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Theorem dchrrcl 23243
Description: Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
Hypotheses
Ref Expression
dchrrcl.g  |-  G  =  (DChr `  N )
dchrrcl.b  |-  D  =  ( Base `  G
)
Assertion
Ref Expression
dchrrcl  |-  ( X  e.  D  ->  N  e.  NN )

Proof of Theorem dchrrcl
Dummy variables  n  b  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dchr 23236 . . 3  |- 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 ) )
>. } )
21dmmptss 5501 . 2  |-  dom DChr  C_  NN
3 n0i 3790 . . 3  |-  ( X  e.  D  ->  -.  D  =  (/) )
4 dchrrcl.g . . . . 5  |-  G  =  (DChr `  N )
5 ndmfv 5888 . . . . 5  |-  ( -.  N  e.  dom DChr  ->  (DChr `  N )  =  (/) )
64, 5syl5eq 2520 . . . 4  |-  ( -.  N  e.  dom DChr  ->  G  =  (/) )
7 fveq2 5864 . . . . 5  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
8 dchrrcl.b . . . . 5  |-  D  =  ( Base `  G
)
9 base0 14525 . . . . 5  |-  (/)  =  (
Base `  (/) )
107, 8, 93eqtr4g 2533 . . . 4  |-  ( G  =  (/)  ->  D  =  (/) )
116, 10syl 16 . . 3  |-  ( -.  N  e.  dom DChr  ->  D  =  (/) )
123, 11nsyl2 127 . 2  |-  ( X  e.  D  ->  N  e.  dom DChr )
132, 12sseldi 3502 1  |-  ( X  e.  D  ->  N  e.  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1379    e. wcel 1767   {crab 2818   [_csb 3435    \ cdif 3473    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   <.cop 4033    X. cxp 4997   dom cdm 4999    |` cres 5001   ` cfv 5586  (class class class)co 6282    oFcof 6520   0cc0 9488    x. cmul 9493   NNcn 10532   ndxcnx 14483   Basecbs 14486   +g cplusg 14551   MndHom cmhm 15775  mulGrpcmgp 16931  Unitcui 17072  ℂfldccnfld 18191  ℤ/nczn 18307  DChrcdchr 23235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-slot 14490  df-base 14491  df-dchr 23236
This theorem is referenced by:  dchrmhm  23244  dchrf  23245  dchrelbas4  23246  dchrzrh1  23247  dchrzrhcl  23248  dchrzrhmul  23249  dchrmul  23251  dchrmulcl  23252  dchrn0  23253  dchrmulid2  23255  dchrinvcl  23256  dchrghm  23259  dchrabs  23263  dchrinv  23264  dchrsum2  23271  dchrsum  23272  dchr2sum  23276
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