dchrn0 - Metamath Proof Explorer

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Theorem dchrn0 27194

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 6828	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴))
2	1	neeq1d 2987	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0))
3		eleq1 2819	. . . . 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 27184	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
12	5, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	6, 7, 8, 9, 12, 10	dchrelbas2 27181	. . . . . 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 3573	. . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))
18	17	imp 406	. 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈)
19		ax-1ne0 11081	. . . . 5 ⊢ 1 ≠ 0
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0)
21	12	nnnn0d 12448	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	7	zncrng 21487	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23		crngring 20169	. . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
24	21, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring)
25		eqid 2731	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
26		eqid 2731	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
27		eqid 2731	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
28	9, 25, 26, 27	unitrinv 20318	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
29	24, 28	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴)) = (1_r‘𝑍))
30	29	fveq2d 6832	. . . . 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 20316	. . . . . . 7 ⊢ ((𝑍 ∈ Ring ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
35	24, 34	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((inv_r‘𝑍)‘𝐴) ∈ 𝐵)
36		eqid 2731	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
37	36, 8	mgpbas 20069	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
38	36, 26	mgpplusg 20068	. . . . . . 7 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
39		eqid 2731	. . . . . . . 8 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
40		cnfldmul 21305	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
41	39, 40	mgpplusg 20068	. . . . . . 7 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
42	37, 38, 41	mhmlin 18707	. . . . . 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 20107	. . . . . . 7 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
45		cnfld1 21336	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
46	39, 45	ringidval 20107	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
47	44, 46	mhm0 18708	. . . . . 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 2774	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 1)
50		cnfldbas 21301	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
51	39, 50	mgpbas 20069	. . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
52	37, 51	mhmf 18703	. . . . . . 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 7024	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((inv_r‘𝑍)‘𝐴)) ∈ ℂ)
55	54	mul02d 11317	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 0)
56	20, 49, 55	3netr4d 3005	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
57		oveq1 7359	. . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
58	57	necon3i 2960	. . 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 800	1 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 ℂcc 11010 0cc0 11012 1c1 11013 · cmul 11017 ℕcn 12131 ℕ₀cn0 12387 Basecbs 17126 ._rcmulr 17168 MndHom cmhm 18695 mulGrpcmgp 20064 1_rcur 20105 Ringcrg 20157 CRingccrg 20158 Unitcui 20279 inv_rcinvr 20311 ℂ_fldccnfld 21297 ℤ/nℤczn 21445 DChrcdchr 27176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 ax-1cn 11070 ax-icn 11071 ax-addcl 11072 ax-addrcl 11073 ax-mulcl 11074 ax-mulrcl 11075 ax-mulcom 11076 ax-addass 11077 ax-mulass 11078 ax-distr 11079 ax-i2m1 11080 ax-1ne0 11081 ax-1rid 11082 ax-rnegex 11083 ax-rrecex 11084 ax-cnre 11085 ax-pre-lttri 11086 ax-pre-lttrn 11087 ax-pre-ltadd 11088 ax-pre-mulgt0 11089 ax-addf 11091 ax-mulf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-ec 8630 df-qs 8634 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9332 df-inf 9333 df-pnf 11154 df-mnf 11155 df-xr 11156 df-ltxr 11157 df-le 11158 df-sub 11352 df-neg 11353 df-nn 12132 df-2 12194 df-3 12195 df-4 12196 df-5 12197 df-6 12198 df-7 12199 df-8 12200 df-9 12201 df-n0 12388 df-z 12475 df-dec 12595 df-uz 12739 df-fz 13414 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17127 df-ress 17148 df-plusg 17180 df-mulr 17181 df-starv 17182 df-sca 17183 df-vsca 17184 df-ip 17185 df-tset 17186 df-ple 17187 df-ds 17189 df-unif 17190 df-0g 17351 df-imas 17418 df-qus 17419 df-mgm 18554 df-sgrp 18633 df-mnd 18649 df-mhm 18697 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19042 df-nsg 19043 df-eqg 19044 df-cmn 19700 df-abl 19701 df-mgp 20065 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20261 df-dvdsr 20281 df-unit 20282 df-invr 20312 df-subrng 20467 df-subrg 20491 df-lmod 20801 df-lss 20871 df-lsp 20911 df-sra 21113 df-rgmod 21114 df-lidl 21151 df-rsp 21152 df-2idl 21193 df-cnfld 21298 df-zring 21390 df-zn 21449 df-dchr 27177

This theorem is referenced by: dchrinvcl 27197 dchrfi 27199 dchrghm 27200 dchreq 27202 dchrabs 27204 dchrabs2 27206 dchr1re 27207 dchrpt 27211 dchrsum 27213 sum2dchr 27218 rpvmasumlem 27431 dchrisum0flblem1 27452

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