dchrn0 - Metamath Proof Explorer

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Theorem dchrn0 27201

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 6832	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴))
2	1	neeq1d 2992	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0))
3		eleq1 2825	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑈 ↔ 𝐴 ∈ 𝑈))
4	2, 3	imbi12d 344	. . . 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 27191	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
12	5, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	6, 7, 8, 9, 12, 10	dchrelbas2 27188	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))))
14	5, 13	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))
15	14	simprd 495	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))
16		dchrn0.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
17	4, 15, 16	rspcdva 3566	. . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))
18	17	imp 406	. 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈)
19		ax-1ne0 11096	. . . . 5 ⊢ 1 ≠ 0
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0)
21	12	nnnn0d 12463	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	7	zncrng 21501	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23		crngring 20184	. . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
24	21, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring)
25		eqid 2737	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
26		eqid 2737	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
27		eqid 2737	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
28	9, 25, 26, 27	unitrinv 20332	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
29	24, 28	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
30	29	fveq2d 6836	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = (𝑋‘(1_r‘𝑍)))
31	14	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
32	31	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂ_fld)))
33	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 𝐴 ∈ 𝐵)
34	9, 25, 8	ringinvcl 20330	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
35	24, 34	sylan 581	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
36		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
37	36, 8	mgpbas 20084	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
38	36, 26	mgpplusg 20083	. . . . . . 7 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
39		eqid 2737	. . . . . . . 8 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
40		cnfldmul 21319	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
41	39, 40	mgpplusg 20083	. . . . . . 7 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
42	37, 38, 41	mhmlin 18719	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
44	36, 27	ringidval 20122	. . . . . . 7 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
45		cnfld1 21350	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
46	39, 45	ringidval 20122	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
47	44, 46	mhm0 18720	. . . . . 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 2780	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 1)
50		cnfldbas 21315	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
51	39, 50	mgpbas 20084	. . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
52	37, 51	mhmf 18715	. . . . . . 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	ffvelcdmd 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((inv_r‘𝑍)‘𝐴)) ∈ ℂ)
55	54	mul02d 11332	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 0)
56	20, 49, 55	3netr4d 3010	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
57		oveq1 7365	. . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
58	57	necon3i 2965	. . 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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 0cc0 11027 1c1 11028 · cmul 11032 ℕcn 12146 ℕ₀cn0 12402 Basecbs 17137 ._rcmulr 17179 MndHom cmhm 18707 mulGrpcmgp 20079 1_rcur 20120 Ringcrg 20172 CRingccrg 20173 Unitcui 20293 inv_rcinvr 20325 ℂ_fldccnfld 21311 ℤ/nℤczn 21459 DChrcdchr 27183

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 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-addf 11106 ax-mulf 11107

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 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-er 8634 df-ec 8636 df-qs 8640 df-map 8766 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9346 df-inf 9347 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-fz 13425 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-0g 17362 df-imas 17430 df-qus 17431 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-nsg 19058 df-eqg 19059 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-subrng 20481 df-subrg 20505 df-lmod 20815 df-lss 20885 df-lsp 20925 df-sra 21127 df-rgmod 21128 df-lidl 21165 df-rsp 21166 df-2idl 21207 df-cnfld 21312 df-zring 21404 df-zn 21463 df-dchr 27184

This theorem is referenced by: dchrinvcl 27204 dchrfi 27206 dchrghm 27207 dchreq 27209 dchrabs 27211 dchrabs2 27213 dchr1re 27214 dchrpt 27218 dchrsum 27220 sum2dchr 27225 rpvmasumlem 27438 dchrisum0flblem1 27459

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