dchrvmaeq0 - Metamath Proof Explorer

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Theorem dchrvmaeq0 27740

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 4748	. . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 ))
4	1, 2, 3	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
5		fveq1 6877	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚)))
6	5	oveq1d 7428	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
7	6	sumeq2sdv 15790	. . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
8	7	eqeq1d 2762	. . . . 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 3649	. . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
11	10	baib 545	. . 3 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
12	4, 11	syl 18	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0))
13		nnuz 12926	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
14		1zzd 12649	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
15		2fveq3 6883	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
16		id 23	. . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
17	15, 16	oveq12d 7431	. . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
18		dchrvmasumif.f	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
19		ovex 7446	. . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
20	17, 18, 19	fvmpt 6986	. . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
21	20	adantl 487	. . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷)
27		nnz 12636	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
29	22, 23, 24, 25, 26, 28	dchrzrhcl 27481	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
30		nncn 12265	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
31	30	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
32		nnne0 12294	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
33	32	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
34	29, 31, 33	divcld 12015	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
35		dchrvmasumif.s	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
36	13, 14, 21, 34, 35	isumclim 15843	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆)
37	36	eqeq1d 2762	. 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0))
38	12, 37	bitrd 282	1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 {crab 3412 ∖ cdif 3896 {csn 4584 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 0cc0 11124 1c1 11125 + caddc 11127 +∞cpnf 11264 ≤ cle 11268 − cmin 11465 / cdiv 11895 ℕcn 12257 ℤcz 12615 [,)cico 13400 ⌊cfl 13851 seqcseq 14065 abscabs 15321 ⇝ cli 15571 Σcsu 15773 Basecbs 17301 0_gc0g 17524 ℤRHomczrh 21712 ℤ/nℤczn 21715 DChrcdchr 27468

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-ec 8698 df-qs 8702 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 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 12529 df-z 12616 df-dec 12737 df-uz 12888 df-rp 13043 df-fz 13562 df-fzo 13710 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-0g 17526 df-imas 17594 df-qus 17595 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-mhm 18891 df-grp 19060 df-minusg 19061 df-sbg 19062 df-mulg 19191 df-subg 19246 df-nsg 19247 df-eqg 19248 df-ghm 19341 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-cring 20375 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-rhm 20613 df-subrng 20708 df-subrg 20732 df-lmod 21046 df-lss 21116 df-lsp 21156 df-sra 21357 df-rgmod 21358 df-lidl 21395 df-rsp 21396 df-2idl 21452 df-cnfld 21586 df-zring 21660 df-zrh 21716 df-zn 21719 df-dchr 27469

This theorem is used by: rpvmasum2 27748 dchrisum0re 27749 dchrisum0lem2 27754 dchrisumn0 27757

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