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Theorem prmcyg 16767
Description: A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypothesis
Ref Expression
cygctb.1  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
prmcyg  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  G  e. CycGrp )

Proof of Theorem prmcyg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 1nprm 14097 . . . 4  |-  -.  1  e.  Prime
2 simpr 461 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  C_ 
{ ( 0g `  G ) } )
3 cygctb.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
4 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 15949 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
65snssd 4178 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  C_  B )
76ad2antrr 725 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  { ( 0g `  G ) }  C_  B )
82, 7eqssd 3526 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  =  { ( 0g `  G ) } )
98fveq2d 5876 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  ( # `  {
( 0g `  G
) } ) )
10 fvex 5882 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
11 hashsng 12418 . . . . . . . 8  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1210, 11ax-mp 5 . . . . . . 7  |-  ( # `  { ( 0g `  G ) } )  =  1
139, 12syl6eq 2524 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  1 )
14 simplr 754 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  e. 
Prime )
1513, 14eqeltrrd 2556 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  1  e.  Prime )
1615ex 434 . . . 4  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  ( B  C_  { ( 0g
`  G ) }  ->  1  e.  Prime ) )
171, 16mtoi 178 . . 3  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  -.  B  C_  { ( 0g
`  G ) } )
18 nss 3567 . . 3  |-  ( -.  B  C_  { ( 0g `  G ) }  <->  E. x ( x  e.  B  /\  -.  x  e.  { ( 0g `  G ) } ) )
1917, 18sylib 196 . 2  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  E. x
( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )
20 eqid 2467 . . 3  |-  ( od
`  G )  =  ( od `  G
)
21 simpll 753 . . 3  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e.  Grp )
22 simprl 755 . . 3  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  x  e.  B
)
23 simprr 756 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  x  e.  { ( 0g `  G
) } )
2420, 4, 3odeq1 16453 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( ( od
`  G ) `  x )  =  1  <-> 
x  =  ( 0g
`  G ) ) )
2521, 22, 24syl2anc 661 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  1  <->  x  =  ( 0g `  G ) ) )
26 elsn 4047 . . . . . 6  |-  ( x  e.  { ( 0g
`  G ) }  <-> 
x  =  ( 0g
`  G ) )
2725, 26syl6bbr 263 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  1  <->  x  e.  { ( 0g `  G ) } ) )
2823, 27mtbird 301 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  ( ( od `  G ) `  x )  =  1 )
29 prmnn 14095 . . . . . . . . . 10  |-  ( (
# `  B )  e.  Prime  ->  ( # `  B
)  e.  NN )
3029ad2antlr 726 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN )
3130nnnn0d 10864 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN0 )
32 fvex 5882 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
333, 32eqeltri 2551 . . . . . . . . 9  |-  B  e. 
_V
34 hashclb 12410 . . . . . . . . 9  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3533, 34ax-mp 5 . . . . . . . 8  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3631, 35sylibr 212 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  B  e.  Fin )
373, 20oddvds2 16459 . . . . . . 7  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  ||  ( # `  B
) )
3821, 36, 22, 37syl3anc 1228 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  ||  ( # `
 B ) )
39 simplr 754 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  Prime )
403, 20odcl2 16458 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  e.  NN )
4121, 36, 22, 40syl3anc 1228 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  e.  NN )
42 dvdsprime 14105 . . . . . . 7  |-  ( ( ( # `  B
)  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  (
( ( od `  G ) `  x
)  ||  ( # `  B
)  <->  ( ( ( od `  G ) `
 x )  =  ( # `  B
)  \/  ( ( od `  G ) `
 x )  =  1 ) ) )
4339, 41, 42syl2anc 661 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  ||  ( # `  B )  <-> 
( ( ( od
`  G ) `  x )  =  (
# `  B )  \/  ( ( od `  G ) `  x
)  =  1 ) ) )
4438, 43mpbid 210 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  ( # `  B
)  \/  ( ( od `  G ) `
 x )  =  1 ) )
4544ord 377 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( -.  (
( od `  G
) `  x )  =  ( # `  B
)  ->  ( ( od `  G ) `  x )  =  1 ) )
4628, 45mt3d 125 . . 3  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  =  (
# `  B )
)
473, 20, 21, 22, 46iscygodd 16762 . 2  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e. CycGrp )
4819, 47exlimddv 1702 1  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  G  e. CycGrp )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3118    C_ wss 3481   {csn 4033   class class class wbr 4453   ` cfv 5594   Fincfn 7528   1c1 9505   NNcn 10548   NN0cn0 10807   #chash 12385    || cdivides 13863   Primecprime 14092   Basecbs 14506   0gc0g 14711   Grpcgrp 15924   odcod 16420  CycGrpccyg 16751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-acn 8335  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-dvds 13864  df-prm 14093  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-eqg 16071  df-od 16424  df-cyg 16752
This theorem is referenced by:  lt6abl  16768
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