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Theorem rpvmasum 26254

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 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑁 ∈ ℕ)
5		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (DChr‘𝑁) = (DChr‘𝑁)
6		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘(DChr‘𝑁)) = (Base‘(DChr‘𝑁))
7		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (0_g‘(DChr‘𝑁)) = (0_g‘(DChr‘𝑁))
8		2fveq3 6673	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑦‘(𝐿‘𝑚)) = (𝑦‘(𝐿‘𝑛)))
9		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛)
10	8, 9	oveq12d 7182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑦‘(𝐿‘𝑛)) / 𝑛))
11	10	cbvsumv 15139	. . . . . . . . . . . . . . . 16 ⊢ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛)
12	11	eqeq1i 2743	. . . . . . . . . . . . . . 15 ⊢ (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0)
13	12	rabbii 3373	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 26248	. . . . . . . . . . . . 13 ⊢ ¬ (𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})
16	15	imnani 404	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})
17	16	eq0rdv 4290	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = ∅)
18	17	fveq2d 6672	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = (♯‘∅))
19		hash0 13813	. . . . . . . . . 10 ⊢ (♯‘∅) = 0
20	18, 19	eqtrdi 2789	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = 0)
21	20	oveq2d 7180	. . . . . . . 8 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = (1 − 0))
22		1m0e1 11830	. . . . . . . 8 ⊢ (1 − 0) = 1
23	21, 22	eqtrdi 2789	. . . . . . 7 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1)
24	23	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0_g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1)
25	24	oveq2d 7180	. . . . 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 25311	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
27	26	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
28	27	recnd 10740	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
29	28	mulid1d 10729	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) · 1) = (log‘𝑥))
30	25, 29	eqtrd 2773	. . . 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 7180	. . 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 5122	. 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 2738	. . 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 26240	. 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 2834	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 399 = wceq 1542 ∈ wcel 2113 {crab 3057 ∖ cdif 3838 ∩ cin 3840 ∅c0 4209 {csn 4513 ↦ cmpt 5107 ^◡ccnv 5518 “ cima 5522 ‘cfv 6333 (class class class)co 7164 ℝcr 10607 0cc0 10608 1c1 10609 · cmul 10613 − cmin 10941 / cdiv 11368 ℕcn 11709 ℝ⁺crp 12465 ...cfz 12974 ⌊cfl 13244 ♯chash 13775 𝑂(1)co1 14926 Σcsu 15128 ϕcphi 16194 Basecbs 16579 0_gc0g 16809 1_rcur 19363 Unitcui 19504 ℤRHomczrh 20313 ℤ/nℤczn 20316 logclog 25290 Λcvma 25821 DChrcdchr 25960

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-disj 4993 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-rpss 7461 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-tpos 7914 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-oadd 8128 df-omul 8129 df-er 8313 df-ec 8315 df-qs 8319 df-map 8432 df-pm 8433 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-dju 9396 df-card 9434 df-acn 9437 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-xnn0 12042 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-ioc 12819 df-ico 12820 df-icc 12821 df-fz 12975 df-fzo 13118 df-fl 13246 df-mod 13322 df-seq 13454 df-exp 13515 df-fac 13719 df-bc 13748 df-hash 13776 df-word 13949 df-concat 14005 df-s1 14032 df-shft 14509 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-limsup 14911 df-clim 14928 df-rlim 14929 df-o1 14930 df-lo1 14931 df-sum 15129 df-ef 15506 df-e 15507 df-sin 15508 df-cos 15509 df-tan 15510 df-pi 15511 df-dvds 15693 df-gcd 15931 df-prm 16106 df-numer 16168 df-denom 16169 df-phi 16196 df-pc 16267 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-qus 16878 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-mhm 18065 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-mulg 18336 df-subg 18387 df-nsg 18388 df-eqg 18389 df-ghm 18467 df-gim 18510 df-ga 18531 df-cntz 18558 df-oppg 18585 df-od 18767 df-gex 18768 df-pgp 18769 df-lsm 18872 df-pj1 18873 df-cmn 19019 df-abl 19020 df-cyg 19109 df-dprd 19229 df-dpj 19230 df-mgp 19352 df-ur 19364 df-ring 19411 df-cring 19412 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-rnghom 19582 df-drng 19616 df-subrg 19645 df-lmod 19748 df-lss 19816 df-lsp 19856 df-sra 20056 df-rgmod 20057 df-lidl 20058 df-rsp 20059 df-2idl 20117 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-fbas 20207 df-fg 20208 df-cnfld 20211 df-zring 20283 df-zrh 20317 df-zn 20320 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-nei 21842 df-lp 21880 df-perf 21881 df-cn 21971 df-cnp 21972 df-haus 22059 df-cmp 22131 df-tx 22306 df-hmeo 22499 df-fil 22590 df-fm 22682 df-flim 22683 df-flf 22684 df-xms 23066 df-ms 23067 df-tms 23068 df-cncf 23623 df-0p 24415 df-limc 24610 df-dv 24611 df-ply 24929 df-idp 24930 df-coe 24931 df-dgr 24932 df-quot 25031 df-ulm 25116 df-log 25292 df-cxp 25293 df-atan 25597 df-em 25722 df-cht 25826 df-vma 25827 df-chp 25828 df-ppi 25829 df-mu 25830 df-dchr 25961

This theorem is referenced by: rplogsum 26255

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