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Theorem prmcyg 17218
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 14429 . . . 4  |-  -.  1  e.  Prime
2 simpr 459 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  C_ 
{ ( 0g `  G ) } )
3 cygctb.1 . . . . . . . . . . . 12  |-  B  =  ( Base `  G
)
4 eqid 2402 . . . . . . . . . . . 12  |-  ( 0g
`  G )  =  ( 0g `  G
)
53, 4grpidcl 16400 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
65snssd 4116 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  C_  B )
76ad2antrr 724 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  { ( 0g `  G ) }  C_  B )
82, 7eqssd 3458 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  B  =  { ( 0g `  G ) } )
98fveq2d 5852 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  ( # `
 B )  =  ( # `  {
( 0g `  G
) } ) )
10 fvex 5858 . . . . . . . 8  |-  ( 0g
`  G )  e. 
_V
11 hashsng 12484 . . . . . . . 8  |-  ( ( 0g `  G )  e.  _V  ->  ( # `
 { ( 0g
`  G ) } )  =  1 )
1210, 11ax-mp 5 . . . . . . 7  |-  ( # `  { ( 0g `  G ) } )  =  1
139, 12syl6eq 2459 . . . . . 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 2491 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  B  C_  { ( 0g `  G ) } )  ->  1  e.  Prime )
1615ex 432 . . . 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 3499 . . 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 2402 . . 3  |-  ( od
`  G )  =  ( od `  G
)
21 simpll 752 . . 3  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e.  Grp )
22 simprl 756 . . 3  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  x  e.  B
)
23 simprr 758 . . . . 5  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  x  e.  { ( 0g `  G
) } )
2420, 4, 3odeq1 16904 . . . . . . 7  |-  ( ( G  e.  Grp  /\  x  e.  B )  ->  ( ( ( od
`  G ) `  x )  =  1  <-> 
x  =  ( 0g
`  G ) ) )
2521, 22, 24syl2anc 659 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( ( od `  G ) `
 x )  =  1  <->  x  =  ( 0g `  G ) ) )
26 elsn 3985 . . . . . 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 299 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  -.  ( ( od `  G ) `  x )  =  1 )
29 prmnn 14427 . . . . . . . . . 10  |-  ( (
# `  B )  e.  Prime  ->  ( # `  B
)  e.  NN )
3029ad2antlr 725 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN )
3130nnnn0d 10892 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( # `  B
)  e.  NN0 )
32 fvex 5858 . . . . . . . . . 10  |-  ( Base `  G )  e.  _V
333, 32eqeltri 2486 . . . . . . . . 9  |-  B  e. 
_V
34 hashclb 12475 . . . . . . . . 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 16910 . . . . . . 7  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  ||  ( # `  B
) )
3821, 36, 22, 37syl3anc 1230 . . . . . 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 16909 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  B  e.  Fin  /\  x  e.  B )  ->  (
( od `  G
) `  x )  e.  NN )
4121, 36, 22, 40syl3anc 1230 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  ( ( od
`  G ) `  x )  e.  NN )
42 dvdsprime 14437 . . . . . . 7  |-  ( ( ( # `  B
)  e.  Prime  /\  (
( od `  G
) `  x )  e.  NN )  ->  (
( ( od `  G ) `  x
)  ||  ( # `  B
)  <->  ( ( ( od `  G ) `
 x )  =  ( # `  B
)  \/  ( ( od `  G ) `
 x )  =  1 ) ) )
4339, 41, 42syl2anc 659 . . . . . 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 375 . . . 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 17213 . 2  |-  ( ( ( G  e.  Grp  /\  ( # `  B
)  e.  Prime )  /\  ( x  e.  B  /\  -.  x  e.  {
( 0g `  G
) } ) )  ->  G  e. CycGrp )
4819, 47exlimddv 1747 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 366    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842   _Vcvv 3058    C_ wss 3413   {csn 3971   class class class wbr 4394   ` cfv 5568   Fincfn 7553   1c1 9522   NNcn 10575   NN0cn0 10835   #chash 12450    || cdvds 14193   Primecprime 14424   Basecbs 14839   0gc0g 15052   Grpcgrp 16375   odcod 16871  CycGrpccyg 17202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-inf2 8090  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-pre-sup 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-disj 4366  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-omul 7171  df-er 7347  df-ec 7349  df-qs 7353  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-sup 7934  df-oi 7968  df-card 8351  df-acn 8354  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-2 10634  df-3 10635  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-fl 11964  df-mod 12033  df-seq 12150  df-exp 12209  df-hash 12451  df-cj 13079  df-re 13080  df-im 13081  df-sqrt 13215  df-abs 13216  df-clim 13458  df-sum 13656  df-dvds 14194  df-prm 14425  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-sbg 16381  df-mulg 16382  df-subg 16520  df-eqg 16522  df-od 16875  df-cyg 17203
This theorem is referenced by:  lt6abl  17219
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