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Theorem dchrval 20489
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 20488 . . . 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 5555 . . . . 5  |-  (ℤ/n `  n
)  e.  _V
54a1i 10 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  e.  _V )
6 ovex 5899 . . . . . . 7  |-  ( (mulGrp `  z ) MndHom  (mulGrp ` fld )
)  e.  _V
76rabex 4181 . . . . . 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 5545 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  =  (ℤ/n `  N
) )
13 dchrval.z . . . . . . . . . . . . . . . 16  |-  Z  =  (ℤ/n `  N )
1412, 13syl6reqr 2347 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  =  N )  ->  Z  =  (ℤ/n `  n ) )
1514eqeq2d 2307 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  =  N )  ->  (
z  =  Z  <->  z  =  (ℤ/n `  n ) ) )
1615biimpar 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
z  =  Z )
1716fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(mulGrp `  z )  =  (mulGrp `  Z )
)
1817oveq1d 5889 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  =  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
1916fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  ( Base `  Z
) )
20 dchrval.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  Z
)
2119, 20syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  B )
2216fveq2d 5545 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  (Unit `  Z )
)
23 dchrval.u . . . . . . . . . . . . . . 15  |-  U  =  (Unit `  Z )
2422, 23syl6eqr 2346 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  U )
2521, 24difeq12d 3308 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( Base `  z
)  \  (Unit `  z
) )  =  ( B  \  U ) )
2625xpeq1d 4728 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( Base `  z )  \  (Unit `  z ) )  X. 
{ 0 } )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
2726sseq1d 3218 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( (
Base `  z )  \  (Unit `  z )
)  X.  { 0 } )  C_  x  <->  ( ( B  \  U
)  X.  { 0 } )  C_  x
) )
2818, 27rabeqbidv 2796 . . . . . . . . . 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 2331 . . . . . . . . 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 2307 . . . . . . . 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 3819 . . . . . 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 4730 . . . . . . . 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 4971 . . . . . . 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 3819 . . . . . 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 3727 . . . . 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 3136 . . . 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 3136 . . 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 4233 . . . 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 5622 . 2  |-  ( ph  ->  (DChr `  N )  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  o F  x.  |`  ( D  X.  D ) )
>. } )
431, 42syl5eq 2340 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   [_csb 3094    \ cdif 3162    C_ wss 3165   {csn 3653   {cpr 3654   <.cop 3656    e. cmpt 4093    X. cxp 4703    |` cres 4707   ` cfv 5271  (class class class)co 5874    o Fcof 6092   0cc0 8753    x. cmul 8758   NNcn 9762   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   MndHom cmhm 14429  mulGrpcmgp 15341  Unitcui 15437  ℂfldccnfld 16393  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrbas  20490  dchrplusg  20502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-dchr 20488
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