dchrghm - Metamath Proof Explorer

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Theorem dchrghm 27613

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 27598	. . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
5		dchrghm.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6	4, 5	sselid 3911	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
7	1, 3	dchrrcl 27597	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
8	5, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12694	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	2	zncrng 21880	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
11	9, 10	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
12		crngring 20502	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
13	11, 12	syl 18	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
14		dchrghm.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑍)
15		eqid 2763	. . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
16	14, 15	unitsubm 20646	. . . . 5 ⊢ (𝑍 ∈ Ring → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍)))
17	13, 16	syl 18	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍)))
18		dchrghm.h	. . . . 5 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾_s 𝑈)
19	18	resmhm 19045	. . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
20	6, 17, 19	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
21		cnring 21730	. . . . 5 ⊢ ℂ_fld ∈ Ring
22		cnfldbas 21712	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
23		cnfld0 21732	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
24		cndrng 21737	. . . . . . 7 ⊢ ℂ_fld ∈ DivRing
25	22, 23, 24	drngui 21016	. . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
26		eqid 2763	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
27	25, 26	unitsubm 20646	. . . . 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 5668	. . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈)
30		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍)
31	1, 2, 3, 30, 5	dchrf 27599	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
32	30, 14	unitss 20636	. . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍)
33	32	sseli 3909	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍))
34		ffvelcdm 7105	. . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
35	31, 33, 34	syl2an 608	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ)
36		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
37	5	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷)
38	33	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍))
39	1, 2, 3, 30, 14, 37, 38	dchrn0 27607	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
40	36, 39	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0)
41		eldifsn 4748	. . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0))
42	35, 40, 41	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
43	42	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
44	31	ffund 6736	. . . . . . 7 ⊢ (𝜑 → Fun 𝑋)
45	31	fdmd 6742	. . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍))
46	32, 45	sseqtrrid 3956	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋)
47		funimass4 6972	. . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
48	44, 46, 47	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
49	43, 48	mpbird 260	. . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}))
50	29, 49	eqsstrrid 3952	. . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0}))
51		dchrghm.m	. . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
52	51	resmhm2b 19047	. . . 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 599	. . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)))
54	20, 53	mpbid 235	. 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))
55	14, 18	unitgrp 20643	. . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
56	13, 55	syl 18	. . 3 ⊢ (𝜑 → 𝐻 ∈ Grp)
57	51	cnmgpabl 21764	. . . 4 ⊢ 𝑀 ∈ Abel
58		ablgrp 20029	. . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp)
59	57, 58	ax-mp 5	. . 3 ⊢ 𝑀 ∈ Grp
60		ghmmhmb 19470	. . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
61	56, 59, 60	sylancl 598	. 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
62	54, 61	eleqtrrd 2867	1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∖ cdif 4048 {csn 4588 dom cdm 5655 ran crn 5656 ↾ cres 5657 “ cima 5658 Fun wfun 6552 ⟶wf 6554 ‘cfv 6558 (class class class)co 7443 ℂcc 11225 0cc0 11227 ℕcn 12362 ℕ₀cn0 12633 Basecbs 17416 ↾_s cress 17437 MndHom cmhm 19005 SubMndcsubmnd 19006 Grpcgrp 19173 GrpHom cghm 19456 Abelcabl 20025 mulGrpcmgp 20390 Ringcrg 20489 CRingccrg 20490 Unitcui 20615 ℂ_fldccnfld 21708 ℤ/nℤczn 21838 DChrcdchr 27589

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-addf 11306 ax-mulf 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-ec 8747 df-qs 8751 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-0g 17641 df-imas 17709 df-qus 17710 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-submnd 19008 df-grp 19176 df-minusg 19177 df-sbg 19178 df-subg 19362 df-nsg 19363 df-eqg 19364 df-ghm 19457 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-unit 20618 df-invr 20648 df-dvr 20661 df-subrng 20828 df-subrg 20852 df-drng 21012 df-lmod 21167 df-lss 21237 df-lsp 21277 df-sra 21478 df-rgmod 21479 df-lidl 21516 df-rsp 21517 df-2idl 21573 df-cnfld 21709 df-zring 21783 df-zn 21842 df-dchr 27590

This theorem is used by: dchrabs 27617 sum2dchr 27631

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