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Theorem cygabl 15177
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 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2283 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg3 15173 . 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 2284 . . . . . 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 2284 . . . . . 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 2289 . . . . . . . . . . . 12  |-  ( y  =  a  ->  (
y  =  ( n (.g `  G ) x )  <->  a  =  ( n (.g `  G ) x ) ) )
87rexbidv 2564 . . . . . . . . . . 11  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  a  =  ( n
(.g `  G ) x ) ) )
9 oveq1 5865 . . . . . . . . . . . . 13  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
109eqeq2d 2294 . . . . . . . . . . . 12  |-  ( n  =  m  ->  (
a  =  ( n (.g `  G ) x )  <->  a  =  ( m (.g `  G ) x ) ) )
1110cbvrexv 2765 . . . . . . . . . . 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 2881 . . . . . . . . 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 2289 . . . . . . . . . . 11  |-  ( y  =  b  ->  (
y  =  ( n (.g `  G ) x )  <->  b  =  ( n (.g `  G ) x ) ) )
1615rexbidv 2564 . . . . . . . . . 10  |-  ( y  =  b  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  b  =  ( n
(.g `  G ) x ) ) )
1716rspccv 2881 . . . . . . . . 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 2707 . . . . . . . . . 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 10029 . . . . . . . . . . . . . . . 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 10029 . . . . . . . . . . . . . . . 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 9014 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( m  +  n )  =  ( n  +  m ) )
2524oveq1d 5873 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . 15  |-  ( +g  `  G )  =  ( +g  `  G )
311, 2, 30mulgdir 14592 . . . . . . . . . . . . . 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 14592 . . . . . . . . . . . . . 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 2323 . . . . . . . . . . . 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 5867 . . . . . . . . . . . . 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 5867 . . . . . . . . . . . . . 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 2297 . . . . . . . . . . . 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 2674 . . . . . . . . . 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 15102 . . . . 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 2667 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740   ZZcz 10024   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362  .gcmg 14366   Abelcabel 15090  CycGrpccyg 15164
This theorem is referenced by:  lt6abl  15181  frgpcyg  16527
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-cmn 15091  df-abl 15092  df-cyg 15165
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