MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cyggexb Structured version   Unicode version

Theorem cyggexb 16380
Description: A finite abelian group is cyclic iff the exponent equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
cygctb.1  |-  B  =  ( Base `  G
)
cyggex.o  |-  E  =  (gEx `  G )
Assertion
Ref Expression
cyggexb  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
 B ) ) )

Proof of Theorem cyggexb
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cygctb.1 . . . . 5  |-  B  =  ( Base `  G
)
2 cyggex.o . . . . 5  |-  E  =  (gEx `  G )
31, 2cyggex 16379 . . . 4  |-  ( ( G  e. CycGrp  /\  B  e. 
Fin )  ->  E  =  ( # `  B
) )
43expcom 435 . . 3  |-  ( B  e.  Fin  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
54adantl 466 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  ->  E  =  ( # `  B
) ) )
6 simpll 753 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Abel )
7 ablgrp 16287 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
87ad2antrr 725 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e.  Grp )
9 simplr 754 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  B  e.  Fin )
101, 2gexcl2 16093 . . . . . 6  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  E  e.  NN )
118, 9, 10syl2anc 661 . . . . 5  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E  e.  NN )
12 eqid 2443 . . . . . 6  |-  ( od
`  G )  =  ( od `  G
)
131, 2, 12gexex 16340 . . . . 5  |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  B  ( ( od `  G ) `  x )  =  E )
146, 11, 13syl2anc 661 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  E. x  e.  B  ( ( od `  G ) `  x
)  =  E )
15 simplr 754 . . . . . . 7  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  E  =  ( # `  B
) )
1615eqeq2d 2454 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  <->  ( ( od `  G ) `  x )  =  (
# `  B )
) )
17 eqid 2443 . . . . . . . . . 10  |-  (.g `  G
)  =  (.g `  G
)
18 eqid 2443 . . . . . . . . . 10  |-  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }
191, 17, 18, 12cyggenod 16366 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  B  e.  Fin )  ->  ( x  e.  {
y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  <->  ( x  e.  B  /\  (
( od `  G
) `  x )  =  ( # `  B
) ) ) )
208, 9, 19syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( x  e. 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  <->  ( x  e.  B  /\  ( ( od `  G ) `  x
)  =  ( # `  B ) ) ) )
21 ne0i 3648 . . . . . . . . 9  |-  ( x  e.  { y  e.  B  |  ran  (
n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  ->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =/=  (/) )
221, 17, 18iscyg2 16364 . . . . . . . . . . 11  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
2322baib 896 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  ( G  e. CycGrp  <->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  =/=  (/) ) )
248, 23syl 16 . . . . . . . . 9  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( G  e. CycGrp  <->  { y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G
) y ) )  =  B }  =/=  (/) ) )
2521, 24syl5ibr 221 . . . . . . . 8  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( x  e. 
{ y  e.  B  |  ran  ( n  e.  ZZ  |->  ( n (.g `  G ) y ) )  =  B }  ->  G  e. CycGrp ) )
2620, 25sylbird 235 . . . . . . 7  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( ( x  e.  B  /\  (
( od `  G
) `  x )  =  ( # `  B
) )  ->  G  e. CycGrp ) )
2726expdimp 437 . . . . . 6  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  ( # `  B )  ->  G  e. CycGrp ) )
2816, 27sylbid 215 . . . . 5  |-  ( ( ( ( G  e. 
Abel  /\  B  e.  Fin )  /\  E  =  (
# `  B )
)  /\  x  e.  B )  ->  (
( ( od `  G ) `  x
)  =  E  ->  G  e. CycGrp ) )
2928rexlimdva 2846 . . . 4  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  ( E. x  e.  B  ( ( od `  G ) `  x )  =  E  ->  G  e. CycGrp )
)
3014, 29mpd 15 . . 3  |-  ( ( ( G  e.  Abel  /\  B  e.  Fin )  /\  E  =  ( # `
 B ) )  ->  G  e. CycGrp )
3130ex 434 . 2  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( E  =  ( # `  B
)  ->  G  e. CycGrp ) )
325, 31impbid 191 1  |-  ( ( G  e.  Abel  /\  B  e.  Fin )  ->  ( G  e. CycGrp  <->  E  =  ( # `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   E.wrex 2721   {crab 2724   (/)c0 3642    e. cmpt 4355   ran crn 4846   ` cfv 5423  (class class class)co 6096   Fincfn 7315   NNcn 10327   ZZcz 10651   #chash 12108   Basecbs 14179   Grpcgrp 15415  .gcmg 15419   odcod 16033  gExcgex 16034   Abelcabel 16283  CycGrpccyg 16359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-disj 4268  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-omul 6930  df-er 7106  df-ec 7108  df-qs 7112  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-acn 8117  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-q 10959  df-rp 10997  df-fz 11443  df-fzo 11554  df-fl 11647  df-mod 11714  df-seq 11812  df-exp 11871  df-fac 12057  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-dvds 13541  df-gcd 13696  df-prm 13769  df-pc 13909  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-mnd 15420  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-eqg 15685  df-od 16037  df-gex 16038  df-cmn 16284  df-abl 16285  df-cyg 16360
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator