Theorem dchrghm 24981

Description: A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
dchrghm.g	|- G = (DChr ` N )
dchrghm.z	|- Z = (ℤ/nℤ ` N )
dchrghm.b	|- D = ( Base ` G )
dchrghm.u	|- U = (Unit ` Z )
dchrghm.h	|- H = ( (mulGrp ` Z ) ↾s U )
dchrghm.m	|- M = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
dchrghm.x	|- ( ph -> X e. D )

Assertion

Ref	Expression
dchrghm	|- ( ph -> ( X |` U ) e. ( H GrpHom M ) )

Proof of Theorem dchrghm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dchrghm.g	. . . . . 6 |- G = (DChr ` N )
2		dchrghm.z	. . . . . 6 |- Z = (ℤ/nℤ ` N )
3		dchrghm.b	. . . . . 6 |- D = ( Base ` G )
4	1, 2, 3	dchrmhm 24966	. . . . 5 |- D C_ ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) )
5		dchrghm.x	. . . . 5 |- ( ph -> X e. D )
6	4, 5	sseldi 3601	. . . 4 |- ( ph -> X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) )
7	1, 3	dchrrcl 24965	. . . . . . . . 9 |- ( X e. D -> N e. NN )
8	5, 7	syl 17	. . . . . . . 8 |- ( ph -> N e. NN )
9	8	nnnn0d 11351	. . . . . . 7 |- ( ph -> N e. NN0 )
10	2	zncrng 19893	. . . . . . 7 |- ( N e. NN0 -> Z e. CRing )
11	9, 10	syl 17	. . . . . 6 |- ( ph -> Z e. CRing )
12		crngring 18558	. . . . . 6 |- ( Z e. CRing -> Z e. Ring )
13	11, 12	syl 17	. . . . 5 |- ( ph -> Z e. Ring )
14		dchrghm.u	. . . . . 6 |- U = (Unit ` Z )
15		eqid 2622	. . . . . 6 |- (mulGrp ` Z ) = (mulGrp ` Z )
16	14, 15	unitsubm 18670	. . . . 5 |- ( Z e. Ring -> U e. (SubMnd ` (mulGrp ` Z ) ) )
17	13, 16	syl 17	. . . 4 |- ( ph -> U e. (SubMnd ` (mulGrp ` Z ) ) )
18		dchrghm.h	. . . . 5 |- H = ( (mulGrp ` Z ) ↾s U )
19	18	resmhm 17359	. . . 4 |- ( ( X e. ( (mulGrp ` Z ) MndHom (mulGrp ` ℂfld ) ) /\ U e. (SubMnd ` (mulGrp ` Z ) ) ) -> ( X |` U ) e. ( H MndHom (mulGrp ` ℂfld ) ) )
20	6, 17, 19	syl2anc 693	. . 3 |- ( ph -> ( X |` U ) e. ( H MndHom (mulGrp ` ℂfld ) ) )
21		cnring 19768	. . . . 5 |- ℂfld e. Ring
22		cnfldbas 19750	. . . . . . 7 |- CC = ( Base ` ℂfld )
23		cnfld0 19770	. . . . . . 7 |- 0 = ( 0g ` ℂfld )
24		cndrng 19775	. . . . . . 7 |- ℂfld e. DivRing
25	22, 23, 24	drngui 18753	. . . . . 6 |- ( CC \ { 0 } ) = (Unit ` ℂfld )
26		eqid 2622	. . . . . 6 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
27	25, 26	unitsubm 18670	. . . . 5 |- (ℂfld e. Ring -> ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) )
28	21, 27	ax-mp 5	. . . 4 |- ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) )
29		df-ima 5127	. . . . 5 |- ( X " U ) = ran ( X |` U )
30		eqid 2622	. . . . . . . . . 10 |- ( Base ` Z ) = ( Base ` Z )
31	1, 2, 3, 30, 5	dchrf 24967	. . . . . . . . 9 |- ( ph -> X : ( Base ` Z ) --> CC )
32	30, 14	unitss 18660	. . . . . . . . . 10 |- U C_ ( Base ` Z )
33	32	sseli 3599	. . . . . . . . 9 |- ( x e. U -> x e. ( Base ` Z ) )
34		ffvelrn 6357	. . . . . . . . 9 |- ( ( X : ( Base ` Z ) --> CC /\ x e. ( Base ` Z ) ) -> ( X ` x ) e. CC )
35	31, 33, 34	syl2an 494	. . . . . . . 8 |- ( ( ph /\ x e. U ) -> ( X ` x ) e. CC )
36		simpr 477	. . . . . . . . 9 |- ( ( ph /\ x e. U ) -> x e. U )
