dchrn0 - Metamath Proof Explorer

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Theorem dchrn0 26085

Description: A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)

**Hypotheses**
Ref	Expression
dchrmhm.g	⊢ 𝐺 = (DChr‘𝑁)
dchrmhm.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
dchrmhm.b	⊢ 𝐷 = (Base‘𝐺)
dchrn0.b	⊢ 𝐵 = (Base‘𝑍)
dchrn0.u	⊢ 𝑈 = (Unit‘𝑍)
dchrn0.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrn0.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
dchrn0	⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈))

Proof of Theorem dchrn0 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6695	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴))
2	1	neeq1d 2991	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0))
3		eleq1 2818	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
4	2, 3	imbi12d 348	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈) ↔ ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈)))
5		dchrn0.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
6		dchrmhm.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
7		dchrmhm.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
8		dchrn0.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑍)
9		dchrn0.u	. . . . . . 7 ⊢ 𝑈 = (Unit‘𝑍)
10		dchrmhm.b	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
11	6, 10	dchrrcl 26075	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
12	5, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	6, 7, 8, 9, 12, 10	dchrelbas2 26072	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))))
14	5, 13	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))
15	14	simprd 499	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))
16		dchrn0.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
17	4, 15, 16	rspcdva 3529	. . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))
18	17	imp 410	. 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈)
19		ax-1ne0 10763	. . . . 5 ⊢ 1 ≠ 0
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0)
21	12	nnnn0d 12115	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	7	zncrng 20463	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23		crngring 19528	. . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
24	21, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring)
25		eqid 2736	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
26		eqid 2736	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
27		eqid 2736	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
28	9, 25, 26, 27	unitrinv 19650	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
29	24, 28	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
30	29	fveq2d 6699	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = (𝑋‘(1_r‘𝑍)))
31	14	simpld 498	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
32	31	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
33	16	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝐵)
34	9, 25, 8	ringinvcl 19648	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
35	24, 34	sylan 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
36		eqid 2736	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
37	36, 8	mgpbas 19464	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
38	36, 26	mgpplusg 19462	. . . . . . 7 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
39		eqid 2736	. . . . . . . 8 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
40		cnfldmul 20323	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
41	39, 40	mgpplusg 19462	. . . . . . 7 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
42	37, 38, 41	mhmlin 18179	. . . . . 6 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ 𝐴 ∈ 𝐵 ∧ ((inv_r‘𝑍)‘𝐴) ∈ 𝐵) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
43	32, 33, 35, 42	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
44	36, 27	ringidval 19472	. . . . . . 7 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
45		cnfld1 20342	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
46	39, 45	ringidval 19472	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
47	44, 46	mhm0 18180	. . . . . 6 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → (𝑋‘(1_r‘𝑍)) = 1)
48	32, 47	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(1_r‘𝑍)) = 1)
49	30, 43, 48	3eqtr3d 2779	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 1)
50		cnfldbas 20321	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
51	39, 50	mgpbas 19464	. . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
52	37, 51	mhmf 18177	. . . . . . 7 ⊢ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) → 𝑋:𝐵⟶ℂ)
53	32, 52	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋:𝐵⟶ℂ)
54	53, 35	ffvelrnd 6883	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((inv_r‘𝑍)‘𝐴)) ∈ ℂ)
55	54	mul02d 10995	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 0)
56	20, 49, 55	3netr4d 3009	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
57		oveq1 7198	. . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
58	57	necon3i 2964	. . 3 ⊢ (((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) → (𝑋‘𝐴) ≠ 0)
59	56, 58	syl 17	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘𝐴) ≠ 0)
60	18, 59	impbida 801	1 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 · cmul 10699 ℕcn 11795 ℕ₀cn0 12055 Basecbs 16666 ._rcmulr 16750 MndHom cmhm 18170 mulGrpcmgp 19458 1_rcur 19470 Ringcrg 19516 CRingccrg 19517 Unitcui 19611 inv_rcinvr 19643 ℂ_fldccnfld 20317 ℤ/nℤczn 20423 DChrcdchr 26067

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-imas 16967 df-qus 16968 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-nsg 18495 df-eqg 18496 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-sra 20163 df-rgmod 20164 df-lidl 20165 df-rsp 20166 df-2idl 20224 df-cnfld 20318 df-zring 20390 df-zn 20427 df-dchr 26068

This theorem is referenced by: dchrinvcl 26088 dchrfi 26090 dchrghm 26091 dchreq 26093 dchrabs 26095 dchrabs2 26097 dchr1re 26098 dchrpt 26102 dchrsum 26104 sum2dchr 26109 rpvmasumlem 26322 dchrisum0flblem1 26343

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