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Theorem dchrrcl 20479
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 20472 . . 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 5169 . 2  |-  dom DChr  C_  NN
3 n0i 3460 . . 3  |-  ( X  e.  D  ->  -.  D  =  (/) )
4 dchrrcl.g . . . . 5  |-  G  =  (DChr `  N )
5 ndmfv 5552 . . . . 5  |-  ( -.  N  e.  dom DChr  ->  (DChr `  N )  =  (/) )
64, 5syl5eq 2327 . . . 4  |-  ( -.  N  e.  dom DChr  ->  G  =  (/) )
7 fveq2 5525 . . . . 5  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
8 dchrrcl.b . . . . 5  |-  D  =  ( Base `  G
)
9 base0 13185 . . . . 5  |-  (/)  =  (
Base `  (/) )
107, 8, 93eqtr4g 2340 . . . 4  |-  ( G  =  (/)  ->  D  =  (/) )
116, 10syl 15 . . 3  |-  ( -.  N  e.  dom DChr  ->  D  =  (/) )
123, 11nsyl2 119 . 2  |-  ( X  e.  D  ->  N  e.  dom DChr )
132, 12sseldi 3178 1  |-  ( X  e.  D  ->  N  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   [_csb 3081    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   <.cop 3643    X. cxp 4687   dom cdm 4689    |` cres 4691   ` cfv 5255  (class class class)co 5858    o Fcof 6076   0cc0 8737    x. cmul 8742   NNcn 9746   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   MndHom cmhm 14413  mulGrpcmgp 15325  Unitcui 15421  ℂfldccnfld 16377  ℤ/nczn 16454  DChrcdchr 20471
This theorem is referenced by:  dchrmhm  20480  dchrf  20481  dchrelbas4  20482  dchrzrh1  20483  dchrzrhcl  20484  dchrzrhmul  20485  dchrmul  20487  dchrmulcl  20488  dchrn0  20489  dchrmulid2  20491  dchrinvcl  20492  dchrghm  20495  dchrabs  20499  dchrinv  20500  dchrsum2  20507  dchrsum  20508  dchr2sum  20512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-slot 13152  df-base 13153  df-dchr 20472
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