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Theorem cyggenod 15171
Description: An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
iscyg.1  |-  B  =  ( Base `  G
)
iscyg.2  |-  .x.  =  (.g
`  G )
iscyg3.e  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
cyggenod.o  |-  O  =  ( od `  G
)
Assertion
Ref Expression
cyggenod  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( X  e.  E  <->  ( X  e.  B  /\  ( O `  X )  =  ( # `  B
) ) ) )
Distinct variable groups:    x, n, B    n, O    n, X, x    n, G, x    .x. , n, x
Allowed substitution hints:    E( x, n)    O( x)

Proof of Theorem cyggenod
StepHypRef Expression
1 iscyg.1 . . 3  |-  B  =  ( Base `  G
)
2 iscyg.2 . . 3  |-  .x.  =  (.g
`  G )
3 iscyg3.e . . 3  |-  E  =  { x  e.  B  |  ran  ( n  e.  ZZ  |->  ( n  .x.  x ) )  =  B }
41, 2, 3iscyggen 15167 . 2  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
5 simplr 731 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  B  e.  Fin )
6 simplll 734 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  G  e.  Grp )
7 simpr 447 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
8 simplr 731 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  X  e.  B )
91, 2mulgcl 14584 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
106, 7, 8, 9syl3anc 1182 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  (
n  .x.  X )  e.  B )
11 eqid 2283 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  ( n  e.  ZZ  |->  ( n  .x.  X ) )
1210, 11fmptd 5684 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B )
13 frn 5395 . . . . . . 7  |-  ( ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )
1412, 13syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) 
C_  B )
15 ssfi 7083 . . . . . 6  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  e. 
Fin )
165, 14, 15syl2anc 642 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin )
17 hashen 11346 . . . . 5  |-  ( ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  e. 
Fin  /\  B  e.  Fin )  ->  ( (
# `  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) )  =  ( # `  B )  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) 
~~  B ) )
1816, 5, 17syl2anc 642 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ( # `
 ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( # `  B
)  <->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B ) )
19 cyggenod.o . . . . . . . 8  |-  O  =  ( od `  G
)
201, 19, 2, 11dfod2 14877 . . . . . . 7  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( O `  X
)  =  if ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  e. 
Fin ,  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) ) ,  0 ) )
2120adantlr 695 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( O `  X )  =  if ( ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin ,  (
# `  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) ) ,  0 ) )
22 iftrue 3571 . . . . . . 7  |-  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  e.  Fin  ->  if ( ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin ,  (
# `  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) ) ,  0 )  =  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) ) )
2316, 22syl 15 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  if ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  e.  Fin , 
( # `  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) ) ) ,  0 )  =  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) ) )
2421, 23eqtr2d 2316 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( O `  X ) )
2524eqeq1d 2291 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ( # `
 ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( # `  B
)  <->  ( O `  X )  =  (
# `  B )
) )
26 fisseneq 7074 . . . . . . 7  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B )  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B )
27263expia 1153 . . . . . 6  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  ( ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) 
~~  B  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
28 enrefg 6893 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
2928adantr 451 . . . . . . 7  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  B  ~~  B )
30 breq1 4026 . . . . . . 7  |-  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B 
<->  B  ~~  B ) )
3129, 30syl5ibrcom 213 . . . . . 6  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  ( ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  B  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B ) )
3227, 31impbid 183 . . . . 5  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  ( ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) 
~~  B  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  B ) )
335, 14, 32syl2anc 642 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B  <->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  B ) )
3418, 25, 333bitr3rd 275 . . 3  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  <->  ( O `  X )  =  (
# `  B )
) )
3534pm5.32da 622 . 2  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B )  <-> 
( X  e.  B  /\  ( O `  X
)  =  ( # `  B ) ) ) )
364, 35syl5bb 248 1  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( X  e.  E  <->  ( X  e.  B  /\  ( O `  X )  =  ( # `  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547    C_ wss 3152   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Fincfn 6863   0cc0 8737   ZZcz 10024   #chash 11337   Basecbs 13148   Grpcgrp 14362  .gcmg 14366   odcod 14840
This theorem is referenced by:  iscygodd  15175  cyggexb  15185
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-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-oadd 6483  df-omul 6484  df-er 6660  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-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-dvds 12532  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-od 14844
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