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Theorem dchrval 20473
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 ) ,  (  o F  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 20472 . . . 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 ) ,  (  o F  x.  |`  ( b  X.  b ) )
>. } )
32a1i 10 . . 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 ) ,  (  o F  x.  |`  ( b  X.  b ) )
>. } ) )
4 fvex 5539 . . . . 5  |-  (ℤ/n `  n
)  e.  _V
54a1i 10 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  e.  _V )
6 ovex 5883 . . . . . . 7  |-  ( (mulGrp `  z ) MndHom  (mulGrp ` fld )
)  e.  _V
76rabex 4165 . . . . . 6  |-  { x  e.  ( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x }  e.  _V
87a1i 10 . . . . 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 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
11 simpr 447 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
1211fveq2d 5529 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  =  (ℤ/n `  N
) )
13 dchrval.z . . . . . . . . . . . . . . . 16  |-  Z  =  (ℤ/n `  N )
1412, 13syl6reqr 2334 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  =  N )  ->  Z  =  (ℤ/n `  n ) )
1514eqeq2d 2294 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  =  N )  ->  (
z  =  Z  <->  z  =  (ℤ/n `  n ) ) )
1615biimpar 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
z  =  Z )
1716fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(mulGrp `  z )  =  (mulGrp `  Z )
)
1817oveq1d 5873 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  =  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
1916fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  ( Base `  Z
) )
20 dchrval.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  Z
)
2119, 20syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  B )
2216fveq2d 5529 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  (Unit `  Z )
)
23 dchrval.u . . . . . . . . . . . . . . 15  |-  U  =  (Unit `  Z )
2422, 23syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  U )
2521, 24difeq12d 3295 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( Base `  z
)  \  (Unit `  z
) )  =  ( B  \  U ) )
2625xpeq1d 4712 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( Base `  z )  \  (Unit `  z ) )  X. 
{ 0 } )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
2726sseq1d 3205 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( (
Base `  z )  \  (Unit `  z )
)  X.  { 0 } )  C_  x  <->  ( ( B  \  U
)  X.  { 0 } )  C_  x
) )
2818, 27rabeqbidv 2783 . . . . . . . . . 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 2318 . . . . . . . . 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 2294 . . . . . . . 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 471 . . . . . . 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 3803 . . . . . 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 4714 . . . . . . . 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 4955 . . . . . . 7  |-  ( ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n
) )  /\  b  =  { x  e.  ( (mulGrp `  z ) MndHom  (mulGrp ` fld ) )  |  ( ( ( Base `  z
)  \  (Unit `  z
) )  X.  {
0 } )  C_  x } )  ->  (  o F  x.  |`  (
b  X.  b ) )  =  (  o F  x.  |`  ( D  X.  D ) ) )
3534opeq2d 3803 . . . . . 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 ) ,  (  o F  x.  |`  ( b  X.  b
) ) >.  =  <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) ) >.
)
3632, 35preq12d 3714 . . . . 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 ) ,  (  o F  x.  |`  ( b  X.  b
) ) >. }  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) )
>. } )
378, 36csbied 3123 . . . 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 ) ,  (  o F  x.  |`  ( b  X.  b ) )
>. }  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D
) ) >. } )
385, 37csbied 3123 . . 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 ) ,  (  o F  x.  |`  ( b  X.  b ) )
>. }  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D
) ) >. } )
39 dchrval.n . . 3  |-  ( ph  ->  N  e.  NN )
40 prex 4217 . . . 4  |-  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D
) ) >. }  e.  _V
4140a1i 10 . . 3  |-  ( ph  ->  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) )
>. }  e.  _V )
423, 38, 39, 41fvmptd 5606 . 2  |-  ( ph  ->  (DChr `  N )  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) )
>. } )
431, 42syl5eq 2327 1  |-  ( ph  ->  G  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D
) ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641   <.cop 3643    e. cmpt 4077    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858    o Fcof 6076   0cc0 8737    x. cmul 8742   NNcn 9746   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   MndHom cmhm 14413  mulGrpcmgp 15325  Unitcui 15421  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrbas  20474  dchrplusg  20486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-dchr 20472
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