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Theorem cyggenod 17086
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 17082 . 2  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
5 simplr 753 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  B  e.  Fin )
6 simplll 757 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  G  e.  Grp )
7 simpr 459 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
8 simplr 753 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  X  e.  B )
91, 2mulgcl 16358 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
106, 7, 8, 9syl3anc 1226 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  (
n  .x.  X )  e.  B )
11 eqid 2454 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  ( n  e.  ZZ  |->  ( n  .x.  X ) )
1210, 11fmptd 6031 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B )
13 frn 5719 . . . . . . 7  |-  ( ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B  ->  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )
1412, 13syl 16 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) ) 
C_  B )
15 ssfi 7733 . . . . . 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 659 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin )
17 hashen 12402 . . . . 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 659 . . . 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 16785 . . . . . . 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 712 . . . . . 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 ) )
2216iftrued 3937 . . . . . 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 ) ) ) )
2321, 22eqtr2d 2496 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( O `  X ) )
2423eqeq1d 2456 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ( # `
 ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( # `  B
)  <->  ( O `  X )  =  (
# `  B )
) )
25 fisseneq 7724 . . . . . . 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 )
26253expia 1196 . . . . . 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 ) )
27 enrefg 7540 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
2827adantr 463 . . . . . . 7  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  B  ~~  B )
29 breq1 4442 . . . . . . 7  |-  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B 
<->  B  ~~  B ) )
3028, 29syl5ibrcom 222 . . . . . 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 ) )
3126, 30impbid 191 . . . . 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 ) )
325, 14, 31syl2anc 659 . . . 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 ) )
3318, 24, 323bitr3rd 284 . . 3  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  <->  ( O `  X )  =  (
# `  B )
) )
3433pm5.32da 639 . 2  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B )  <-> 
( X  e.  B  /\  ( O `  X
)  =  ( # `  B ) ) ) )
354, 34syl5bb 257 1  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( X  e.  E  <->  ( X  e.  B  /\  ( O `  X )  =  ( # `  B
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   ifcif 3929   class class class wbr 4439    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570  (class class class)co 6270    ~~ cen 7506   Fincfn 7509   0cc0 9481   ZZcz 10860   #chash 12387   Basecbs 14716   Grpcgrp 16252  .gcmg 16255   odcod 16748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-omul 7127  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-acn 8314  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fl 11910  df-mod 11979  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-dvds 14071  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-od 16752
This theorem is referenced by:  iscygodd  17090  cyggexb  17100
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