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Theorem cygabl 15193
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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2296 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg3 15189 . 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 2297 . . . . . 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 2297 . . . . . 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 730 . . . . . 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 2302 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  =  ( n (.g `  G ) x )  <->  a  =  ( n (.g `  G ) x ) ) )
87rexbidv 2577 . . . . . . . . . . 11  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  a  =  ( n
(.g `  G ) x ) ) )
9 oveq1 5881 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
109eqeq2d 2307 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
a  =  ( n (.g `  G ) x )  <->  a  =  ( m (.g `  G ) x ) ) )
1110cbvrexv 2778 . . . . . . . . . . 11  |-  ( E. n  e.  ZZ  a  =  ( n (.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) )
128, 11syl6bb 252 . . . . . . . . . 10  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) ) )
1312rspccv 2894 . . . . . . . . 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 452 . . . . . . . 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 2302 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
y  =  ( n (.g `  G ) x )  <->  b  =  ( n (.g `  G ) x ) ) )
1615rexbidv 2577 . . . . . . . . . 10  |-  ( y  =  b  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  b  =  ( n
(.g `  G ) x ) ) )
1716rspccv 2894 . . . . . . . . 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 452 . . . . . . . 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 2720 . . . . . . . . . 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 10045 . . . . . . . . . . . . . . . 16  |-  ( m  e.  ZZ  ->  m  e.  CC )
2120ad2antrl 708 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  CC )
22 zcn 10045 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322ad2antll 709 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  CC )
2421, 23addcomd 9030 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( m  +  n )  =  ( n  +  m ) )
2524oveq1d 5889 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  G  e.  Grp )
27 simprl 732 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  ZZ )
28 simprr 733 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  ZZ )
29 simplr 731 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  x  e.  ( Base `  G )
)
30 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( +g  `  G )  =  ( +g  `  G )
311, 2, 30mulgdir 14608 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 14608 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 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 2336 . . . . . . . . . . . 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 5883 . . . . . . . . . . . . 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 5883 . . . . . . . . . . . . . 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 439 . . . . . . . . . . . . 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 2310 . . . . . . . . . . . 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 213 . . . . . . . . . . 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 2687 . . . . . . . . . 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 209 . . . . . . . . 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 451 . . . . . . . 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 469 . . . . . . 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 1149 . . . . . 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 15118 . . . . 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 423 . . . 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 2680 . . 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 418 . 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 187 1  |-  ( G  e. CycGrp  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756   ZZcz 10040   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378  .gcmg 14382   Abelcabel 15106  CycGrpccyg 15180
This theorem is referenced by:  lt6abl  15197  frgpcyg  16543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-cmn 15107  df-abl 15108  df-cyg 15181
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