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Theorem prmcyg 16389
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 13787 . . . 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 2443 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 15585 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
65snssd 4037 . . . . . . . . . 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 3392 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  =  { ( 0g `  G ) } )
98fveq2d 5714 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  ( # `  {
( 0g `  G
) } ) )
10 fvex 5720 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
11 hashsng 12155 . . . . . . . 8  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1210, 11ax-mp 5 . . . . . . 7  |-  ( # `  { ( 0g `  G ) } )  =  1
139, 12syl6eq 2491 . . . . . 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 2518 . . . . 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 3433 . . 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 2443 . . 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 16080 . . . . . . 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 3910 . . . . . 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 13785 . . . . . . . . . 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 10655 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN0 )
32 fvex 5720 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
333, 32eqeltri 2513 . . . . . . . . 9  |-  B  e. 
_V
34 hashclb 12147 . . . . . . . . 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 16086 . . . . . . 7  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  ||  ( # `  B
) )
3821, 36, 22, 37syl3anc 1218 . . . . . 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 16085 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  e.  NN )
4121, 36, 22, 40syl3anc 1218 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  e.  NN )
42 dvdsprime 13795 . . . . . . 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 16384 . 2  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e. CycGrp )
4819, 47exlimddv 1692 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 1369   E.wex 1586    e. wcel 1756   _Vcvv 2991    C_ wss 3347   {csn 3896   class class class wbr 4311   ` cfv 5437   Fincfn 7329   1c1 9302   NNcn 10341   NN0cn0 10598   #chash 12122    || cdivides 13554   Primecprime 13782   Basecbs 14193   0gc0g 14397   Grpcgrp 15429   odcod 16047  CycGrpccyg 16373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391  ax-inf2 7866  ax-cnex 9357  ax-resscn 9358  ax-1cn 9359  ax-icn 9360  ax-addcl 9361  ax-addrcl 9362  ax-mulcl 9363  ax-mulrcl 9364  ax-mulcom 9365  ax-addass 9366  ax-mulass 9367  ax-distr 9368  ax-i2m1 9369  ax-1ne0 9370  ax-1rid 9371  ax-rnegex 9372  ax-rrecex 9373  ax-cnre 9374  ax-pre-lttri 9375  ax-pre-lttrn 9376  ax-pre-ltadd 9377  ax-pre-mulgt0 9378  ax-pre-sup 9379
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-pss 3363  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-tp 3901  df-op 3903  df-uni 4111  df-int 4148  df-iun 4192  df-disj 4282  df-br 4312  df-opab 4370  df-mpt 4371  df-tr 4405  df-eprel 4651  df-id 4655  df-po 4660  df-so 4661  df-fr 4698  df-se 4699  df-we 4700  df-ord 4741  df-on 4742  df-lim 4743  df-suc 4744  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-isom 5446  df-riota 6071  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-om 6496  df-1st 6596  df-2nd 6597  df-recs 6851  df-rdg 6885  df-1o 6939  df-2o 6940  df-oadd 6943  df-omul 6944  df-er 7120  df-ec 7122  df-qs 7126  df-map 7235  df-en 7330  df-dom 7331  df-sdom 7332  df-fin 7333  df-sup 7710  df-oi 7743  df-card 8128  df-acn 8131  df-pnf 9439  df-mnf 9440  df-xr 9441  df-ltxr 9442  df-le 9443  df-sub 9616  df-neg 9617  df-div 10013  df-nn 10342  df-2 10399  df-3 10400  df-n0 10599  df-z 10666  df-uz 10881  df-rp 11011  df-fz 11457  df-fzo 11568  df-fl 11661  df-mod 11728  df-seq 11826  df-exp 11885  df-hash 12123  df-cj 12607  df-re 12608  df-im 12609  df-sqr 12743  df-abs 12744  df-clim 12985  df-sum 13183  df-dvds 13555  df-prm 13783  df-ndx 14196  df-slot 14197  df-base 14198  df-sets 14199  df-ress 14200  df-plusg 14270  df-0g 14399  df-mnd 15434  df-grp 15564  df-minusg 15565  df-sbg 15566  df-mulg 15567  df-subg 15697  df-eqg 15699  df-od 16051  df-cyg 16374
This theorem is referenced by:  lt6abl  16390
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