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Theorem cyggexb 15185
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 15184 . . . 4  |-  ( ( G  e. CycGrp  /\  B  e. 
Fin )  ->  E  =  ( # `  B
) )
43expcom 424 . . 3  |-  ( B  e.  Fin  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
54adantl 452 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
6 simpll 730 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Abel )
7 ablgrp 15094 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 706 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Grp )
9 simplr 731 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  B  e.  Fin )
101, 2gexcl2 14900 . . . . . 6  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
118, 9, 10syl2anc 642 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E  e.  NN )
12 eqid 2283 . . . . . 6  |-  ( od
`  G )  =  ( od `  G
)
131, 2, 12gexex 15145 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  B  ( ( od `  G ) `  x )  =  E )
146, 11, 13syl2anc 642 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E. x  e.  B  ( ( od `  G ) `  x
)  =  E )
15 simplr 731 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  E  =  ( # `  B
) )
1615eqeq2d 2294 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  <->  ( ( od `  G ) `  x )  =  (
# `  B )
) )
17 eqid 2283 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
18 eqid 2283 . . . . . . . . . 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 15171 . . . . . . . . 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 642 . . . . . . . 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 3461 . . . . . . . . 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 15169 . . . . . . . . . . 11  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
2322baib 871 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  ( G  e. CycGrp  <->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
248, 23syl 15 . . . . . . . . 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 212 . . . . . . . 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 226 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( ( x  e.  B  /\  (
( od `  G
) `  x )  =  ( # `  B
) )  ->  G  e. CycGrp ) )
2726expdimp 426 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  ( # `  B )  ->  G  e. CycGrp ) )
2816, 27sylbid 206 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  ->  G  e. CycGrp ) )
2928rexlimdva 2667 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( E. x  e.  B  ( ( od `  G ) `  x )  =  E  ->  G  e. CycGrp )
)
3014, 29mpd 14 . . 3  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e. CycGrp )
3130ex 423 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( E  =  ( # `  B
)  ->  G  e. CycGrp ) )
325, 31impbid 183 1  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   {crab 2547   (/)c0 3455    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   Fincfn 6863   NNcn 9746   ZZcz 10024   #chash 11337   Basecbs 13148   Grpcgrp 14362  .gcmg 14366   odcod 14840  gExcgex 14841   Abelcabel 15090  CycGrpccyg 15164
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-disj 3994  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-se 4353  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-isom 5264  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-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  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-fac 11289  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-od 14844  df-gex 14845  df-cmn 15091  df-abl 15092  df-cyg 15165
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