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Theorem cyggexb 15498
Description: A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cygctb.1  |-  B  =  ( Base `  G
)
cyggex.o  |-  E  =  (gEx `  G )
Assertion
Ref Expression
cyggexb  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
 B ) ) )

Proof of Theorem cyggexb
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cygctb.1 . . . . 5  |-  B  =  ( Base `  G
)
2 cyggex.o . . . . 5  |-  E  =  (gEx `  G )
31, 2cyggex 15497 . . . 4  |-  ( ( G  e. CycGrp  /\  B  e. 
Fin )  ->  E  =  ( # `  B
) )
43expcom 425 . . 3  |-  ( B  e.  Fin  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
54adantl 453 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
6 simpll 731 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Abel )
7 ablgrp 15407 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 707 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Grp )
9 simplr 732 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  B  e.  Fin )
101, 2gexcl2 15213 . . . . . 6  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
118, 9, 10syl2anc 643 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E  e.  NN )
12 eqid 2435 . . . . . 6  |-  ( od
`  G )  =  ( od `  G
)
131, 2, 12gexex 15458 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  B  ( ( od `  G ) `  x )  =  E )
146, 11, 13syl2anc 643 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E. x  e.  B  ( ( od `  G ) `  x
)  =  E )
15 simplr 732 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  E  =  ( # `  B
) )
1615eqeq2d 2446 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  <->  ( ( od `  G ) `  x )  =  (
# `  B )
) )
17 eqid 2435 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
18 eqid 2435 . . . . . . . . . 10  |-  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }
191, 17, 18, 12cyggenod 15484 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( x  e.  {
y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  <->  ( x  e.  B  /\  (
( od `  G
) `  x )  =  ( # `  B
) ) ) )
208, 9, 19syl2anc 643 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( x  e. 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  <->  ( x  e.  B  /\  ( ( od `  G ) `  x
)  =  ( # `  B ) ) ) )
21 ne0i 3626 . . . . . . . . 9  |-  ( x  e.  { y  e.  B  |  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  ->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =/=  (/) )
221, 17, 18iscyg2 15482 . . . . . . . . . . 11  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
2322baib 872 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  ( G  e. CycGrp  <->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
248, 23syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( G  e. CycGrp  <->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =/=  (/) ) )
2521, 24syl5ibr 213 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( x  e. 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  ->  G  e. CycGrp ) )
2620, 25sylbird 227 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( ( x  e.  B  /\  (
( od `  G
) `  x )  =  ( # `  B
) )  ->  G  e. CycGrp ) )
2726expdimp 427 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  ( # `  B )  ->  G  e. CycGrp ) )
2816, 27sylbid 207 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  ->  G  e. CycGrp ) )
2928rexlimdva 2822 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( E. x  e.  B  ( ( od `  G ) `  x )  =  E  ->  G  e. CycGrp )
)
3014, 29mpd 15 . . 3  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e. CycGrp )
3130ex 424 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( E  =  ( # `  B
)  ->  G  e. CycGrp ) )
325, 31impbid 184 1  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   {crab 2701   (/)c0 3620    e. cmpt 4258   ran crn 4871   ` cfv 5446  (class class class)co 6073   Fincfn 7101   NNcn 9990   ZZcz 10272   #chash 11608   Basecbs 13459   Grpcgrp 14675  .gcmg 14679   odcod 15153  gExcgex 15154   Abelcabel 15403  CycGrpccyg 15477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7469  df-card 7816  df-acn 7819  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-q 10565  df-rp 10603  df-fz 11034  df-fzo 11126  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-fac 11557  df-hash 11609  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-clim 12272  df-sum 12470  df-dvds 12843  df-gcd 12997  df-prm 13070  df-pc 13201  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-mnd 14680  df-grp 14802  df-minusg 14803  df-sbg 14804  df-mulg 14805  df-subg 14931  df-eqg 14933  df-od 15157  df-gex 15158  df-cmn 15404  df-abl 15405  df-cyg 15478
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