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Theorem dchrrcl 21025
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 21018 . . 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 ) ,  (  o F  x.  |`  ( b  X.  b ) )
>. } )
21dmmptss 5367 . 2  |-  dom DChr  C_  NN
3 n0i 3634 . . 3  |-  ( X  e.  D  ->  -.  D  =  (/) )
4 dchrrcl.g . . . . 5  |-  G  =  (DChr `  N )
5 ndmfv 5756 . . . . 5  |-  ( -.  N  e.  dom DChr  ->  (DChr `  N )  =  (/) )
64, 5syl5eq 2481 . . . 4  |-  ( -.  N  e.  dom DChr  ->  G  =  (/) )
7 fveq2 5729 . . . . 5  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
8 dchrrcl.b . . . . 5  |-  D  =  ( Base `  G
)
9 base0 13507 . . . . 5  |-  (/)  =  (
Base `  (/) )
107, 8, 93eqtr4g 2494 . . . 4  |-  ( G  =  (/)  ->  D  =  (/) )
116, 10syl 16 . . 3  |-  ( -.  N  e.  dom DChr  ->  D  =  (/) )
123, 11nsyl2 122 . 2  |-  ( X  e.  D  ->  N  e.  dom DChr )
132, 12sseldi 3347 1  |-  ( X  e.  D  ->  N  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2710   [_csb 3252    \ cdif 3318    C_ wss 3321   (/)c0 3629   {csn 3815   {cpr 3816   <.cop 3818    X. cxp 4877   dom cdm 4879    |` cres 4881   ` cfv 5455  (class class class)co 6082    o Fcof 6304   0cc0 8991    x. cmul 8996   NNcn 10001   ndxcnx 13467   Basecbs 13470   +g cplusg 13530   MndHom cmhm 14737  mulGrpcmgp 15649  Unitcui 15745  ℂfldccnfld 16704  ℤ/nczn 16782  DChrcdchr 21017
This theorem is referenced by:  dchrmhm  21026  dchrf  21027  dchrelbas4  21028  dchrzrh1  21029  dchrzrhcl  21030  dchrzrhmul  21031  dchrmul  21033  dchrmulcl  21034  dchrn0  21035  dchrmulid2  21037  dchrinvcl  21038  dchrghm  21041  dchrabs  21045  dchrinv  21046  dchrsum2  21053  dchrsum  21054  dchr2sum  21058
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-slot 13474  df-base 13475  df-dchr 21018
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