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Theorem cygabl 17512
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 2422 . . 3  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2422 . . 3  |-  (.g `  G
)  =  (.g `  G
)
31, 2iscyg3 17508 . 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 2423 . . . 4  |-  ( ( ( 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 2423 . . . 4  |-  ( ( ( 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 758 . . . 4  |-  ( ( ( 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 2426 . . . . . . . . . 10  |-  ( y  =  a  ->  (
y  =  ( n (.g `  G ) x )  <->  a  =  ( n (.g `  G ) x ) ) )
87rexbidv 2939 . . . . . . . . 9  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  a  =  ( n
(.g `  G ) x ) ) )
9 oveq1 6308 . . . . . . . . . . 11  |-  ( n  =  m  ->  (
n (.g `  G ) x )  =  ( m (.g `  G ) x ) )
109eqeq2d 2436 . . . . . . . . . 10  |-  ( n  =  m  ->  (
a  =  ( n (.g `  G ) x )  <->  a  =  ( m (.g `  G ) x ) ) )
1110cbvrexv 3056 . . . . . . . . 9  |-  ( E. n  e.  ZZ  a  =  ( n (.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) )
128, 11syl6bb 264 . . . . . . . 8  |-  ( y  =  a  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. m  e.  ZZ  a  =  ( m
(.g `  G ) x ) ) )
1312rspccv 3179 . . . . . . 7  |-  ( 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 467 . . . . . 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 )  ->  E. m  e.  ZZ  a  =  ( m (.g `  G ) x ) ) )
15 eqeq1 2426 . . . . . . . . 9  |-  ( y  =  b  ->  (
y  =  ( n (.g `  G ) x )  <->  b  =  ( n (.g `  G ) x ) ) )
1615rexbidv 2939 . . . . . . . 8  |-  ( y  =  b  ->  ( E. n  e.  ZZ  y  =  ( n
(.g `  G ) x )  <->  E. n  e.  ZZ  b  =  ( n
(.g `  G ) x ) ) )
1716rspccv 3179 . . . . . . 7  |-  ( 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 467 . . . . . 6  |-  ( ( ( 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 2996 . . . . . . . 8  |-  ( 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 10942 . . . . . . . . . . . . . 14  |-  ( m  e.  ZZ  ->  m  e.  CC )
2120ad2antrl 732 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  CC )
22 zcn 10942 . . . . . . . . . . . . . 14  |-  ( n  e.  ZZ  ->  n  e.  CC )
2322ad2antll 733 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  CC )
2421, 23addcomd 9835 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  ( m  +  n )  =  ( n  +  m ) )
2524oveq1d 6316 . . . . . . . . . . 11  |-  ( ( ( 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 758 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  G  e.  Grp )
27 simprl 762 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  m  e.  ZZ )
28 simprr 764 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  n  e.  ZZ )
29 simplr 760 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  ( m  e.  ZZ  /\  n  e.  ZZ ) )  ->  x  e.  ( Base `  G )
)
30 eqid 2422 . . . . . . . . . . . . 13  |-  ( +g  `  G )  =  ( +g  `  G )
311, 2, 30mulgdir 16770 . . . . . . . . . . . 12  |-  ( ( 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 1266 . . . . . . . . . . 11  |-  ( ( ( 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 16770 . . . . . . . . . . . 12  |-  ( ( 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 1266 . . . . . . . . . . 11  |-  ( ( ( 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 2471 . . . . . . . . . 10  |-  ( ( ( 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 6310 . . . . . . . . . . 11  |-  ( ( 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 6310 . . . . . . . . . . . 12  |-  ( ( 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 454 . . . . . . . . . . 11  |-  ( ( 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 2444 . . . . . . . . . 10  |-  ( ( 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 225 . . . . . . . . 9  |-  ( ( ( 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 2924 . . . . . . . 8  |-  ( ( 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 221 . . . . . . 7  |-  ( ( 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 466 . . . . . 6  |-  ( ( ( 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 485 . . . . 5  |-  ( ( ( 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 1203 . . . 4  |-  ( ( ( ( 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 17430 . . 3  |-  ( ( ( G  e.  Grp  /\  x  e.  ( Base `  G ) )  /\  A. y  e.  ( Base `  G ) E. n  e.  ZZ  y  =  ( n (.g `  G ) x ) )  ->  G  e.  Abel )
4746r19.29an 2969 . 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 )
483, 47sylbi 198 1  |-  ( G  e. CycGrp  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   ` cfv 5597  (class class class)co 6301   CCcc 9537    + caddc 9542   ZZcz 10937   Basecbs 15108   +g cplusg 15177   Grpcgrp 16656  .gcmg 16659   Abelcabl 17418  CycGrpccyg 17499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-1st 6803  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-seq 12213  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-grp 16660  df-minusg 16661  df-mulg 16663  df-cmn 17419  df-abl 17420  df-cyg 17500
This theorem is referenced by:  lt6abl  17516  frgpcyg  19130
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