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Theorem dchrrcl 20495
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 20488 . . 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 5185 . 2  |-  dom DChr  C_  NN
3 n0i 3473 . . 3  |-  ( X  e.  D  ->  -.  D  =  (/) )
4 dchrrcl.g . . . . 5  |-  G  =  (DChr `  N )
5 ndmfv 5568 . . . . 5  |-  ( -.  N  e.  dom DChr  ->  (DChr `  N )  =  (/) )
64, 5syl5eq 2340 . . . 4  |-  ( -.  N  e.  dom DChr  ->  G  =  (/) )
7 fveq2 5541 . . . . 5  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
8 dchrrcl.b . . . . 5  |-  D  =  ( Base `  G
)
9 base0 13201 . . . . 5  |-  (/)  =  (
Base `  (/) )
107, 8, 93eqtr4g 2353 . . . 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 3191 1  |-  ( X  e.  D  ->  N  e.  NN )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   [_csb 3094    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656    X. cxp 4703   dom cdm 4705    |` cres 4707   ` cfv 5271  (class class class)co 5874    o Fcof 6092   0cc0 8753    x. cmul 8758   NNcn 9762   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   MndHom cmhm 14429  mulGrpcmgp 15341  Unitcui 15437  ℂfldccnfld 16393  ℤ/nczn 16470  DChrcdchr 20487
This theorem is referenced by:  dchrmhm  20496  dchrf  20497  dchrelbas4  20498  dchrzrh1  20499  dchrzrhcl  20500  dchrzrhmul  20501  dchrmul  20503  dchrmulcl  20504  dchrn0  20505  dchrmulid2  20507  dchrinvcl  20508  dchrghm  20511  dchrabs  20515  dchrinv  20516  dchrsum2  20523  dchrsum  20524  dchr2sum  20528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-slot 13168  df-base 13169  df-dchr 20488
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