37	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ x e. U ) -> X e. D )
38	33	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ x e. U ) -> x e. ( Base ` Z ) )
39	1, 2, 3, 30, 14, 37, 38	dchrn0 24975	. . . . . . . . 9 |- ( ( ph /\ x e. U ) -> ( ( X ` x ) =/= 0 <-> x e. U ) )
40	36, 39	mpbird 247	. . . . . . . 8 |- ( ( ph /\ x e. U ) -> ( X ` x ) =/= 0 )
41		eldifsn 4317	. . . . . . . 8 |- ( ( X ` x ) e. ( CC \ { 0 } ) <-> ( ( X ` x ) e. CC /\ ( X ` x ) =/= 0 ) )
42	35, 40, 41	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ x e. U ) -> ( X ` x ) e. ( CC \ { 0 } ) )
43	42	ralrimiva 2966	. . . . . 6 |- ( ph -> A. x e. U ( X ` x ) e. ( CC \ { 0 } ) )
44		ffun 6048	. . . . . . . 8 |- ( X : ( Base ` Z ) --> CC -> Fun X )
45	31, 44	syl 17	. . . . . . 7 |- ( ph -> Fun X )
46		fdm 6051	. . . . . . . . 9 |- ( X : ( Base ` Z ) --> CC -> dom X = ( Base ` Z ) )
47	31, 46	syl 17	. . . . . . . 8 |- ( ph -> dom X = ( Base ` Z ) )
48	32, 47	syl5sseqr 3654	. . . . . . 7 |- ( ph -> U C_ dom X )
49		funimass4 6247	. . . . . . 7 |- ( ( Fun X /\ U C_ dom X ) -> ( ( X " U ) C_ ( CC \ { 0 } ) <-> A. x e. U ( X ` x ) e. ( CC \ { 0 } ) ) )
50	45, 48, 49	syl2anc 693	. . . . . 6 |- ( ph -> ( ( X " U ) C_ ( CC \ { 0 } ) <-> A. x e. U ( X ` x ) e. ( CC \ { 0 } ) ) )
51	43, 50	mpbird 247	. . . . 5 |- ( ph -> ( X " U ) C_ ( CC \ { 0 } ) )
52	29, 51	syl5eqssr 3650	. . . 4 |- ( ph -> ran ( X |` U ) C_ ( CC \ { 0 } ) )
53		dchrghm.m	. . . . 5 |- M = ( (mulGrp ` ℂfld ) ↾s ( CC \ { 0 } ) )
54	53	resmhm2b 17361	. . . 4 |- ( ( ( CC \ { 0 } ) e. (SubMnd ` (mulGrp ` ℂfld ) ) /\ ran ( X |` U ) C_ ( CC \ { 0 } ) ) -> ( ( X |` U ) e. ( H MndHom (mulGrp ` ℂfld ) ) <-> ( X |` U ) e. ( H MndHom M ) ) )
55	28, 52, 54	sylancr 695	. . 3 |- ( ph -> ( ( X |` U ) e. ( H MndHom (mulGrp ` ℂfld ) ) <-> ( X |` U ) e. ( H MndHom M ) ) )
56	20, 55	mpbid 222	. 2 |- ( ph -> ( X |` U ) e. ( H MndHom M ) )
57	14, 18	unitgrp 18667	. . . 4 |- ( Z e. Ring -> H e. Grp )
58	13, 57	syl 17	. . 3 |- ( ph -> H e. Grp )
59	53	cnmgpabl 19807	. . . 4 |- M e. Abel
60		ablgrp 18198	. . . 4 |- ( M e. Abel -> M e. Grp )
61	59, 60	ax-mp 5	. . 3 |- M e. Grp
62		ghmmhmb 17671	. . 3 |- ( ( H e. Grp /\ M e. Grp ) -> ( H GrpHom M ) = ( H MndHom M ) )
63	58, 61, 62	sylancl 694	. 2 |- ( ph -> ( H GrpHom M ) = ( H MndHom M ) )
64	56, 63	eleqtrrd 2704	1 |- ( ph -> ( X |` U ) e. ( H GrpHom M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   C_ wss 3574   {csn 4177   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   NNcn 11020   NN0cn0 11292   Basecbs 15857   ↾_s cress 15858   MndHom cmhm 17333  SubMndcsubmnd 17334   Grpcgrp 17422   GrpHom cghm 17657   Abelcabl 18194  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548  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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-cnfld 19747  df-zring 19819  df-zn 19855  df-dchr 24958

This theorem is referenced by:  dchrabs  24985  sum2dchr  24999