gexex 16335
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 16040 . . . . . . 7  |-  O : X
--> NN0
4 frn 5565 . . . . . . 7  |-  ( O : X --> NN0  ->  ran 
O  C_  NN0 )
53, 4ax-mp 5 . . . . . 6  |-  ran  O  C_ 
NN0
6 nn0ssz 10667 . . . . . 6  |-  NN0  C_  ZZ
75, 6sstri 3365 . . . . 5  |-  ran  O  C_  ZZ
87a1i 11 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O 
C_  ZZ )
9 ablgrp 16282 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
109adantr 465 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  G  e.  Grp )
111grpbn0 15567 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
1210, 11syl 16 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  X  =/=  (/) )
133fdmi 5564 . . . . . . . 8  |-  dom  O  =  X
1413eqeq1i 2450 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  X  =  (/) )
15 dm0rn0 5056 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  ran  O  =  (/) )
1614, 15bitr3i 251 . . . . . 6  |-  ( X  =  (/)  <->  ran  O  =  (/) )
1716necon3bii 2640 . . . . 5  |-  ( X  =/=  (/)  <->  ran  O  =/=  (/) )
1812, 17sylib 196 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O  =/=  (/) )
19 nnz 10668 . . . . . 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 16085 . . . . . . . . 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 16039 . . . . . . . . . . 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 10745 . . . . . . . . 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 13578 . . . . . . . . 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 2799 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. x  e.  X  ( O `  x )  <_  E
)
32 ffn 5559 . . . . . . . 8  |-  ( O : X --> NN0  ->  O  Fn  X )
333, 32ax-mp 5 . . . . . . 7  |-  O  Fn  X
34 breq1 4295 . . . . . . . 8  |-  ( y  =  ( O `  x )  ->  (
y  <_  E  <->  ( O `  x )  <_  E
) )
3534ralrn 5846 . . . . . . 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 4296 . . . . . . 7  |-  ( n  =  E  ->  (
y  <_  n  <->  y  <_  E ) )
3938ralbidv 2735 . . . . . 6  |-  ( n  =  E  ->  ( A. y  e.  ran  O  y  <_  n  <->  A. y  e.  ran  O  y  <_  E ) )
4039rspcev 3073 . . . . 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 10945 . . . 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 1218 . . 3  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  sup ( ran  O ,  RR ,  <  )  e.  ran  O )
44 fvelrnb 5739 . . . 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 5840 . . . . . . . 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 10946 . . . . . . 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 1218 . . . . . 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 4318 . . . . 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 16334 . . . 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 2828 . 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 1369    e. wcel 1756    =/= wne 2606   A.wral 2715   E.wrex 2716    C_ wss 3328   (/)c0 3637   class class class wbr 4292   dom cdm 4840   ran crn 4841    Fn wfn 5413   -->wf 5414   ` cfv 5418   supcsup 7690   RRcr 9281    < clt 9418    <_ cle 9419   NNcn 10322   NN0cn0 10579   ZZcz 10646    || cdivides 13535   Basecbs 14174   Grpcgrp 15410   odcod 16028  gExcgex 16029   Abelcabel 16278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-q 10954  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-prm 13764  df-pc 13904  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-od 16032  df-gex 16033  df-cmn 16279  df-abl 16280
This theorem is referenced by:  cyggexb  16375  pgpfaclem3  16584
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