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Theorem dchrvmaeq0 27442

Description: The set 𝑊 is the collection of all non-principal Dirichlet characters such that the sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasumif.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrvmasumif.c	⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
dchrvmasumif.s	⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrvmasumif.1	⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
dchrvmaeq0.w	⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}

**Assertion**
Ref	Expression
dchrvmaeq0	⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0))

Distinct variable groups: 𝑦,𝑚, 1 𝐶,𝑚,𝑦 𝑦,𝐹 𝑚,𝑎,𝑦 𝑚,𝑁,𝑦 𝜑,𝑚 𝑆,𝑚,𝑦 𝑚,𝑍,𝑦 𝐷,𝑚,𝑦 𝐿,𝑎,𝑚,𝑦 𝑋,𝑎,𝑚,𝑦 𝑚,𝐹 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑆(𝑎) 1 (𝑎) 𝐹(𝑎) 𝐺(𝑦,𝑚,𝑎) 𝑁(𝑎) 𝑊(𝑦,𝑚,𝑎) 𝑍(𝑎)

**Proof of Theorem dchrvmaeq0**
Step	Hyp	Ref	Expression
1		dchrisum.b	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		dchrisum.n1	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 1 )
3		eldifsn 4735	. . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 ))
4	1, 2, 3	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
5		fveq1 6821	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚)))
6	5	oveq1d 7361	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
7	6	sumeq2sdv 15610	. . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
8	7	eqeq1d 2733	. . . . 5 ⊢ (𝑦 = 𝑋 → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
9		dchrvmaeq0.w	. . . . 5 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
10	8, 9	elrab2 3645	. . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
11	10	baib 535	. . 3 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
12	4, 11	syl 17	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
13		nnuz 12775	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
14		1zzd 12503	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
15		2fveq3 6827	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
16		id 22	. . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
17	15, 16	oveq12d 7364	. . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
18		dchrvmasumif.f	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
19		ovex 7379	. . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
20	17, 18, 19	fvmpt 6929	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
21	20	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
22		rpvmasum.g	. . . . . 6 ⊢ 𝐺 = (DChr‘𝑁)
23		rpvmasum.z	. . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
24		rpvmasum.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐺)
25		rpvmasum.l	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍)
26	1	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
27		nnz 12489	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
28	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
29	22, 23, 24, 25, 26, 28	dchrzrhcl 27183	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
30		nncn 12133	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
31	30	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
32		nnne0 12159	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
33	32	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
34	29, 31, 33	divcld 11897	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
35		dchrvmasumif.s	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
36	13, 14, 21, 34, 35	isumclim 15664	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆)
37	36	eqeq1d 2733	. 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0))
38	12, 37	bitrd 279	1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 {crab 3395 ∖ cdif 3894 {csn 4573 class class class wbr 5089 ↦ cmpt 5170 ‘cfv 6481 (class class class)co 7346 ℂcc 11004 0cc0 11006 1c1 11007 + caddc 11009 +∞cpnf 11143 ≤ cle 11147 − cmin 11344 / cdiv 11774 ℕcn 12125 ℤcz 12468 [,)cico 13247 ⌊cfl 13694 seqcseq 13908 abscabs 15141 ⇝ cli 15391 Σcsu 15593 Basecbs 17120 0_gc0g 17343 ℤRHomczrh 21436 ℤ/nℤczn 21439 DChrcdchr 27170

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 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

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 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-0g 17345 df-imas 17412 df-qus 17413 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-nsg 19037 df-eqg 19038 df-ghm 19125 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-rhm 20390 df-subrng 20461 df-subrg 20485 df-lmod 20795 df-lss 20865 df-lsp 20905 df-sra 21107 df-rgmod 21108 df-lidl 21145 df-rsp 21146 df-2idl 21187 df-cnfld 21292 df-zring 21384 df-zrh 21440 df-zn 21443 df-dchr 27171

This theorem is referenced by: rpvmasum2 27450 dchrisum0re 27451 dchrisum0lem2 27456 dchrisumn0 27459

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