dchrghm - Metamath Proof Explorer

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Theorem dchrghm 27219

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 27204	. . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
5		dchrghm.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6	4, 5	sselid 3920	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
7	1, 3	dchrrcl 27203	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
8	5, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12498	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	2	zncrng 21524	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
12		crngring 20226	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
14		dchrghm.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑍)
15		eqid 2737	. . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
16	14, 15	unitsubm 20366	. . . . 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 18788	. . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
20	6, 17, 19	syl2anc 585	. . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
21		cnring 21374	. . . . 5 ⊢ ℂ_fld ∈ Ring
22		cnfldbas 21356	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
23		cnfld0 21376	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
24		cndrng 21381	. . . . . . 7 ⊢ ℂ_fld ∈ DivRing
25	22, 23, 24	drngui 20712	. . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
26		eqid 2737	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
27	25, 26	unitsubm 20366	. . . . 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 5644	. . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈)
30		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍)
31	1, 2, 3, 30, 5	dchrf 27205	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
32	30, 14	unitss 20356	. . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍)
33	32	sseli 3918	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍))
34		ffvelcdm 7034	. . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
35	31, 33, 34	syl2an 597	. . . . . . . 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 27213	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
40	36, 39	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0)
41		eldifsn 4732	. . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0))
42	35, 40, 41	sylanbrc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
43	42	ralrimiva 3130	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
44	31	ffund 6673	. . . . . . 7 ⊢ (𝜑 → Fun 𝑋)
45	31	fdmd 6679	. . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍))
46	32, 45	sseqtrrid 3966	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋)
47		funimass4 6905	. . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
48	44, 46, 47	syl2anc 585	. . . . . 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 3962	. . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0}))
51		dchrghm.m	. . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
52	51	resmhm2b 18790	. . . 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 588	. . 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 20363	. . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
56	13, 55	syl 17	. . 3 ⊢ (𝜑 → 𝐻 ∈ Grp)
57	51	cnmgpabl 21408	. . . 4 ⊢ 𝑀 ∈ Abel
58		ablgrp 19760	. . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp)
59	57, 58	ax-mp 5	. . 3 ⊢ 𝑀 ∈ Grp
60		ghmmhmb 19202	. . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
61	56, 59, 60	sylancl 587	. 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
62	54, 61	eleqtrrd 2840	1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∖ cdif 3887 ⊆ wss 3890 {csn 4568 dom cdm 5631 ran crn 5632 ↾ cres 5633 “ cima 5634 Fun wfun 6493 ⟶wf 6495 ‘cfv 6499 (class class class)co 7367 ℂcc 11036 0cc0 11038 ℕcn 12174 ℕ₀cn0 12437 Basecbs 17179 ↾_s cress 17200 MndHom cmhm 18749 SubMndcsubmnd 18750 Grpcgrp 18909 GrpHom cghm 19187 Abelcabl 19756 mulGrpcmgp 20121 Ringcrg 20214 CRingccrg 20215 Unitcui 20335 ℂ_fldccnfld 21352 ℤ/nℤczn 21482 DChrcdchr 27195

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6266 df-ord 6327 df-on 6328 df-lim 6329 df-suc 6330 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-subrng 20523 df-subrg 20547 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zn 21486 df-dchr 27196

This theorem is referenced by: dchrabs 27223 sum2dchr 27237

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