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Theorem dchrval 24160
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 24159 . . . 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 5891 . . . . 5  |-  (ℤ/n `  n
)  e.  _V
54a1i 11 . . . 4  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  e.  _V )
6 ovex 6333 . . . . . . 7  |-  ( (mulGrp `  z ) MndHom  (mulGrp ` fld )
)  e.  _V
76rabex 4575 . . . . . 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 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
11 simpr 462 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  =  N )  ->  n  =  N )
1211fveq2d 5885 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  =  N )  ->  (ℤ/n `  n
)  =  (ℤ/n `  N
) )
13 dchrval.z . . . . . . . . . . . . . . . 16  |-  Z  =  (ℤ/n `  N )
1412, 13syl6reqr 2482 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  =  N )  ->  Z  =  (ℤ/n `  n ) )
1514eqeq2d 2436 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  n  =  N )  ->  (
z  =  Z  <->  z  =  (ℤ/n `  n ) ) )
1615biimpar 487 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
z  =  Z )
1716fveq2d 5885 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(mulGrp `  z )  =  (mulGrp `  Z )
)
1817oveq1d 6320 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( (mulGrp `  z
) MndHom  (mulGrp ` fld ) )  =  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) ) )
1916fveq2d 5885 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  ( Base `  Z
) )
20 dchrval.b . . . . . . . . . . . . . . 15  |-  B  =  ( Base `  Z
)
2119, 20syl6eqr 2481 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( Base `  z )  =  B )
2216fveq2d 5885 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  (Unit `  Z )
)
23 dchrval.u . . . . . . . . . . . . . . 15  |-  U  =  (Unit `  Z )
2422, 23syl6eqr 2481 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
(Unit `  z )  =  U )
2521, 24difeq12d 3584 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( Base `  z
)  \  (Unit `  z
) )  =  ( B  \  U ) )
2625xpeq1d 4876 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( Base `  z )  \  (Unit `  z ) )  X. 
{ 0 } )  =  ( ( B 
\  U )  X. 
{ 0 } ) )
2726sseq1d 3491 . . . . . . . . . . 11  |-  ( ( ( ph  /\  n  =  N )  /\  z  =  (ℤ/n `  n ) )  -> 
( ( ( (
Base `  z )  \  (Unit `  z )
)  X.  { 0 } )  C_  x  <->  ( ( B  \  U
)  X.  { 0 } )  C_  x
) )
2818, 27rabeqbidv 3075 . . . . . . . . . 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 2466 . . . . . . . . 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 2436 . . . . . . . 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 487 . . . . . . 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 4194 . . . . . 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 >. )
3331sqxpeqd 4879 . . . . . . . 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 5124 . . . . . . 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 4194 . . . . . 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 4087 . . . . 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 3422 . . . 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 3422 . . 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 4663 . . . 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 5970 . 2  |-  ( ph  ->  (DChr `  N )  =  { <. ( Base `  ndx ) ,  D >. , 
<. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D ) )
>. } )
431, 42syl5eq 2475 1  |-  ( ph  ->  G  =  { <. (
Base `  ndx ) ,  D >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( D  X.  D
) ) >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872   {crab 2775   _Vcvv 3080   [_csb 3395    \ cdif 3433    C_ wss 3436   {csn 3998   {cpr 4000   <.cop 4004    |-> cmpt 4482    X. cxp 4851    |` cres 4855   ` cfv 5601  (class class class)co 6305    oFcof 6543   0cc0 9546    x. cmul 9551   NNcn 10616   ndxcnx 15117   Basecbs 15120   +g cplusg 15189   MndHom cmhm 16579  mulGrpcmgp 17722  Unitcui 17866  ℂfldccnfld 18969  ℤ/nczn 19072  DChrcdchr 24158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-dchr 24159
This theorem is referenced by:  dchrbas  24161  dchrplusg  24173
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