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Theorem dchrbas 23891
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 2403 . . . 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 23890 . . 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 5853 . 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 6306 . . . 4  |-  ( (mulGrp `  Z ) MndHom  (mulGrp ` fld )
)  e.  _V
1110rabex 4545 . . 3  |-  { x  e.  ( (mulGrp `  Z
) MndHom  (mulGrp ` fld ) )  |  ( ( B  \  U
)  X.  { 0 } )  C_  x }  e.  _V
12 eqid 2402 . . . 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 14953 . . 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 2468 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 1405    e. wcel 1842   {crab 2758   _Vcvv 3059    \ cdif 3411    C_ wss 3414   {csn 3972   {cpr 3974   <.cop 3978    X. cxp 4821    |` cres 4825   ` cfv 5569  (class class class)co 6278    oFcof 6519   0cc0 9522    x. cmul 9527   NNcn 10576   ndxcnx 14838   Basecbs 14841   +g cplusg 14909   MndHom cmhm 16288  mulGrpcmgp 17461  Unitcui 17608  ℂfldccnfld 18740  ℤ/nczn 18840  DChrcdchr 23888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-dchr 23889
This theorem is referenced by:  dchrelbas  23892  dchrplusg  23903
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