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Theorem dchrval 22578
Description: Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
Hypotheses
Ref Expression
dchrval.g  |-  G  =  (DChr `  N )
dchrval.z  |-  Z  =  (ℤ/n `  N )
dchrval.b  |-  B  =  ( Base `  Z
)
dchrval.u  |-  U  =  (Unit `  Z )
dchrval.n  |-  ( ph  ->  N  e.  NN )
dchrval.d  |-  ( ph  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
Assertion
Ref Expression
dchrval  |-  ( ph  ->  G  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. } )
Distinct variable groups:    x, B    x, N    x, U    ph, x    x, Z
Allowed substitution hints:    D( x)    G( x)

Proof of Theorem dchrval
Dummy variables  z  n  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dchrval.g . 2  |-  G  =  (DChr `  N )
2 df-dchr 22577 . . . 4  |- DChr  =  ( n  e.  NN  |->  [_ (ℤ/n `  n )  /  z ]_ [_ { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b ) )
>. } )
32a1i 11 . . 3  |-  ( ph  -> DChr  =  ( n  e.  NN  |->  [_ (ℤ/n `  n )  /  z ]_ [_ { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b ) )
>. } ) )
4 fvex 5706 . . . . 5  |-  (ℤ/n `  n
)  e.  _V
54a1i 11 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  e.  _V )
6 ovex 6121 . . . . . . 7  |-  ( (mulGrp `  z ) MndHom  (mulGrp ` fld )
)  e.  _V
76rabex 4448 . . . . . 6  |-  { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  e.  _V
87a1i 11 . . . . 5  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  { x  e.  (
(mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  e.  _V )
9 dchrval.d . . . . . . . . . . 11  |-  ( ph  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
109ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
11 simpr 461 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
1211fveq2d 5700 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  =  (ℤ/n `  N
) )
13 dchrval.z . . . . . . . . . . . . . . . 16  |-  Z  =  (ℤ/n `  N )
1412, 13syl6reqr 2494 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  =  N )  ->  Z  =  (ℤ/n `  n ) )
1514eqeq2d 2454 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  =  N )  ->  (
z  =  Z  <->  z  =  (ℤ/n `  n ) ) )
1615biimpar 485 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
z  =  Z )
1716fveq2d 5700 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(mulGrp `  z )  =  (mulGrp `  Z )
)
1817oveq1d 6111 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  =  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
1916fveq2d 5700 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  ( Base `  Z
) )
20 dchrval.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  Z
)
2119, 20syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  B )
2216fveq2d 5700 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  (Unit `  Z )
)
23 dchrval.u . . . . . . . . . . . . . . 15  |-  U  =  (Unit `  Z )
2422, 23syl6eqr 2493 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  U )
2521, 24difeq12d 3480 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( Base `  z
)  \  (Unit `  z
) )  =  ( B  \  U ) )
2625xpeq1d 4868 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( Base `  z )  \  (Unit `  z ) )  X. 
{ 0 } )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
2726sseq1d 3388 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( (
Base `  z )  \  (Unit `  z )
)  X.  { 0 } )  C_  x  <->  ( ( B  \  U
)  X.  { 0 } )  C_  x
) )
2818, 27rabeqbidv 2972 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  { x  e.  (
(mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  =  {
x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  |  ( ( B  \  U )  X.  { 0 } )  C_  x }
)
2910, 28eqtr4d 2478 . . . . . . . . 9  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  D  =  { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )
3029eqeq2d 2454 . . . . . . . 8  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( b  =  D  <-> 
b  =  { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } ) )
3130biimpar 485 . . . . . . 7  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  b  =  D )
3231opeq2d 4071 . . . . . 6  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  D >. )
3331, 31xpeq12d 4870 . . . . . . . 8  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  (
b  X.  b )  =  ( D  X.  D ) )
3433reseq2d 5115 . . . . . . 7  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  (  oF  x.  |`  (
b  X.  b ) )  =  (  oF  x.  |`  ( D  X.  D ) ) )
3534opeq2d 4071 . . . . . 6  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b
) ) >.  =  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D ) ) >.
)
3632, 35preq12d 3967 . . . . 5  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  { <. (
Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b
) ) >. }  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D ) )
>. } )
378, 36csbied 3319 . . . 4  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  [_ { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b ) )
>. }  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. } )
385, 37csbied 3319 . . 3  |-  ( (
ph  /\  n  =  N )  ->  [_ (ℤ/n `  n
)  /  z ]_ [_ { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  /  b ]_ { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( b  X.  b ) )
>. }  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. } )
39 dchrval.n . . 3  |-  ( ph  ->  N  e.  NN )
40 prex 4539 . . . 4  |-  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. }  e.  _V
4140a1i 11 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D ) )
>. }  e.  _V )
423, 38, 39, 41fvmptd 5784 . 2  |-  ( ph  ->  (DChr `  N )  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D ) )
>. } )
431, 42syl5eq 2487 1  |-  ( ph  ->  G  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2724   _Vcvv 2977   [_csb 3293    \ cdif 3330    C_ wss 3333   {csn 3882   {cpr 3884   <.cop 3888    e. cmpt 4355    X. cxp 4843    |` cres 4847   ` cfv 5423  (class class class)co 6096    oFcof 6323   0cc0 9287    x. cmul 9292   NNcn 10327   ndxcnx 14176   Basecbs 14179   +g cplusg 14243   MndHom cmhm 15467  mulGrpcmgp 16596  Unitcui 16736  ℂfldccnfld 17823  ℤ/nczn 17939  DChrcdchr 22576
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-dchr 22577
This theorem is referenced by:  dchrbas  22579  dchrplusg  22591
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