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Theorem cygabl 16367
Description: A cyclic group is abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
Assertion
Ref Expression
cygabl  |-  ( G  e. CycGrp  ->  G  e.  Abel )

Proof of Theorem cygabl
Dummy variables  m  n  x  y  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2443 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg3 16363 . 2  |-  ( G  e. CycGrp 
<->  ( G  e.  Grp  /\ 
E. x  e.  (
Base `  G ) A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) ) )
4 eqidd 2444 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  ( Base `  G )  =  ( Base `  G
) )
5 eqidd 2444 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  ( +g  `  G )  =  ( +g  `  G
) )
6 simpll 753 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  G  e.  Grp )
7 eqeq1 2449 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  =  ( n (.g `  G ) x )  <->  a  =  ( n (.g `  G ) x ) ) )
87rexbidv 2736 . . . . . . . . . . 11  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  a  =  ( n
(.g `  G ) x ) ) )
9 oveq1 6098 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
109eqeq2d 2454 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
a  =  ( n (.g `  G ) x )  <->  a  =  ( m (.g `  G ) x ) ) )
1110cbvrexv 2948 . . . . . . . . . . 11  |-  ( E. n  e.  ZZ  a  =  ( n (.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) )
128, 11syl6bb 261 . . . . . . . . . 10  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) ) )
1312rspccv 3070 . . . . . . . . 9  |-  ( A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x )  ->  ( a  e.  ( Base `  G
)  ->  E. m  e.  ZZ  a  =  ( m (.g `  G ) x ) ) )
1413adantl 466 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  (
a  e.  ( Base `  G )  ->  E. m  e.  ZZ  a  =  ( m (.g `  G ) x ) ) )
15 eqeq1 2449 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
y  =  ( n (.g `  G ) x )  <->  b  =  ( n (.g `  G ) x ) ) )
1615rexbidv 2736 . . . . . . . . . 10  |-  ( y  =  b  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  b  =  ( n
(.g `  G ) x ) ) )
1716rspccv 3070 . . . . . . . . 9  |-  ( A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x )  ->  ( b  e.  ( Base `  G
)  ->  E. n  e.  ZZ  b  =  ( n (.g `  G ) x ) ) )
1817adantl 466 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  (
b  e.  ( Base `  G )  ->  E. n  e.  ZZ  b  =  ( n (.g `  G ) x ) ) )
19 reeanv 2888 . . . . . . . . . 10  |-  ( E. m  e.  ZZ  E. n  e.  ZZ  (
a  =  ( m (.g `  G ) x )  /\  b  =  ( n (.g `  G
) x ) )  <-> 
( E. m  e.  ZZ  a  =  ( m (.g `  G ) x )  /\  E. n  e.  ZZ  b  =  ( n (.g `  G ) x ) ) )
20 zcn 10651 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ZZ  ->  m  e.  CC )
2120ad2antrl 727 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  CC )
22 zcn 10651 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322ad2antll 728 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  CC )
2421, 23addcomd 9571 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( m  +  n )  =  ( n  +  m ) )
2524oveq1d 6106 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
m  +  n ) (.g `  G ) x )  =  ( ( n  +  m ) (.g `  G ) x ) )
26 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  G  e.  Grp )
27 simprl 755 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  ZZ )
28 simprr 756 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  ZZ )
29 simplr 754 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  x  e.  ( Base `  G )
)
30 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( +g  `  G )  =  ( +g  `  G )
311, 2, 30mulgdir 15652 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( m  e.  ZZ  /\  n  e.  ZZ  /\  x  e.  ( Base `  G ) ) )  ->  ( ( m  +  n ) (.g `  G ) x )  =  ( ( m (.g `  G ) x ) ( +g  `  G
) ( n (.g `  G ) x ) ) )
3226, 27, 28, 29, 31syl13anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
m  +  n ) (.g `  G ) x )  =  ( ( m (.g `  G ) x ) ( +g  `  G
