dchrvmaeq0 - Metamath Proof Explorer

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Theorem dchrvmaeq0 27723

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 4755	. . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 ))
4	1, 2, 3	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 }))
5		fveq1 6884	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚)))
6	5	oveq1d 7434	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
7	6	sumeq2sdv 15782	. . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚))
8	7	eqeq1d 2767	. . . . 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 3656	. . . 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 12921	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
14		1zzd 12644	. . . 4 ⊢ (𝜑 → 1 ∈ ℤ)
15		2fveq3 6890	. . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚)))
16		id 23	. . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚)
17	15, 16	oveq12d 7437	. . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚))
18		dchrvmasumif.f	. . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
19		ovex 7452	. . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V
20	17, 18, 19	fvmpt 6993	. . . . 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 12631	. . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ)
28	27	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ)
29	22, 23, 24, 25, 26, 28	dchrzrhcl 27464	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
30		nncn 12260	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ)
31	30	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ)
32		nnne0 12289	. . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0)
33	32	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0)
34	29, 31, 33	divcld 12010	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ)
35		dchrvmasumif.s	. . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
36	13, 14, 21, 34, 35	isumclim 15835	. . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆)
37	36	eqeq1d 2767	. 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 2146 ≠ wne 2960 ∀wral 3081 {crab 3418 ∖ cdif 3903 {csn 4591 class class class wbr 5111 ↦ cmpt 5194 ‘cfv 6540 (class class class)co 7419 ℂcc 11117 0cc0 11119 1c1 11120 + caddc 11122 +∞cpnf 11259 ≤ cle 11263 − cmin 11460 / cdiv 11890 ℕcn 12252 ℤcz 12610 [,)cico 13394 ⌊cfl 13845 seqcseq 14059 abscabs 15313 ⇝ cli 15563 Σcsu 15765 Basecbs 17295 0_gc0g 17518 ℤRHomczrh 21703 ℤ/nℤczn 21706 DChrcdchr 27451

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196 ax-pre-sup 11197 ax-addf 11198 ax-mulf 11199

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-sup 9409 df-inf 9410 df-oi 9479 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-div 11891 df-nn 12253 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12524 df-z 12611 df-dec 12732 df-uz 12883 df-rp 13037 df-fz 13556 df-fzo 13704 df-seq 14060 df-exp 14120 df-hash 14389 df-cj 15178 df-re 15179 df-im 15180 df-sqrt 15314 df-abs 15315 df-clim 15567 df-sum 15766 df-struct 17233 df-sets 17250 df-slot 17268 df-ndx 17280 df-base 17296 df-ress 17317 df-plusg 17349 df-mulr 17350 df-starv 17351 df-sca 17352 df-vsca 17353 df-ip 17354 df-tset 17355 df-ple 17356 df-ds 17358 df-unif 17359 df-0g 17520 df-imas 17588 df-qus 17589 df-mgm 18724 df-sgrp 18813 df-mnd 18829 df-mhm 18882 df-grp 19051 df-minusg 19052 df-sbg 19053 df-mulg 19182 df-subg 19237 df-nsg 19238 df-eqg 19239 df-ghm 19332 df-cmn 19900 df-abl 19901 df-mgp 20265 df-rng 20279 df-ur 20312 df-ring 20365 df-cring 20366 df-oppr 20469 df-dvdsr 20489 df-unit 20490 df-rhm 20604 df-subrng 20699 df-subrg 20723 df-lmod 21037 df-lss 21107 df-lsp 21147 df-sra 21348 df-rgmod 21349 df-lidl 21386 df-rsp 21387 df-2idl 21443 df-cnfld 21577 df-zring 21651 df-zrh 21707 df-zn 21710 df-dchr 27452

This theorem is used by: rpvmasum2 27731 dchrisum0re 27732 dchrisum0lem2 27737 dchrisumn0 27740

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