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Theorem dchrabl 20991
Description: The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypothesis
Ref Expression
dchrabl.g  |-  G  =  (DChr `  N )
Assertion
Ref Expression
dchrabl  |-  ( N  e.  NN  ->  G  e.  Abel )

Proof of Theorem dchrabl
Dummy variables  x  a  b  c  k 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2405 . 2  |-  ( N  e.  NN  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2405 . 2  |-  ( N  e.  NN  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 dchrabl.g . . . 4  |-  G  =  (DChr `  N )
4 eqid 2404 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
5 eqid 2404 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2404 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
7 simp2 958 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x  e.  ( Base `  G
) )
8 simp3 959 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y  e.  ( Base `  G
) )
93, 4, 5, 6, 7, 8dchrmulcl 20986 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
10 fvex 5701 . . . . . . 7  |-  ( Base `  (ℤ/n `  N ) )  e. 
_V
1110a1i 11 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( Base `  (ℤ/n `  N ) )  e. 
_V )
12 eqid 2404 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
133, 4, 5, 12, 7dchrf 20979 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x : ( Base `  (ℤ/n `  N
) ) --> CC )
14133adant3r3 1164 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x : ( Base `  (ℤ/n `  N
) ) --> CC )
153, 4, 5, 12, 8dchrf 20979 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y : ( Base `  (ℤ/n `  N
) ) --> CC )
16153adant3r3 1164 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y : ( Base `  (ℤ/n `  N ) ) --> CC )
17 simpr3 965 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z  e.  ( Base `  G ) )
183, 4, 5, 12, 17dchrf 20979 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z : ( Base `  (ℤ/n `  N ) ) --> CC )
19 mulass 9034 . . . . . . 7  |-  ( ( a  e.  CC  /\  b  e.  CC  /\  c  e.  CC )  ->  (
( a  x.  b
)  x.  c )  =  ( a  x.  ( b  x.  c
) ) )
2019adantl 453 . . . . . 6  |-  ( ( ( N  e.  NN  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  /\  ( a  e.  CC  /\  b  e.  CC  /\  c  e.  CC ) )  -> 
( ( a  x.  b )  x.  c
)  =  ( a  x.  ( b  x.  c ) ) )
2111, 14, 16, 18, 20caofass 6297 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x  o F  x.  y )  o F  x.  z
)  =  ( x  o F  x.  (
y  o F  x.  z ) ) )
22 simpr1 963 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x  e.  ( Base `  G ) )
23 simpr2 964 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y  e.  ( Base `  G ) )
243, 4, 5, 6, 22, 23dchrmul 20985 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x  o F  x.  y ) )
2524oveq1d 6055 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  o F  x.  z )  =  ( ( x  o F  x.  y )  o F  x.  z
) )
263, 4, 5, 6, 23, 17dchrmul 20985 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( y  o F  x.  z ) )
2726oveq2d 6056 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x  o F  x.  ( y ( +g  `  G ) z ) )  =  ( x  o F  x.  ( y  o F  x.  z ) ) )
2821, 25, 273eqtr4d 2446 . . . 4  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  o F  x.  z )  =  ( x  o F  x.  ( y ( +g  `  G ) z ) ) )
2993adant3r3 1164 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  e.  ( Base `  G
) )
303, 4, 5, 6, 29, 17dchrmul 20985 . . . 4  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y ) ( +g  `  G ) z )  =  ( ( x ( +g  `  G
) y )  o F  x.  z ) )
313, 4, 5, 6, 23, 17dchrmulcl 20986 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  e.  ( Base `  G
) )
323, 4, 5, 6, 22, 31dchrmul 20985 . . . 4  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) ( y ( +g  `  G
) z ) )  =  ( x  o F  x.  ( y ( +g  `  G
) z ) ) )
3328, 30, 323eqtr4d 2446 . . 3  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y ) ( +g  `  G ) z )  =  ( x ( +g  `  G ) ( y ( +g  `  G ) z ) ) )
34 eqid 2404 . . . 4  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
35 eqid 2404 . . . 4  |-  ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) )  =  ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) )
36 id 20 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN )
373, 4, 5, 12, 34, 35, 36dchr1cl 20988 . . 3  |-  ( N  e.  NN  ->  (
k  e.  ( Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) )  e.  ( Base `  G ) )
38 simpr 448 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  ->  x  e.  ( Base `  G ) )
393, 4, 5, 12, 34, 35, 6, 38dchrmulid2 20989 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  -> 
( ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) ) ( +g  `  G
) x )  =  x )
40 eqid 2404 . . . . 5  |-  ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) )  =  ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) )
413, 4, 5, 12, 34, 35, 6, 38, 40dchrinvcl 20990 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  -> 
( ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) )  e.  ( Base `  G )  /\  (
( k  e.  (
Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) ) ( +g  `  G
) x )  =  ( k  e.  (
Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) ) ) )
4241simpld 446 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  -> 
( k  e.  (
Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) )  e.  ( Base `  G ) )
4341simprd 450 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  -> 
( ( k  e.  ( Base `  (ℤ/n `  N
) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  ( 1  /  (
x `  k )
) ,  0 ) ) ( +g  `  G
) x )  =  ( k  e.  (
Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) ) )
441, 2, 9, 33, 37, 39, 42, 43isgrpd 14785 . 2  |-  ( N  e.  NN  ->  G  e.  Grp )
4510a1i 11 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  ( Base `  (ℤ/n `  N ) )  e. 
_V )
46 mulcom 9032 . . . . 5  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  x.  b
)  =  ( b  x.  a ) )
4746adantl 453 . . . 4  |-  ( ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  /\  (
a  e.  CC  /\  b  e.  CC )
)  ->  ( a  x.  b )  =  ( b  x.  a ) )
4845, 13, 15, 47caofcom 6295 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x  o F  x.  y )  =  ( y  o F  x.  x ) )
493, 4, 5, 6, 7, 8dchrmul 20985 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( x  o F  x.  y ) )
503, 4, 5, 6, 8, 7dchrmul 20985 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) x )  =  ( y  o F  x.  x ) )
5148, 49, 503eqtr4d 2446 . 2  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
521, 2, 44, 51isabld 15380 1  |-  ( N  e.  NN  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   ifcif 3699    e. cmpt 4226   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    / cdiv 9633   NNcn 9956   Basecbs 13424   +g cplusg 13484   Abelcabel 15368  Unitcui 15699  ℤ/nczn 16736  DChrcdchr 20969
This theorem is referenced by:  dchr1  20994  dchrinv  20998  dchr1re  21000  dchrpt  21004  dchrsum2  21005  sumdchr2  21007  dchrhash  21008  dchr2sum  21010  rpvmasumlem  21134  rpvmasum2  21159  dchrisum0re  21160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-nsg 14897  df-eqg 14898  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-cnfld 16659  df-zn 16740  df-dchr 20970
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