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Theorem cyggenod 16690
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 16686 . 2  |-  ( X  e.  E  <->  ( X  e.  B  /\  ran  (
n  e.  ZZ  |->  ( n  .x.  X ) )  =  B ) )
5 simplr 754 . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  n  e.  ZZ )
8 simplr 754 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  X  e.  B )
91, 2mulgcl 15969 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  n  e.  ZZ  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
106, 7, 8, 9syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  B  e.  Fin )  /\  X  e.  B )  /\  n  e.  ZZ )  ->  (
n  .x.  X )  e.  B )
11 eqid 2467 . . . . . . . 8  |-  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  =  ( n  e.  ZZ  |->  ( n  .x.  X ) )
1210, 11fmptd 6045 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( n  e.  ZZ  |->  ( n  .x.  X ) ) : ZZ --> B )
13 frn 5737 . . . . . . 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 7740 . . . . . 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 661 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ran  ( n  e.  ZZ  |->  ( n 
.x.  X ) )  e.  Fin )
17 hashen 12388 . . . . 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 661 . . . 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 16392 . . . . . . 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 714 . . . . . 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 3945 . . . . . . 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 2509 . . . . 5  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( # `  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( O `  X ) )
2524eqeq1d 2469 . . . 4  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ( # `
 ran  ( n  e.  ZZ  |->  ( n  .x.  X ) ) )  =  ( # `  B
)  <->  ( O `  X )  =  (
# `  B )
) )
26 fisseneq 7731 . . . . . . 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 1198 . . . . . 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 7547 . . . . . . . 8  |-  ( B  e.  Fin  ->  B  ~~  B )
2928adantr 465 . . . . . . 7  |-  ( ( B  e.  Fin  /\  ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  C_  B )  ->  B  ~~  B )
30 breq1 4450 . . . . . . 7  |-  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  -> 
( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  ~~  B 
<->  B  ~~  B ) )
3129, 30syl5ibrcom 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 ) )
3227, 31impbid 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 ) )
335, 14, 32syl2anc 661 . . . 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 284 . . 3  |-  ( ( ( G  e.  Grp  /\  B  e.  Fin )  /\  X  e.  B
)  ->  ( ran  ( n  e.  ZZ  |->  ( n  .x.  X ) )  =  B  <->  ( O `  X )  =  (
# `  B )
) )
3534pm5.32da 641 . 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 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 369    = wceq 1379    e. wcel 1767   {crab 2818    C_ wss 3476   ifcif 3939   class class class wbr 4447    |-> cmpt 4505   ran crn 5000   -->wf 5584   ` cfv 5588  (class class class)co 6284    ~~ cen 7513   Fincfn 7516   0cc0 9492   ZZcz 10864   #chash 12373   Basecbs 14490   Grpcgrp 15727  .gcmg 15731   odcod 16355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-omul 7135  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-acn 8323  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fl 11897  df-mod 11965  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-dvds 13848  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-od 16359
This theorem is referenced by:  iscygodd  16694  cyggexb  16704
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