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Theorem cyggexb 15437
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 15436 . . . 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 15346 . . . . . . 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 15152 . . . . . 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 2389 . . . . . 6  |-  ( od
`  G )  =  ( od `  G
)
131, 2, 12gexex 15397 . . . . 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 2400 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  <->  ( ( od `  G ) `  x )  =  (
# `  B )
) )
17 eqid 2389 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
18 eqid 2389 . . . . . . . . . 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 15423 . . . . . . . . 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 3579 . . . . . . . . 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 15421 . . . . . . . . . . 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 2775 . . . 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 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652   {crab 2655   (/)c0 3573    e. cmpt 4209   ran crn 4821   ` cfv 5396  (class class class)co 6022   Fincfn 7047   NNcn 9934   ZZcz 10216   #chash 11547   Basecbs 13398   Grpcgrp 14614  .gcmg 14618   odcod 15092  gExcgex 15093   Abelcabel 15342  CycGrpccyg 15416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-disj 4126  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-omul 6667  df-er 6843  df-ec 6845  df-qs 6849  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-sup 7383  df-oi 7414  df-card 7761  df-acn 7764  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-fz 10978  df-fzo 11068  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-fac 11496  df-hash 11548  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-clim 12211  df-sum 12409  df-dvds 12782  df-gcd 12936  df-prm 13009  df-pc 13140  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-eqg 14872  df-od 15096  df-gex 15097  df-cmn 15343  df-abl 15344  df-cyg 15417
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