) ( n (.g `  G ) x ) ) )
331, 2, 30mulgdir 15652 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Grp  /\  ( n  e.  ZZ  /\  m  e.  ZZ  /\  x  e.  ( Base `  G ) ) )  ->  ( ( n  +  m ) (.g `  G ) x )  =  ( ( n (.g `  G ) x ) ( +g  `  G
) ( m (.g `  G ) x ) ) )
3426, 28, 27, 29, 33syl13anc 1220 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
n  +  m ) (.g `  G ) x )  =  ( ( n (.g `  G ) x ) ( +g  `  G
) ( m (.g `  G ) x ) ) )
3525, 32, 343eqtr3d 2483 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
m (.g `  G ) x ) ( +g  `  G
) ( n (.g `  G ) x ) )  =  ( ( n (.g `  G ) x ) ( +g  `  G
) ( m (.g `  G ) x ) ) )
36 oveq12 6100 . . . . . . . . . . . . 13  |-  ( ( a  =  ( m (.g `  G ) x )  /\  b  =  ( n (.g `  G
) x ) )  ->  ( a ( +g  `  G ) b )  =  ( ( m (.g `  G
) x ) ( +g  `  G ) ( n (.g `  G
) x ) ) )
37 oveq12 6100 . . . . . . . . . . . . . 14  |-  ( ( b  =  ( n (.g `  G ) x )  /\  a  =  ( m (.g `  G
) x ) )  ->  ( b ( +g  `  G ) a )  =  ( ( n (.g `  G
) x ) ( +g  `  G ) ( m (.g `  G
) x ) ) )
3837ancoms 453 . . . . . . . . . . . . 13  |-  ( ( a  =  ( m (.g `  G ) x )  /\  b  =  ( n (.g `  G
) x ) )  ->  ( b ( +g  `  G ) a )  =  ( ( n (.g `  G
) x ) ( +g  `  G ) ( m (.g `  G
) x ) ) )
3936, 38eqeq12d 2457 . . . . . . . . . . . 12  |-  ( ( a  =  ( m (.g `  G ) x )  /\  b  =  ( n (.g `  G
) x ) )  ->  ( ( a ( +g  `  G
) b )  =  ( b ( +g  `  G ) a )  <-> 
( ( m (.g `  G ) x ) ( +g  `  G
) ( n (.g `  G ) x ) )  =  ( ( n (.g `  G ) x ) ( +g  `  G
) ( m (.g `  G ) x ) ) ) )
4035, 39syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( (
a  =  ( m (.g `  G ) x )  /\  b  =  ( n (.g `  G
) x ) )  ->  ( a ( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) ) )
4140rexlimdvva 2848 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  -> 
( E. m  e.  ZZ  E. n  e.  ZZ  ( a  =  ( m (.g `  G
) x )  /\  b  =  ( n
(.g `  G ) x ) )  ->  (
a ( +g  `  G
) b )  =  ( b ( +g  `  G ) a ) ) )
4219, 41syl5bir 218 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  -> 
( ( E. m  e.  ZZ  a  =  ( m (.g `  G ) x )  /\  E. n  e.  ZZ  b  =  ( n (.g `  G ) x ) )  ->  (
a ( +g  `  G
) b )  =  ( b ( +g  `  G ) a ) ) )
4342adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  (
( E. m  e.  ZZ  a  =  ( m (.g `  G ) x )  /\  E. n  e.  ZZ  b  =  ( n (.g `  G ) x ) )  ->  (
a ( +g  `  G
) b )  =  ( b ( +g  `  G ) a ) ) )
4414, 18, 43syl2and 483 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  (
( a  e.  (
Base `  G )  /\  b  e.  ( Base `  G ) )  ->  ( a ( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) ) )
45443impib 1185 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  x  e.  ( Base `  G )
)  /\  A. y  e.  ( Base `  G
) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  /\  a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
)  ->  ( a
( +g  `  G ) b )  =  ( b ( +g  `  G
) a ) )
464, 5, 6, 45isabld 16290 . . . . 5  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  G  e.  Abel )
4746ex 434 . . . 4  |-  ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  -> 
( A. y  e.  ( Base `  G
) E. n  e.  ZZ  y  =  ( n (.g `  G ) x )  ->  G  e.  Abel ) )
4847rexlimdva 2841 . . 3  |-  ( G  e.  Grp  ->  ( E. x  e.  ( Base `  G ) A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x )  ->  G  e.  Abel ) )
4948imp 429 . 2  |-  ( ( G  e.  Grp  /\  E. x  e.  ( Base `  G ) A. y  e.  ( Base `  G
) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  G  e.  Abel )
503, 49sylbi 195 1  |-  ( G  e. CycGrp  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   ` cfv 5418  (class class class)co 6091   CCcc 9280    + caddc 9285   ZZcz 10646   Basecbs 14174   +g cplusg 14238   Grpcgrp 15410  .gcmg 15414   Abelcabel 16278  CycGrpccyg 16354
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-mulg 15548  df-cmn 16279  df-abl 16280  df-cyg 16355
This theorem is referenced by:  lt6abl  16371  frgpcyg  18006
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