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Theorem dchrabl 22719
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 2452 . 2  |-  ( N  e.  NN  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2452 . 2  |-  ( N  e.  NN  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 dchrabl.g . . . 4  |-  G  =  (DChr `  N )
4 eqid 2451 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
5 eqid 2451 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2451 . . . 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 22714 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
10 fvex 5802 . . . . . . 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 2451 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
133, 4, 5, 12, 7dchrf 22707 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  x : ( Base `  (ℤ/n `  N
) ) --> CC )
14133adant3r3 1199 . . . . . 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 22707 . . . . . . 7  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  y : ( Base `  (ℤ/n `  N
) ) --> CC )
16153adant3r3 1199 . . . . . 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 22707 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z : ( Base `  (ℤ/n `  N ) ) --> CC )
19 mulass 9474 . . . . . . 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 6457 . . . . 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 22713 . . . . . 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 6208 . . . . 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 22713 . . . . . 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 6209 . . . . 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 2502 . . . 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 1199 . . . . 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 22713 . . . 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 22714 . . . . 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 22713 . . . 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 2502 . . 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 2451 . . . 4  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
35 eqid 2451 . . . 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 22716 . . 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 22717 . . 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 2451 . . . . 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 22718 . . . 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 15674 . 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 9472 . . . . 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 6455 . . 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 22713 . . 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 22713 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) x )  =  ( y  oF  x.  x ) )
5148, 49, 503eqtr4d 2502 . 2  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
521, 2, 44, 51isabld 16403 1  |-  ( N  e.  NN  ->  G  e.  Abel )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3071   ifcif 3892    |-> cmpt 4451   -->wf 5515   ` cfv 5519  (class class class)co 6193    oFcof 6421   CCcc 9384   0cc0 9386   1c1 9387    x. cmul 9391    / cdiv 10097   NNcn 10426   Basecbs 14285   +g cplusg 14349   Abelcabel 16391  Unitcui 16846  ℤ/nczn 18052  DChrcdchr 22697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-ec 7206  df-qs 7210  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-0g 14491  df-imas 14557  df-divs 14558  df-mnd 15526  df-mhm 15575  df-grp 15656  df-minusg 15657  df-sbg 15658  df-subg 15789  df-nsg 15790  df-eqg 15791  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-cring 16763  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-subrg 16978  df-lmod 17065  df-lss 17129  df-lsp 17168  df-sra 17368  df-rgmod 17369  df-lidl 17370  df-rsp 17371  df-2idl 17429  df-cnfld 17937  df-zring 18002  df-zn 18056  df-dchr 22698
This theorem is referenced by:  dchr1  22722  dchrinv  22726  dchr1re  22728  dchrpt  22732  dchrsum2  22733  sumdchr2  22735  dchrhash  22736  dchr2sum  22738  rpvmasumlem  22862  rpvmasum2  22887  dchrisum0re  22888
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