MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prmcyg Unicode version

Theorem prmcyg 15180
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 12763 . . . 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 2283 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 14510 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
65snssd 3760 . . . . . . . . . 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 3196 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  =  { ( 0g `  G ) } )
98fveq2d 5529 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  ( # `  {
( 0g `  G
) } ) )
10 fvex 5539 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
11 hashsng 11356 . . . . . . . 8  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1210, 11ax-mp 8 . . . . . . 7  |-  ( # `  { ( 0g `  G ) } )  =  1
139, 12syl6eq 2331 . . . . . 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 2358 . . . . 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 3236 . . 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 2283 . . . . 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 14873 . . . . . . . . 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 3655 . . . . . . . 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 12761 . . . . . . . . . . . 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 10018 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN0 )
32 fvex 5539 . . . . . . . . . . . 12  |-  ( Base `  G )  e.  _V
333, 32eqeltri 2353 . . . . . . . . . . 11  |-  B  e. 
_V
34 hashclb 11352 . . . . . . . . . . 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 14879 . . . . . . . . 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 14878 . . . . . . . . . 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 12771 . . . . . . . . 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 15175 . . . 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 1664 . 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 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   {csn 3640   class class class wbr 4023   ` cfv 5255   Fincfn 6863   1c1 8738   NNcn 9746   NN0cn0 9965   #chash 11337    || cdivides 12531   Primecprime 12758   Basecbs 13148   0gc0g 13400   Grpcgrp 14362   odcod 14840  CycGrpccyg 15164
This theorem is referenced by:  lt6abl  15181
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-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  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-prm 12759  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-cyg 15165
  Copyright terms: Public domain W3C validator