Theorem dchrval 24959

Description: Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)

Hypotheses

Ref	Expression
dchrval.g	|- G = (DChr ` N )
dchrval.z	|- Z = (ℤ/nℤ ` N )
dchrval.b	|- B = ( Base ` Z )
dchrval.u	|- U = (Unit ` Z )
dchrval.n	|- ( ph -> N e. NN )
dchrval.d	|- ( ph -> D = { x e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )

Assertion

Ref	Expression
dchrval	|- ( ph -> G = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )

Distinct variable groups:   x, B   x, N   x, U   ph, x   x, Z

Allowed substitution hints:   D( x)   G( x)

Proof of Theorem dchrval

Dummy variables z n b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrval.g	. 2 |- G = (DChr ` N )
2		df-dchr 24958	. . . 4 |- 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 ) ) >. } )
3	2	a1i 11	. . 3 |- ( ph -> 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 ) ) >. } ) )
4		fvexd 6203	. . . 4 |- ( ( ph /\ n = N ) -> (ℤ/nℤ ` n ) e. _V )
5		ovex 6678	. . . . . . 7 |- ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) e. _V
6	5	rabex 4813	. . . . . 6 |- { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } e. _V
7	6	a1i 11	. . . . 5 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } e. _V )
8		dchrval.d	. . . . . . . . . . 11 |- ( ph -> D = { x e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )
9	8	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> D = { x e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )
10		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ n = N ) -> n = N )
11	10	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ n = N ) -> (ℤ/nℤ ` n ) = (ℤ/nℤ ` N ) )
12		dchrval.z	. . . . . . . . . . . . . . . 16 |- Z = (ℤ/nℤ ` N )
13	11, 12	syl6reqr 2675	. . . . . . . . . . . . . . 15 |- ( ( ph /\ n = N ) -> Z = (ℤ/nℤ ` n ) )
14	13	eqeq2d 2632	. . . . . . . . . . . . . 14 |- ( ( ph /\ n = N ) -> ( z = Z <-> z = (ℤ/nℤ ` n ) ) )
15	14	biimpar 502	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> z = Z )
16	15	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> (mulGrp ` z ) = (mulGrp ` Z ) )
17	16	oveq1d 6665	. . . . . . . . . . 11 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) = ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
18	15	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( Base ` z ) = ( Base ` Z ) )
19		dchrval.b	. . . . . . . . . . . . . . 15 |- B = ( Base ` Z )
20	18, 19	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( Base ` z ) = B )
21	15	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> (Unit ` z ) = (Unit ` Z ) )
22		dchrval.u	. . . . . . . . . . . . . . 15 |- U = (Unit ` Z )
23	21, 22	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> (Unit ` z ) = U )
24	20, 23	difeq12d 3729	. . . . . . . . . . . . 13 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( ( Base ` z ) \ (Unit ` z ) ) = ( B \ U ) )
25	24	xpeq1d 5138	. . . . . . . . . . . 12 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) = ( ( B \ U ) X. { 0 } ) )
26	25	sseq1d 3632	. . . . . . . . . . 11 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x <-> ( ( B \ U ) X. { 0 } ) C_ x ) )
27	17, 26	rabeqbidv 3195	. . . . . . . . . 10 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } = { x e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) | ( ( B \ U ) X. { 0 } ) C_ x } )
28	9, 27	eqtr4d 2659	. . . . . . . . 9 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> D = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } )
29	28	eqeq2d 2632	. . . . . . . 8 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> ( b = D <-> b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) )
30	29	biimpar 502	. . . . . . 7 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> b = D )
31	30	opeq2d 4409	. . . . . 6 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> <. ( Base ` ndx ) , b >. = <. ( Base ` ndx ) , D >. )
32	30	sqxpeqd 5141	. . . . . . . 8 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> ( b X. b ) = ( D X. D ) )
33	32	reseq2d 5396	. . . . . . 7 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> ( oF x. |` ( b X. b ) ) = ( oF x. |` ( D X. D ) ) )
34	33	opeq2d 4409	. . . . . 6 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> <. ( +g ` ndx ) , ( oF x. |` ( b X. b ) ) >. = <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. )
35	31, 34	preq12d 4276	. . . . 5 |- ( ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) /\ b = { x e. ( (mulGrp ` z ) MndHom (mulGrp ` ℂfld ) ) | ( ( ( Base ` z ) \ (Unit ` z ) ) X. { 0 } ) C_ x } ) -> { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF x. |` ( b X. b ) ) >. } = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )
36	7, 35	csbied 3560	. . . 4 |- ( ( ( ph /\ n = N ) /\ z = (ℤ/nℤ ` n ) ) -> [_ { 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 ) ) >. } = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )
37	4, 36	csbied 3560	. . 3 |- ( ( ph /\ n = N ) -> [_ (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 ) ) >. } = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )
38		dchrval.n	. . 3 |- ( ph -> N e. NN )
39		prex 4909	. . . 4 |- { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } e. _V
40	39	a1i 11	. . 3 |- ( ph -> { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } e. _V )
41	3, 37, 38, 40	fvmptd 6288	. 2 |- ( ph -> (DChr ` N ) = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )
42	1, 41	syl5eq 2668	1 |- ( ph -> G = { <. ( Base ` ndx ) , D >. , <. ( +g ` ndx ) , ( oF x. |` ( D X. D ) ) >. } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [_csb 3533   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   |-> cmpt 4729   X. cxp 5112   |` 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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-csb 3534  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-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-dchr 24958

This theorem is referenced by:  dchrbas  24960  dchrplusg  24972