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Theorem cyggenod 15496
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 15492 . 2  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
5 simplr 733 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  B  e.  Fin )
6 simplll 736 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  G  e.  Grp )
7 simpr 449 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
8 simplr 733 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  X  e.  B )
91, 2mulgcl 14909 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
106, 7, 8, 9syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  (
n  .x.  X )  e.  B )
11 eqid 2438 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  ( n  e.  ZZ  |->  ( n  .x.  X ) )
1210, 11fmptd 5895 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B )
13 frn 5599 . . . . . . 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 7331 . . . . . 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 644 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin )
17 hashen 11633 . . . . 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 644 . . . 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 15202 . . . . . . 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 697 . . . . . 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 3747 . . . . . . 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 16 . . . . . 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 2471 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( O `  X ) )
2524eqeq1d 2446 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ( # `
 ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( # `  B
)  <->  ( O `  X )  =  (
# `  B )
) )
26 fisseneq 7322 . . . . . . 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 1156 . . . . . 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 7141 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
2928adantr 453 . . . . . . 7  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  B  ~~  B )
30 breq1 4217 . . . . . . 7  |-  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B 
<->  B  ~~  B ) )
3129, 30syl5ibrcom 215 . . . . . 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 185 . . . . 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 644 . . . 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 277 . . 3  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  <->  ( O `  X )  =  (
# `  B )
) )
3534pm5.32da 624 . 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 250 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ifcif 3741   class class class wbr 4214    e. cmpt 4268   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083    ~~ cen 7108   Fincfn 7111   0cc0 8992   ZZcz 10284   #chash 11620   Basecbs 13471   Grpcgrp 14687  .gcmg 14691   odcod 15165
This theorem is referenced by:  iscygodd  15500  cyggexb  15510
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-omul 6731  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-oi 7481  df-card 7828  df-acn 7831  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-hash 11621  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-od 15169
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