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Theorem cygabl 18292
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 2622 . . 3 |- ( Base ` G ) = ( Base ` G )
2 eqid 2622 . . 3 |- (.g ` G ) = (.g ` G )
31, 2iscyg3 18288 . 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 2623 . . . 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 2623 . . . 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 790 . . . 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 2626 . . . . . . . . . 10 |- ( y = a -> ( y = ( n (.g ` G ) x ) <-> a = ( n (.g ` G ) x ) ) )
87rexbidv 3052 . . . . . . . . 9 |- ( y = a -> ( E. n e. ZZ y = ( n (.g ` G ) x ) <-> E. n e. ZZ a = ( n (.g ` G ) x ) ) )
9 oveq1 6657 . . . . . . . . . . 11 |- ( n = m -> ( n (.g ` G ) x ) = ( m (.g ` G ) x ) )
109eqeq2d 2632 . . . . . . . . . 10 |- ( n = m -> ( a = ( n (.g ` G ) x ) <-> a = ( m (.g ` G ) x ) ) )
1110cbvrexv 3172 . . . . . . . . 9 |- ( E. n e. ZZ a = ( n (.g ` G ) x ) <-> E. m e. ZZ a = ( m (.g ` G ) x ) )
128, 11syl6bb 276 . . . . . . . 8 |- ( y = a -> ( E. n e. ZZ y = ( n (.g ` G ) x ) <-> E. m e. ZZ a = ( m (.g ` G ) x ) ) )
1312rspccv 3306 . . . . . . 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 482 . . . . . 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 2626 . . . . . . . . 9 |- ( y = b -> ( y = ( n (.g ` G ) x ) <-> b = ( n (.g ` G ) x ) ) )
1615rexbidv 3052 . . . . . . . 8 |- ( y = b -> ( E. n e. ZZ y = ( n (.g ` G ) x ) <-> E. n e. ZZ b = ( n (.g ` G ) x ) ) )
1716rspccv 3306 . . . . . . 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 482 . . . . . 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 3107 . . . . . . . 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 11382 . . . . . . . . . . . . . 14 |- ( m e. ZZ -> m e. CC )
2120ad2antrl 764 . . . . . . . . . . . . 13 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> m e. CC )
22 zcn 11382 . . . . . . . . . . . . . 14 |- ( n e. ZZ -> n e. CC )
2322ad2antll 765 . . . . . . . . . . . . 13 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> n e. CC )
2421, 23addcomd 10238 . . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( m + n ) = ( n + m ) )
2524oveq1d 6665 . . . . . . . . . . 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 790 . . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> G e. Grp )
27 simprl 794 . . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> m e. ZZ )
28 simprr 796 . . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> n e. ZZ )
29 simplr 792 . . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ x e. ( Base ` G ) ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> x e. ( Base ` G ) )
30 eqid 2622 . . . . . . . . . . . . 13 |- ( +g ` G ) = ( +g ` G )
311, 2, 30mulgdir 17573 . . . . . . . . . . . 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 1328 . . . . . . . . . . 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 17573 . . . . . . . . . . . 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 1328 . . . . . . . . . . 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 2664 . . . . . . . . . 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 6659 . . . . . . . . . . 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 6659 . . . . . . . . . . . 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 469 . . . . . . . . . . 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 2637 . . . . . . . . . 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 237 . . . . . . . . 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 3038 . . . . . . . 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 233 . . . . . . 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 481 . . . . . 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 500 . . . . 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 1262 . . . 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 18206 . . 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 3077 . 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 207 1 |- ( G e. CycGrp -> G e. Abel )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   ` cfv 5888  (class class class)co 6650   CCcc 9934   + caddc 9939   ZZcz 11377   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422  .gcmg 17540   Abelcabl 18194  CycGrpccyg 18279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mulg 17541  df-cmn 18195  df-abl 18196  df-cyg 18280
This theorem is referenced by:  lt6abl  18296  frgpcyg  19922
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