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Theorem dchrrcl 22695
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 22688 . . 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 5432 . 2  |-  dom DChr  C_  NN
3 n0i 3740 . . 3  |-  ( X  e.  D  ->  -.  D  =  (/) )
4 dchrrcl.g . . . . 5  |-  G  =  (DChr `  N )
5 ndmfv 5813 . . . . 5  |-  ( -.  N  e.  dom DChr  ->  (DChr `  N )  =  (/) )
64, 5syl5eq 2504 . . . 4  |-  ( -.  N  e.  dom DChr  ->  G  =  (/) )
7 fveq2 5789 . . . . 5  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
8 dchrrcl.b . . . . 5  |-  D  =  ( Base `  G
)
9 base0 14315 . . . . 5  |-  (/)  =  (
Base `  (/) )
107, 8, 93eqtr4g 2517 . . . 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 3452 1  |-  ( X  e.  D  ->  N  e.  NN )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2799   [_csb 3386    \ cdif 3423    C_ wss 3426   (/)c0 3735   {csn 3975   {cpr 3977   <.cop 3981    X. cxp 4936   dom cdm 4938    |` cres 4940   ` cfv 5516  (class class class)co 6190    oFcof 6418   0cc0 9383    x. cmul 9388   NNcn 10423   ndxcnx 14273   Basecbs 14276   +g cplusg 14340   MndHom cmhm 15564  mulGrpcmgp 16696  Unitcui 16837  ℂfldccnfld 17927  ℤ/nczn 18043  DChrcdchr 22687
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fv 5524  df-slot 14280  df-base 14281  df-dchr 22688
This theorem is referenced by:  dchrmhm  22696  dchrf  22697  dchrelbas4  22698  dchrzrh1  22699  dchrzrhcl  22700  dchrzrhmul  22701  dchrmul  22703  dchrmulcl  22704  dchrn0  22705  dchrmulid2  22707  dchrinvcl  22708  dchrghm  22711  dchrabs  22715  dchrinv  22716  dchrsum2  22723  dchrsum  22724  dchr2sum  22728
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