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Theorem dchrabl 22568
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 2439 . 2  |-  ( N  e.  NN  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2439 . 2  |-  ( N  e.  NN  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 dchrabl.g . . . 4  |-  G  =  (DChr `  N )
4 eqid 2438 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
5 eqid 2438 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2438 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
7 simp2 989 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x  e.  ( Base `  G
) )
8 simp3 990 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y  e.  ( Base `  G
) )
93, 4, 5, 6, 7, 8dchrmulcl 22563 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
10 fvex 5696 . . . . . . 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 2438 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
133, 4, 5, 12, 7dchrf 22556 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x : ( Base `  (ℤ/n `  N
) ) --> CC )
14133adant3r3 1198 . . . . . 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 22556 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y : ( Base `  (ℤ/n `  N
) ) --> CC )
16153adant3r3 1198 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y : ( Base `  (ℤ/n `  N ) ) --> CC )
17 simpr3 996 . . . . . . 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 22556 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z : ( Base `  (ℤ/n `  N ) ) --> CC )
19 mulass 9362 . . . . . . 7  |-  ( ( a  e.  CC  /\  b  e.  CC  /\  c  e.  CC )  ->  (
( a  x.  b
)  x.  c )  =  ( a  x.  ( b  x.  c
) ) )
2019adantl 466 . . . . . 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 6349 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x  oF  x.  y )  oF  x.  z
)  =  ( x  oF  x.  (
y  oF  x.  z ) ) )
22 simpr1 994 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x  e.  ( Base `  G ) )
23 simpr2 995 . . . . . . 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 22562 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) y )  =  ( x  oF  x.  y ) )
2524oveq1d 6101 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  oF  x.  z )  =  ( ( x  oF  x.  y )  oF  x.  z
) )
263, 4, 5, 6, 23, 17dchrmul 22562 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( y ( +g  `  G ) z )  =  ( y  oF  x.  z ) )
2726oveq2d 6102 . . . . 5  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x  oF  x.  ( y ( +g  `  G ) z ) )  =  ( x  oF  x.  ( y  oF  x.  z ) ) )
2821, 25, 273eqtr4d 2480 . . . 4  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( ( x ( +g  `  G ) y )  oF  x.  z )  =  ( x  oF  x.  ( y ( +g  `  G ) z ) ) )
2993adant3r3 1198 . . . . 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 22562 . . . 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 )  oF  x.  z ) )
313, 4, 5, 6, 23, 17dchrmulcl 22563 . . . . 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 22562 . . . 4  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
( x ( +g  `  G ) ( y ( +g  `  G
) z ) )  =  ( x  oF  x.  ( y ( +g  `  G
) z ) ) )
3328, 30, 323eqtr4d 2480 . . 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 2438 . . . 4  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
35 eqid 2438 . . . 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 22 . . . 4  |-  ( N  e.  NN  ->  N  e.  NN )
373, 4, 5, 12, 34, 35, 36dchr1cl 22565 . . 3  |-  ( N  e.  NN  ->  (
k  e.  ( Base `  (ℤ/n `  N ) )  |->  if ( k  e.  (Unit `  (ℤ/n `  N ) ) ,  1 ,  0 ) )  e.  ( Base `  G ) )
38 simpr 461 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G ) )  ->  x  e.  ( Base `  G ) )
393, 4, 5, 12, 34, 35, 6, 38dchrmulid2 22566 . . 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 2438 . . . . 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 22567 . . . 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 459 . . 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 463 . . 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 15554 . 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 9360 . . . . 5  |-  ( ( a  e.  CC  /\  b  e.  CC )  ->  ( a  x.  b
)  =  ( b  x.  a ) )
4746adantl 466 . . . 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 6347 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x  oF  x.  y )  =  ( y  oF  x.  x ) )
493, 4, 5, 6, 7, 8dchrmul 22562 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( x  oF  x.  y ) )
503, 4, 5, 6, 8, 7dchrmul 22562 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
5148, 49, 503eqtr4d 2480 . 2  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
521, 2, 44, 51isabld 16281 1  |-  ( N  e.  NN  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967   ifcif 3786    e. cmpt 4345   -->wf 5409   ` cfv 5413  (class class class)co 6086    oFcof 6313   CCcc 9272   0cc0 9274   1c1 9275    x. cmul 9279    / cdiv 9985   NNcn 10314   Basecbs 14166   +g cplusg 14230   Abelcabel 16269  Unitcui 16719  ℤ/nczn 17909  DChrcdchr 22546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-0g 14372  df-imas 14438  df-divs 14439  df-mnd 15407  df-mhm 15456  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-nsg 15670  df-eqg 15671  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-cring 16636  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-subrg 16841  df-lmod 16928  df-lss 16991  df-lsp 17030  df-sra 17230  df-rgmod 17231  df-lidl 17232  df-rsp 17233  df-2idl 17291  df-cnfld 17794  df-zring 17859  df-zn 17913  df-dchr 22547
This theorem is referenced by:  dchr1  22571  dchrinv  22575  dchr1re  22577  dchrpt  22581  dchrsum2  22582  sumdchr2  22584  dchrhash  22585  dchr2sum  22587  rpvmasumlem  22711  rpvmasum2  22736  dchrisum0re  22737
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