Theorem dchrrcl 24965

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.

Step	Hyp	Ref	Expression
1		df-dchr 24958	. . 3 |- DChr = ( n e. NN |-> [_ (ℤ/nℤ ` n ) / z ]_ [_ { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } / b ]_ { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF x. |` ( b X. b ) ) >. } )
2	1	dmmptss 5631	. 2 |- dom DChr C_ NN
3		n0i 3920	. . 3 |- ( X e. D -> -. D = (/) )
4		dchrrcl.g	. . . . 5 |- G = (DChr ` N )
5		ndmfv 6218	. . . . 5 |- ( -. N e. dom DChr -> (DChr ` N ) = (/) )
6	4, 5	syl5eq 2668	. . . 4 |- ( -. N e. dom DChr -> G = (/) )
7		fveq2 6191	. . . . 5 |- ( G = (/) -> ( Base ` G ) = ( Base ` (/) ) )
8		dchrrcl.b	. . . . 5 |- D = ( Base ` G )
9		base0 15912	. . . . 5 |- (/) = ( Base ` (/) )
10	7, 8, 9	3eqtr4g 2681	. . . 4 |- ( G = (/) -> D = (/) )
11	6, 10	syl 17	. . 3 |- ( -. N e. dom DChr -> D = (/) )
12	3, 11	nsyl2 142	. 2 |- ( X e. D -> N e. dom DChr )
13	2, 12	sseldi 3601	1 |- ( X e. D -> N e. NN )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   [_csb 3533   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   <.cop 4183   X. cxp 5112   dom cdm 5114   |` cres 5116   ` cfv 5888  (class class class)co 6650   oFcof 6895   0cc0 9936   x. cmul 9941   NNcn 11020   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   MndHom cmhm 17333  mulGrpcmgp 18489  Unitcui 18639  ℂ_fldccnfld 19746  ℤ/nℤczn 19851  DChrcdchr 24957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-dchr 24958

This theorem is referenced by:  dchrmhm  24966  dchrf  24967  dchrelbas4  24968  dchrzrh1  24969  dchrzrhcl  24970  dchrzrhmul  24971  dchrmul  24973  dchrmulcl  24974  dchrn0  24975  dchrmulid2  24977  dchrinvcl  24978  dchrghm  24981  dchrabs  24985  dchrinv  24986  dchrsum2  24993  dchrsum  24994  dchr2sum  24998