rpvmasum - Metamath Proof Explorer

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Theorem rpvmasum 27494

Description: The sum of the von Mangoldt function over those integers 𝑛≡𝐴 (mod 𝑁) is asymptotic to log𝑥 / ϕ(𝑥) + 𝑂(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.u	⊢ 𝑈 = (Unit‘𝑍)
rpvmasum.b	⊢ (𝜑 → 𝐴 ∈ 𝑈)
rpvmasum.t	⊢ 𝑇 = (^◡𝐿 “ {𝐴})

**Assertion**
Ref	Expression
rpvmasum	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑛,𝐴 𝑛,𝑁,𝑥 𝜑,𝑛,𝑥 𝑇,𝑛,𝑥 𝑈,𝑛,𝑥 𝑛,𝑍,𝑥 𝑛,𝐿,𝑥 𝐴,𝑛

Proof of Theorem rpvmasum Dummy variables 𝑚 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
2		rpvmasum.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (ℤRHom‘𝑍)
3		rpvmasum.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑁 ∈ ℕ)
5		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (DChr‘𝑁) = (DChr‘𝑁)
6		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘(DChr‘𝑁)) = (Base‘(DChr‘𝑁))
7		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0_g‘(DChr‘𝑁)) = (0_g‘(DChr‘𝑁))
8		2fveq3 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑦‘(𝐿‘𝑚)) = (𝑦‘(𝐿‘𝑛)))
9		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
10	8, 9	oveq12d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑦‘(𝐿‘𝑛)) / 𝑛))
11	10	cbvsumv 15717	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛)
12	11	eqeq1i 2741	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0)
13	12	rabbii 3426	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0}
14		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})
15	1, 2, 4, 5, 6, 7, 13, 14	dchrisum0 27488	. . . . . . . . . . . . 13 ⊢ ¬ (𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})
16	15	imnani 400	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})
17	16	eq0rdv 4387	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = ∅)
18	17	fveq2d 6885	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = (♯‘∅))
19		hash0 14390	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
20	18, 19	eqtrdi 2787	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = 0)
21	20	oveq2d 7426	. . . . . . . 8 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = (1 − 0))
22		1m0e1 12366	. . . . . . . 8 ⊢ (1 − 0) = 1
23	21, 22	eqtrdi 2787	. . . . . . 7 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1)
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1)
25	24	oveq2d 7426	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = ((log‘𝑥) · 1))
26		relogcl 26541	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
28	27	recnd 11268	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29	28	mulridd 11257	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) = (log‘𝑥))
30	25, 29	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = (log‘𝑥))
31	30	oveq2d 7426	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})))) = (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))
32	31	mpteq2dva 5219	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))))) = (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))))
33		eqid 2736	. . 3 ⊢ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}
34		rpvmasum.u	. . 3 ⊢ 𝑈 = (Unit‘𝑍)
35		rpvmasum.b	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
36		rpvmasum.t	. . 3 ⊢ 𝑇 = (^◡𝐿 “ {𝐴})
37	15	pm2.21i 119	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝐴 = (1_r‘𝑍))
38	1, 2, 3, 5, 6, 7, 33, 34, 35, 36, 37	rpvmasum2 27480	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))))) ∈ 𝑂(1))
39	32, 38	eqeltrrd 2836	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 {crab 3420 ∖ cdif 3928 ∩ cin 3930 ∅c0 4313 {csn 4606 ↦ cmpt 5206 ^◡ccnv 5658 “ cima 5662 ‘cfv 6536 (class class class)co 7410 ℝcr 11133 0cc0 11134 1c1 11135 · cmul 11139 − cmin 11471 / cdiv 11899 ℕcn 12245 ℝ⁺crp 13013 ...cfz 13529 ⌊cfl 13812 ♯chash 14353 𝑂(1)co1 15507 Σcsu 15707 ϕcphi 16788 Basecbs 17233 0_gc0g 17458 1_rcur 20146 Unitcui 20320 ℤRHomczrh 21465 ℤ/nℤczn 21468 logclog 26520 Λcvma 27059 DChrcdchr 27200

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-inf2 9660 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-disj 5092 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-rpss 7722 df-om 7867 df-1st 7993 df-2nd 7994 df-supp 8165 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8724 df-ec 8726 df-qs 8730 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9379 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9529 df-dju 9920 df-card 9958 df-acn 9961 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-xnn0 12580 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13371 df-ioc 13372 df-ico 13373 df-icc 13374 df-fz 13530 df-fzo 13677 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-fac 14297 df-bc 14326 df-hash 14354 df-word 14537 df-concat 14594 df-s1 14619 df-shft 15091 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-limsup 15492 df-clim 15509 df-rlim 15510 df-o1 15511 df-lo1 15512 df-sum 15708 df-ef 16088 df-e 16089 df-sin 16090 df-cos 16091 df-tan 16092 df-pi 16093 df-dvds 16278 df-gcd 16519 df-prm 16696 df-numer 16759 df-denom 16760 df-phi 16790 df-pc 16862 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-hom 17300 df-cco 17301 df-rest 17441 df-topn 17442 df-0g 17460 df-gsum 17461 df-topgen 17462 df-pt 17463 df-prds 17466 df-xrs 17521 df-qtop 17526 df-imas 17527 df-qus 17528 df-xps 17529 df-mre 17603 df-mrc 17604 df-acs 17606 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-nsg 19112 df-eqg 19113 df-ghm 19201 df-gim 19247 df-ga 19278 df-cntz 19305 df-oppg 19334 df-od 19514 df-gex 19515 df-pgp 19516 df-lsm 19622 df-pj1 19623 df-cmn 19768 df-abl 19769 df-cyg 19864 df-dprd 19983 df-dpj 19984 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrng 20511 df-subrg 20535 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-sra 21136 df-rgmod 21137 df-lidl 21174 df-rsp 21175 df-2idl 21216 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-zring 21413 df-zrh 21469 df-zn 21472 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-0p 25628 df-limc 25824 df-dv 25825 df-ply 26150 df-idp 26151 df-coe 26152 df-dgr 26153 df-quot 26256 df-ulm 26343 df-log 26522 df-cxp 26523 df-atan 26834 df-em 26960 df-cht 27064 df-vma 27065 df-chp 27066 df-ppi 27067 df-mu 27068 df-dchr 27201

This theorem is referenced by: rplogsum 27495

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