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Theorem dchrabl 21038
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 2437 . 2  |-  ( N  e.  NN  ->  ( Base `  G )  =  ( Base `  G
) )
2 eqidd 2437 . 2  |-  ( N  e.  NN  ->  ( +g  `  G )  =  ( +g  `  G
) )
3 dchrabl.g . . . 4  |-  G  =  (DChr `  N )
4 eqid 2436 . . . 4  |-  (ℤ/n `  N
)  =  (ℤ/n `  N
)
5 eqid 2436 . . . 4  |-  ( Base `  G )  =  (
Base `  G )
6 eqid 2436 . . . 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 21033 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  e.  ( Base `  G
) )
10 fvex 5742 . . . . . . 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 2436 . . . . . . . 8  |-  ( Base `  (ℤ/n `  N ) )  =  ( Base `  (ℤ/n `  N
) )
133, 4, 5, 12, 7dchrf 21026 . . . . . . 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 21026 . . . . . . 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 21026 . . . . . 6  |-  ( ( N  e.  NN  /\  ( x  e.  ( Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) ) )  -> 
z : ( Base `  (ℤ/n `  N ) ) --> CC )
19 mulass 9078 . . . . . . 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 6338 . . . . 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 21032 . . . . . 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 6096 . . . . 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 21032 . . . . . 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 6097 . . . . 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 2478 . . . 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 21032 . . . 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 21033 . . . . 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 21032 . . . 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 2478 . . 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 2436 . . . 4  |-  (Unit `  (ℤ/n `  N ) )  =  (Unit `  (ℤ/n `  N ) )
35 eqid 2436 . . . 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 21035 . . 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 21036 . . 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 2436 . . . . 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 21037 . . . 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 14830 . 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 9076 . . . . 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 6336 . . 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 21032 . . 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 21032 . . 3  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
y ( +g  `  G
) x )  =  ( y  o F  x.  x ) )
5148, 49, 503eqtr4d 2478 . 2  |-  ( ( N  e.  NN  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( +g  `  G
) y )  =  ( y ( +g  `  G ) x ) )
521, 2, 44, 51isabld 15425 1  |-  ( N  e.  NN  ->  G  e.  Abel )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956   ifcif 3739    e. cmpt 4266   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677   NNcn 10000   Basecbs 13469   +g cplusg 13529   Abelcabel 15413  Unitcui 15744  ℤ/nczn 16781  DChrcdchr 21016
This theorem is referenced by:  dchr1  21041  dchrinv  21045  dchr1re  21047  dchrpt  21051  dchrsum2  21052  sumdchr2  21054  dchrhash  21055  dchr2sum  21057  rpvmasumlem  21181  rpvmasum2  21206  dchrisum0re  21207
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-imas 13734  df-divs 13735  df-mnd 14690  df-mhm 14738  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-nsg 14942  df-eqg 14943  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-2idl 16303  df-cnfld 16704  df-zn 16785  df-dchr 21017
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