dchrn0 - Metamath Proof Explorer

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Theorem dchrn0 27309

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 6907	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑋‘𝑥) = (𝑋‘𝐴))
2	1	neeq1d 2998	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑋‘𝑥) ≠ 0 ↔ (𝑋‘𝐴) ≠ 0))
3		eleq1 2827	. . . . 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 27299	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ)
12	5, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
13	6, 7, 8, 9, 12, 10	dchrelbas2 27296	. . . . . 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 3623	. . 3 ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 → 𝐴 ∈ 𝑈))
18	17	imp 406	. 2 ⊢ ((𝜑 ∧ (𝑋‘𝐴) ≠ 0) → 𝐴 ∈ 𝑈)
19		ax-1ne0 11222	. . . . 5 ⊢ 1 ≠ 0
20	19	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → 1 ≠ 0)
21	12	nnnn0d 12585	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
22	7	zncrng 21581	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → 𝑍 ∈ CRing)
23		crngring 20263	. . . . . . . 8 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring)
24	21, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ Ring)
25		eqid 2735	. . . . . . . 8 ⊢ (inv_r‘𝑍) = (inv_r‘𝑍)
26		eqid 2735	. . . . . . . 8 ⊢ (._r‘𝑍) = (._r‘𝑍)
27		eqid 2735	. . . . . . . 8 ⊢ (1_r‘𝑍) = (1_r‘𝑍)
28	9, 25, 26, 27	unitrinv 20411	. . . . . . 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 6911	. . . . 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 20409	. . . . . . 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 2735	. . . . . . . 8 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍)
37	36, 8	mgpbas 20158	. . . . . . 7 ⊢ 𝐵 = (Base‘(mulGrp‘𝑍))
38	36, 26	mgpplusg 20156	. . . . . . 7 ⊢ (._r‘𝑍) = (+_g‘(mulGrp‘𝑍))
39		eqid 2735	. . . . . . . 8 ⊢ (mulGrp‘ℂ_fld) = (mulGrp‘ℂ_fld)
40		cnfldmul 21390	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
41	39, 40	mgpplusg 20156	. . . . . . 7 ⊢ · = (+_g‘(mulGrp‘ℂ_fld))
42	37, 38, 41	mhmlin 18819	. . . . . 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 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘(𝐴(._r‘𝑍)((inv_r‘𝑍)‘𝐴))) = ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))))
44	36, 27	ringidval 20201	. . . . . . 7 ⊢ (1_r‘𝑍) = (0_g‘(mulGrp‘𝑍))
45		cnfld1 21424	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
46	39, 45	ringidval 20201	. . . . . . 7 ⊢ 1 = (0_g‘(mulGrp‘ℂ_fld))
47	44, 46	mhm0 18820	. . . . . 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 2783	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 1)
50		cnfldbas 21386	. . . . . . . . 9 ⊢ ℂ = (Base‘ℂ_fld)
51	39, 50	mgpbas 20158	. . . . . . . 8 ⊢ ℂ = (Base‘(mulGrp‘ℂ_fld))
52	37, 51	mhmf 18815	. . . . . . 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 7105	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (𝑋‘((inv_r‘𝑍)‘𝐴)) ∈ ℂ)
55	54	mul02d 11457	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))) = 0)
56	20, 49, 55	3netr4d 3016	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑈) → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) ≠ (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
57		oveq1 7438	. . . 4 ⊢ ((𝑋‘𝐴) = 0 → ((𝑋‘𝐴) · (𝑋‘((inv_r‘𝑍)‘𝐴))) = (0 · (𝑋‘((inv_r‘𝑍)‘𝐴))))
58	57	necon3i 2971	. . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 ℕcn 12264 ℕ₀cn0 12524 Basecbs 17245 ._rcmulr 17299 MndHom cmhm 18807 mulGrpcmgp 20152 1_rcur 20199 Ringcrg 20251 CRingccrg 20252 Unitcui 20372 inv_rcinvr 20404 ℂ_fldccnfld 21382 ℤ/nℤczn 21531 DChrcdchr 27291

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-eqg 19156 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-subrng 20563 df-subrg 20587 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zn 21535 df-dchr 27292

This theorem is referenced by: dchrinvcl 27312 dchrfi 27314 dchrghm 27315 dchreq 27317 dchrabs 27319 dchrabs2 27321 dchr1re 27322 dchrpt 27326 dchrsum 27328 sum2dchr 27333 rpvmasumlem 27546 dchrisum0flblem1 27567

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