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Theorem gexex 15145
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 14852 . . . . . . 7  |-  O : X
--> NN0
4 frn 5395 . . . . . . 7  |-  ( O : X --> NN0  ->  ran 
O  C_  NN0 )
53, 4ax-mp 8 . . . . . 6  |-  ran  O  C_ 
NN0
6 nn0ssz 10044 . . . . . 6  |-  NN0  C_  ZZ
75, 6sstri 3188 . . . . 5  |-  ran  O  C_  ZZ
87a1i 10 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O 
C_  ZZ )
9 ablgrp 15094 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
109adantr 451 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  G  e.  Grp )
111grpbn0 14511 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
1210, 11syl 15 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  X  =/=  (/) )
133fdmi 5394 . . . . . . . 8  |-  dom  O  =  X
1413eqeq1i 2290 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  X  =  (/) )
15 dm0rn0 4895 . . . . . . 7  |-  ( dom 
O  =  (/)  <->  ran  O  =  (/) )
1614, 15bitr3i 242 . . . . . 6  |-  ( X  =  (/)  <->  ran  O  =  (/) )
1716necon3bii 2478 . . . . 5  |-  ( X  =/=  (/)  <->  ran  O  =/=  (/) )
1812, 17sylib 188 . . . 4  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  ran  O  =/=  (/) )
19 nnz 10045 . . . . . 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 14897 . . . . . . . . 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 14851 . . . . . . . . . . 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 10115 . . . . . . . . 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 12574 . . . . . . . . 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 2626 . . . . . 6  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  A. x  e.  X  ( O `  x )  <_  E
)
32 ffn 5389 . . . . . . . 8  |-  ( O : X --> NN0  ->  O  Fn  X )
333, 32ax-mp 8 . . . . . . 7  |-  O  Fn  X
34 breq1 4026 . . . . . . . 8  |-  ( y  =  ( O `  x )  ->  (
y  <_  E  <->  ( O `  x )  <_  E
) )
3534ralrn 5668 . . . . . . 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 4027 . . . . . . 7  |-  ( n  =  E  ->  (
y  <_  n  <->  y  <_  E ) )
3938ralbidv 2563 . . . . . 6  |-  ( n  =  E  ->  ( A. y  e.  ran  O  y  <_  n  <->  A. y  e.  ran  O  y  <_  E ) )
4039rspcev 2884 . . . . 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 10308 . . . 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 5570 . . . 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 5662 . . . . . . . 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 10309 . . . . . . 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 4049 . . . . 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 15144 . . . 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 2655 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255   supcsup 7193   RRcr 8736    < clt 8867    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024    || cdivides 12531   Basecbs 13148   Grpcgrp 14362   odcod 14840  gExcgex 14841   Abelcabel 15090
This theorem is referenced by:  cyggexb  15185  pgpfaclem3  15318
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-int 3863  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-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-od 14844  df-gex 14845  df-cmn 15091  df-abl 15092
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