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Theorem dchrabl 23401
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 2444 . 2  |-  ( N  e.  NN  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2444 . 2  |-  ( N  e.  NN  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 dchrabl.g . . . 4  |-  G  =  (DChr `  N )
4 eqid 2443 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
5 eqid 2443 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2443 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
7 simp2 998 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x  e.  ( Base `  G
) )
8 simp3 999 . . . 4  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y  e.  ( Base `  G
) )
93, 4, 5, 6, 7, 8dchrmulcl 23396 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
10 fvex 5866 . . . . . . 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 2443 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
133, 4, 5, 12, 7dchrf 23389 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x : ( Base `  (ℤ/n `  N
) ) --> CC )
14133adant3r3 1208 . . . . . 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 23389 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y : ( Base `  (ℤ/n `  N
) ) --> CC )
16153adant3r3 1208 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
y : ( Base `  (ℤ/n `  N ) ) --> CC )
17 simpr3 1005 . . . . . . 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 23389 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z : ( Base `  (ℤ/n `  N ) ) --> CC )
19 mulass 9583 . . . . . . 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 6559 . . . . 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 1003 . . . . . . 7  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  ->  x  e.  ( Base `  G ) )
23 simpr2 1004 . . . . . . 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 23395 . . . . . 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 6296 . . . . 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 23395 . . . . . 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 6297 . . . . 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 2494 . . . 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 1208 . . . . 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 23395 . . . 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 23396 . . . . 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 23395 . . . 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 2494 . . 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 2443 . . . 4  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
35 eqid 2443 . . . 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 23398 . . 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 23399 . . 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 2443 . . . . 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 23400 . . . 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 15949 . 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 9581 . . . . 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 6557 . . 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 23395 . . 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 23395 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
5148, 49, 503eqtr4d 2494 . 2  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
521, 2, 44, 51isabld 16685 1  |-  ( N  e.  NN  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   _Vcvv 3095   ifcif 3926    |-> cmpt 4495   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523   CCcc 9493   0cc0 9495   1c1 9496    x. cmul 9500    / cdiv 10212   NNcn 10542   Basecbs 14509   +g cplusg 14574   Abelcabl 16673  Unitcui 17162  ℤ/nczn 18413  DChrcdchr 23379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-0g 14716  df-imas 14782  df-qus 14783  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-nsg 16073  df-eqg 16074  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-subrg 17301  df-lmod 17388  df-lss 17453  df-lsp 17492  df-sra 17692  df-rgmod 17693  df-lidl 17694  df-rsp 17695  df-2idl 17754  df-cnfld 18295  df-zring 18363  df-zn 18417  df-dchr 23380
This theorem is referenced by:  dchr1  23404  dchrinv  23408  dchr1re  23410  dchrpt  23414  dchrsum2  23415  sumdchr2  23417  dchrhash  23418  dchr2sum  23420  rpvmasumlem  23544  rpvmasum2  23569  dchrisum0re  23570
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