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Theorem prmcyg 15196
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 12779 . . . 4  |-  -.  1  e.  Prime
2 simpr 447 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  C_ 
{ ( 0g `  G ) } )
3 cygctb.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
4 eqid 2296 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 14526 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
65snssd 3776 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  C_  B )
76ad2antrr 706 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  { ( 0g `  G ) }  C_  B )
82, 7eqssd 3209 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  =  { ( 0g `  G ) } )
98fveq2d 5545 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  ( # `  {
( 0g `  G
) } ) )
10 fvex 5555 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
11 hashsng 11372 . . . . . . . 8  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1210, 11ax-mp 8 . . . . . . 7  |-  ( # `  { ( 0g `  G ) } )  =  1
139, 12syl6eq 2344 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  1 )
14 simplr 731 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  e. 
Prime )
1513, 14eqeltrrd 2371 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  1  e.  Prime )
1615ex 423 . . . 4  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  ( B  C_  { ( 0g
`  G ) }  ->  1  e.  Prime ) )
171, 16mtoi 169 . . 3  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  -.  B  C_  { ( 0g
`  G ) } )
18 nss 3249 . . 3  |-  ( -.  B  C_  { ( 0g `  G ) }  <->  E. x ( x  e.  B  /\  -.  x  e.  { ( 0g `  G ) } ) )
1917, 18sylib 188 . 2  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  E. x
( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )
20 eqid 2296 . . . . 5  |-  ( od
`  G )  =  ( od `  G
)
21 simpll 730 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e.  Grp )
22 simprl 732 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  x  e.  B
)
23 simprr 733 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  x  e.  { ( 0g `  G
) } )
2420, 4, 3odeq1 14889 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( ( od
`  G ) `  x )  =  1  <-> 
x  =  ( 0g
`  G ) ) )
2521, 22, 24syl2anc 642 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  1  <->  x  =  ( 0g `  G ) ) )
26 elsn 3668 . . . . . . . 8  |-  ( x  e.  { ( 0g
`  G ) }  <-> 
x  =  ( 0g
`  G ) )
2725, 26syl6bbr 254 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  1  <->  x  e.  { ( 0g `  G ) } ) )
2823, 27mtbird 292 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  ( ( od `  G ) `  x )  =  1 )
29 prmnn 12777 . . . . . . . . . . . 12  |-  ( (
# `  B )  e.  Prime  ->  ( # `  B
)  e.  NN )
3029ad2antlr 707 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN )
3130nnnn0d 10034 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN0 )
32 fvex 5555 . . . . . . . . . . . 12  |-  ( Base `  G )  e.  _V
333, 32eqeltri 2366 . . . . . . . . . . 11  |-  B  e. 
_V
34 hashclb 11368 . . . . . . . . . . 11  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3533, 34ax-mp 8 . . . . . . . . . 10  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3631, 35sylibr 203 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  B  e.  Fin )
373, 20oddvds2 14895 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  ||  ( # `  B
) )
3821, 36, 22, 37syl3anc 1182 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  ||  ( # `
 B ) )
39 simplr 731 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  Prime )
403, 20odcl2 14894 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  e.  NN )
4121, 36, 22, 40syl3anc 1182 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  e.  NN )
42 dvdsprime 12787 . . . . . . . . 9  |-  ( ( ( # `  B
)  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  (
( ( od `  G ) `  x
)  ||  ( # `  B
)  <->  ( ( ( od `  G ) `
 x )  =  ( # `  B
)  \/  ( ( od `  G ) `
 x )  =  1 ) ) )
4339, 41, 42syl2anc 642 . . . . . . . 8  |-  ( ( ( 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 201 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  ( # `  B
)  \/  ( ( od `  G ) `
 x )  =  1 ) )
4544ord 366 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( -.  (
( od `  G
) `  x )  =  ( # `  B
)  ->  ( ( od `  G ) `  x )  =  1 ) )
4628, 45mt3d 117 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  =  (
# `  B )
)
473, 20, 21, 22, 46iscygodd 15191 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e. CycGrp )
4847ex 423 . . 3  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  (
( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } )  ->  G  e. CycGrp ) )
4948exlimdv 1626 . 2  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  ( E. x ( x  e.  B  /\  -.  x  e.  { ( 0g `  G ) } )  ->  G  e. CycGrp )
)
5019, 49mpd 14 1  |-  ( ( G  e.  Grp  /\  ( # `  B )  e.  Prime )  ->  G  e. CycGrp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   class class class wbr 4039   ` cfv 5271   Fincfn 6879   1c1 8754   NNcn 9762   NN0cn0 9981   #chash 11353    || cdivides 12547   Primecprime 12774   Basecbs 13164   0gc0g 13416   Grpcgrp 14378   odcod 14856  CycGrpccyg 15180
This theorem is referenced by:  lt6abl  15197
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-prm 12775  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-od 14860  df-cyg 15181
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