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Theorem rpvmasum 27588

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 2740	. . . . . . . . . . . . . 14 ⊢ (DChr‘𝑁) = (DChr‘𝑁)
6		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (Base‘(DChr‘𝑁)) = (Base‘(DChr‘𝑁))
7		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (0_g‘(DChr‘𝑁)) = (0_g‘(DChr‘𝑁))
8		2fveq3 6925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑦‘(𝐿‘𝑚)) = (𝑦‘(𝐿‘𝑛)))
9		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
10	8, 9	oveq12d 7466	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑦‘(𝐿‘𝑛)) / 𝑛))
11	10	cbvsumv 15744	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛)
12	11	eqeq1i 2745	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0)
13	12	rabbii 3449	. . . . . . . . . . . . . 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 27582	. . . . . . . . . . . . 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 4430	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = ∅)
18	17	fveq2d 6924	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = (♯‘∅))
19		hash0 14416	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
20	18, 19	eqtrdi 2796	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = 0)
21	20	oveq2d 7464	. . . . . . . 8 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = (1 − 0))
22		1m0e1 12414	. . . . . . . 8 ⊢ (1 − 0) = 1
23	21, 22	eqtrdi 2796	. . . . . . 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 7464	. . . . 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 26635	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
27	26	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
28	27	recnd 11318	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29	28	mulridd 11307	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) = (log‘𝑥))
30	25, 29	eqtrd 2780	. . . 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 7464	. . 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 5266	. 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 2740	. . 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 27574	. 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 2845	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 1537 ∈ wcel 2108 {crab 3443 ∖ cdif 3973 ∩ cin 3975 ∅c0 4352 {csn 4648 ↦ cmpt 5249 ^◡ccnv 5699 “ cima 5703 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 − cmin 11520 / cdiv 11947 ℕcn 12293 ℝ⁺crp 13057 ...cfz 13567 ⌊cfl 13841 ♯chash 14379 𝑂(1)co1 15532 Σcsu 15734 ϕcphi 16811 Basecbs 17258 0_gc0g 17499 1_rcur 20208 Unitcui 20381 ℤRHomczrh 21533 ℤ/nℤczn 21536 logclog 26614 Λcvma 27153 DChrcdchr 27294

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263 ax-mulf 11264

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-rpss 7758 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-ec 8765 df-qs 8769 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-word 14563 df-concat 14619 df-s1 14644 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-o1 15536 df-lo1 15537 df-sum 15735 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-numer 16782 df-denom 16783 df-phi 16813 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-qus 17569 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-mhm 18818 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-nsg 19164 df-eqg 19165 df-ghm 19253 df-gim 19299 df-ga 19330 df-cntz 19357 df-oppg 19386 df-od 19570 df-gex 19571 df-pgp 19572 df-lsm 19678 df-pj1 19679 df-cmn 19824 df-abl 19825 df-cyg 19920 df-dprd 20039 df-dpj 20040 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrng 20572 df-subrg 20597 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-sra 21195 df-rgmod 21196 df-lidl 21241 df-rsp 21242 df-2idl 21283 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-zring 21481 df-zrh 21537 df-zn 21540 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-0p 25724 df-limc 25921 df-dv 25922 df-ply 26247 df-idp 26248 df-coe 26249 df-dgr 26250 df-quot 26351 df-ulm 26438 df-log 26616 df-cxp 26617 df-atan 26928 df-em 27054 df-cht 27158 df-vma 27159 df-chp 27160 df-ppi 27161 df-mu 27162 df-dchr 27295

This theorem is referenced by: rplogsum 27589

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