dchrghm - Metamath Proof Explorer

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Theorem dchrghm 25528

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 25513	. . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld))
5		dchrghm.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6	4, 5	sseldi 3855	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
7	1, 3	dchrrcl 25512	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
8	5, 7	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 11764	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10	2	zncrng 20387	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing)
12		crngring 19025	. . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring)
14		dchrghm.u	. . . . . 6 ⊢ 𝑈 = (Unit‘𝑍)
15		eqid 2775	. . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
16	14, 15	unitsubm 19137	. . . . 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 17821	. . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
20	6, 17, 19	syl2anc 576	. . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)))
21		cnring 20263	. . . . 5 ⊢ ℂ_fld ∈ Ring
22		cnfldbas 20245	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
23		cnfld0 20265	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
24		cndrng 20270	. . . . . . 7 ⊢ ℂ_fld ∈ DivRing
25	22, 23, 24	drngui 19225	. . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂ_fld)
26		eqid 2775	. . . . . 6 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
27	25, 26	unitsubm 19137	. . . . 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 5417	. . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈)
30		eqid 2775	. . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍)
31	1, 2, 3, 30, 5	dchrf 25514	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ)
32	30, 14	unitss 19127	. . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍)
33	32	sseli 3853	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍))
34		ffvelrn 6672	. . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ)
35	31, 33, 34	syl2an 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ)
36		simpr 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈)
37	5	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷)
38	33	adantl 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍))
39	1, 2, 3, 30, 14, 37, 38	dchrn0 25522	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈))
40	36, 39	mpbird 249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0)
41		eldifsn 4591	. . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0))
42	35, 40, 41	sylanbrc 575	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
43	42	ralrimiva 3129	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))
44	31	ffund 6346	. . . . . . 7 ⊢ (𝜑 → Fun 𝑋)
45	31	fdmd 6351	. . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍))
46	32, 45	syl5sseqr 3909	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋)
47		funimass4 6558	. . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
48	44, 46, 47	syl2anc 576	. . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})))
49	43, 48	mpbird 249	. . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}))
50	29, 49	syl5eqssr 3905	. . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0}))
51		dchrghm.m	. . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂ_fld) ↾_s (ℂ ∖ {0}))
52	51	resmhm2b 17823	. . . 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 578	. . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂ_fld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)))
54	20, 53	mpbid 224	. 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))
55	14, 18	unitgrp 19134	. . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp)
56	13, 55	syl 17	. . 3 ⊢ (𝜑 → 𝐻 ∈ Grp)
57	51	cnmgpabl 20302	. . . 4 ⊢ 𝑀 ∈ Abel
58		ablgrp 18665	. . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp)
59	57, 58	ax-mp 5	. . 3 ⊢ 𝑀 ∈ Grp
60		ghmmhmb 18134	. . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
61	56, 59, 60	sylancl 577	. 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀))
62	54, 61	eleqtrrd 2866	1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ≠ wne 2964 ∀wral 3085 ∖ cdif 3825 ⊆ wss 3828 {csn 4439 dom cdm 5404 ran crn 5405 ↾ cres 5406 “ cima 5407 Fun wfun 6180 ⟶wf 6182 ‘cfv 6186 (class class class)co 6974 ℂcc 10329 0cc0 10331 ℕcn 11435 ℕ₀cn0 11704 Basecbs 16333 ↾_s cress 16334 MndHom cmhm 17795 SubMndcsubmnd 17796 Grpcgrp 17885 GrpHom cghm 18120 Abelcabl 18661 mulGrpcmgp 18956 Ringcrg 19014 CRingccrg 19015 Unitcui 19106 ℂ_fldccnfld 20241 ℤ/nℤczn 20346 DChrcdchr 25504

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-rep 5047 ax-sep 5058 ax-nul 5065 ax-pow 5117 ax-pr 5184 ax-un 7277 ax-cnex 10387 ax-resscn 10388 ax-1cn 10389 ax-icn 10390 ax-addcl 10391 ax-addrcl 10392 ax-mulcl 10393 ax-mulrcl 10394 ax-mulcom 10395 ax-addass 10396 ax-mulass 10397 ax-distr 10398 ax-i2m1 10399 ax-1ne0 10400 ax-1rid 10401 ax-rnegex 10402 ax-rrecex 10403 ax-cnre 10404 ax-pre-lttri 10405 ax-pre-lttrn 10406 ax-pre-ltadd 10407 ax-pre-mulgt0 10408 ax-addf 10410 ax-mulf 10411

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ne 2965 df-nel 3071 df-ral 3090 df-rex 3091 df-reu 3092 df-rmo 3093 df-rab 3094 df-v 3414 df-sbc 3681 df-csb 3786 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-pss 3844 df-nul 4178 df-if 4349 df-pw 4422 df-sn 4440 df-pr 4442 df-tp 4444 df-op 4446 df-uni 4711 df-int 4748 df-iun 4792 df-br 4928 df-opab 4990 df-mpt 5007 df-tr 5029 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7498 df-2nd 7499 df-tpos 7692 df-wrecs 7747 df-recs 7809 df-rdg 7847 df-1o 7901 df-oadd 7905 df-er 8085 df-ec 8087 df-qs 8091 df-map 8204 df-en 8303 df-dom 8304 df-sdom 8305 df-fin 8306 df-sup 8697 df-inf 8698 df-pnf 10472 df-mnf 10473 df-xr 10474 df-ltxr 10475 df-le 10476 df-sub 10668 df-neg 10669 df-div 11095 df-nn 11436 df-2 11500 df-3 11501 df-4 11502 df-5 11503 df-6 11504 df-7 11505 df-8 11506 df-9 11507 df-n0 11705 df-z 11791 df-dec 11909 df-uz 12056 df-fz 12706 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-0g 16565 df-imas 16631 df-qus 16632 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-mhm 17797 df-submnd 17798 df-grp 17888 df-minusg 17889 df-sbg 17890 df-subg 18054 df-nsg 18055 df-eqg 18056 df-ghm 18121 df-cmn 18662 df-abl 18663 df-mgp 18957 df-ur 18969 df-ring 19016 df-cring 19017 df-oppr 19090 df-dvdsr 19108 df-unit 19109 df-invr 19139 df-dvr 19150 df-drng 19221 df-subrg 19250 df-lmod 19352 df-lss 19420 df-lsp 19460 df-sra 19660 df-rgmod 19661 df-lidl 19662 df-rsp 19663 df-2idl 19720 df-cnfld 20242 df-zring 20314 df-zn 20350 df-dchr 25505

This theorem is referenced by: dchrabs 25532 sum2dchr 25546

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