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Theorem dchrbas 22717
Description: Base set 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 )
dchrbas.b  |-  D  =  ( Base `  G
)
Assertion
Ref Expression
dchrbas  |-  ( ph  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
Distinct variable groups:    x, B    x, N    x, U    ph, x    x, Z
Allowed substitution hints:    D( x)    G( x)

Proof of Theorem dchrbas
StepHypRef Expression
1 dchrval.g . . . 4  |-  G  =  (DChr `  N )
2 dchrval.z . . . 4  |-  Z  =  (ℤ/n `  N )
3 dchrval.b . . . 4  |-  B  =  ( Base `  Z
)
4 dchrval.u . . . 4  |-  U  =  (Unit `  Z )
5 dchrval.n . . . 4  |-  ( ph  ->  N  e.  NN )
6 eqidd 2455 . . . 4  |-  ( ph  ->  { x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
71, 2, 3, 4, 5, 6dchrval 22716 . . 3  |-  ( ph  ->  G  =  { <. (
Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. } )
87fveq2d 5806 . 2  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  { <. ( Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. } ) )
9 dchrbas.b . 2  |-  D  =  ( Base `  G
)
10 ovex 6228 . . . 4  |-  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  e.  _V
1110rabex 4554 . . 3  |-  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  e.  _V
12 eqid 2454 . . . 4  |-  { <. (
Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. }  =  { <. ( Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. }
1312grpbase 14401 . . 3  |-  ( { x  e.  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  |  ( ( B  \  U )  X.  { 0 } )  C_  x }  e.  _V  ->  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  =  ( Base `  { <. ( Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. } ) )
1411, 13ax-mp 5 . 2  |-  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  =  ( Base `  { <. ( Base `  ndx ) ,  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } >. ,  <. ( +g  `  ndx ) ,  (  oF  x.  |`  ( { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  X.  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } ) ) >. } )
158, 9, 143eqtr4g 2520 1  |-  ( ph  ->  D  =  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    \ cdif 3436    C_ wss 3439   {csn 3988   {cpr 3990   <.cop 3994    X. cxp 4949    |` cres 4953   ` cfv 5529  (class class class)co 6203    oFcof 6431   0cc0 9397    x. cmul 9402   NNcn 10437   ndxcnx 14293   Basecbs 14296   +g cplusg 14361   MndHom cmhm 15585  mulGrpcmgp 16723  Unitcui 16864  ℂfldccnfld 17953  ℤ/nczn 18069  DChrcdchr 22714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-struct 14298  df-ndx 14299  df-slot 14300  df-base 14301  df-plusg 14374  df-dchr 22715
This theorem is referenced by:  dchrelbas  22718  dchrplusg  22729
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