dchrghm - Metamath Proof Explorer

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Theorem dchrghm 27194

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‘ℂ_fld) ↾_s (ℂ ∖ {0}))
dchrghm.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)

**Assertion**
Ref	Expression
dchrghm	⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (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 27179	. . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
5		dchrghm.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6	4, 5	sselid 3927	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
7	1, 3	dchrrcl 27178	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
8	5, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12442	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	2	zncrng 21481	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
12		crngring 20163	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
14		dchrghm.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑍)
15		eqid 2731	. . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
16	14, 15	unitsubm 20304	. . . . 5 ⊢ (𝑍 ∈ Ring → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍)))
17	13, 16	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍)))
18		dchrghm.h	. . . . 5 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
19	18	resmhm 18728	. . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
20	6, 17, 19	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
21		cnring 21327	. . . . 5 ⊢ ℂ_fld ∈ Ring
22		cnfldbas 21295	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
23		cnfld0 21329	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
24		cndrng 21335	. . . . . . 7 ⊢ ℂ_fld ∈ DivRing
25	22, 23, 24	drngui 20650	. . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
26		eqid 2731	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
27	25, 26	unitsubm 20304	. . . . 5 ⊢ (ℂ_fld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld)))
28	21, 27	ax-mp 5	. . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld))
29		df-ima 5627	. . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈)
30		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍)
31	1, 2, 3, 30, 5	dchrf 27180	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
32	30, 14	unitss 20294	. . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍)
33	32	sseli 3925	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍))
34		ffvelcdm 7014	. . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
35	31, 33, 34	syl2an 596	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ)
36		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
37	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷)
38	33	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍))
39	1, 2, 3, 30, 14, 37, 38	dchrn0 27188	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
40	36, 39	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0)
41		eldifsn 4735	. . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0))
42	35, 40, 41	sylanbrc 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
43	42	ralrimiva 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
44	31	ffund 6655	. . . . . . 7 ⊢ (𝜑 → Fun 𝑋)
45	31	fdmd 6661	. . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍))
46	32, 45	sseqtrrid 3973	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋)
47		funimass4 6886	. . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
48	44, 46, 47	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
49	43, 48	mpbird 257	. . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}))
50	29, 49	eqsstrrid 3969	. . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0}))
51		dchrghm.m	. . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
52	51	resmhm2b 18730	. . . 4 ⊢ (((ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂ_fld)) ∧ ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)))
53	28, 50, 52	sylancr 587	. . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)))
54	20, 53	mpbid 232	. 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))
55	14, 18	unitgrp 20301	. . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
56	13, 55	syl 17	. . 3 ⊢ (𝜑 → 𝐻 ∈ Grp)
57	51	cnmgpabl 21365	. . . 4 ⊢ 𝑀 ∈ Abel
58		ablgrp 19697	. . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp)
59	57, 58	ax-mp 5	. . 3 ⊢ 𝑀 ∈ Grp
60		ghmmhmb 19139	. . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
61	56, 59, 60	sylancl 586	. 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
62	54, 61	eleqtrrd 2834	1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∖ cdif 3894 ⊆ wss 3897 {csn 4573 dom cdm 5614 ran crn 5615 ↾ cres 5616 “ cima 5617 Fun wfun 6475 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 0cc0 11006 ℕcn 12125 ℕ₀cn0 12381 Basecbs 17120 ↾_s cress 17141 MndHom cmhm 18689 SubMndcsubmnd 18690 Grpcgrp 18846 GrpHom cghm 19124 Abelcabl 19693 mulGrpcmgp 20058 Ringcrg 20151 CRingccrg 20152 Unitcui 20273 ℂ_fldccnfld 21291 ℤ/nℤczn 21439 DChrcdchr 27170

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-imas 17412 df-qus 17413 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-subrng 20461 df-subrg 20485 df-drng 20646 df-lmod 20795 df-lss 20865 df-lsp 20905 df-sra 21107 df-rgmod 21108 df-lidl 21145 df-rsp 21146 df-2idl 21187 df-cnfld 21292 df-zring 21384 df-zn 21443 df-dchr 27171

This theorem is referenced by: dchrabs 27198 sum2dchr 27212

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