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| Mirrors > Home > MPE Home > Th. List > rpvmasum | Structured version Visualization version GIF version | ||
| 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.) | 
| 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}) | 
| Ref | Expression | 
|---|---|
| rpvmasum | ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1)) | 
| 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)) ∖ {(0g‘(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 ⊢ (0g‘(DChr‘𝑁)) = (0g‘(DChr‘𝑁)) | |
| 8 | 2fveq3 6910 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → (𝑦‘(𝐿‘𝑚)) = (𝑦‘(𝐿‘𝑛))) | |
| 9 | id 22 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛) | |
| 10 | 8, 9 | oveq12d 7450 | . . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = 𝑛 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑦‘(𝐿‘𝑛)) / 𝑛)) | 
| 11 | 10 | cbvsumv 15733 | . . . . . . . . . . . . . . . 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 3441 | . . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0} | 
| 14 | simpr 484 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) | |
| 15 | 1, 2, 4, 5, 6, 7, 13, 14 | dchrisum0 27565 | . . . . . . . . . . . . 13 ⊢ ¬ (𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) | 
| 16 | 15 | imnani 400 | . . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) | 
| 17 | 16 | eq0rdv 4406 | . . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = ∅) | 
| 18 | 17 | fveq2d 6909 | . . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = (♯‘∅)) | 
| 19 | hash0 14407 | . . . . . . . . . 10 ⊢ (♯‘∅) = 0 | |
| 20 | 18, 19 | eqtrdi 2792 | . . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = 0) | 
| 21 | 20 | oveq2d 7448 | . . . . . . . 8 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = (1 − 0)) | 
| 22 | 1m0e1 12388 | . . . . . . . 8 ⊢ (1 − 0) = 1 | |
| 23 | 21, 22 | eqtrdi 2792 | . . . . . . 7 ⊢ (𝜑 → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1) | 
| 24 | 23 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1) | 
| 25 | 24 | oveq2d 7448 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = ((log‘𝑥) · 1)) | 
| 26 | relogcl 26618 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ) | |
| 27 | 26 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ) | 
| 28 | 27 | recnd 11290 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) | 
| 29 | 28 | mulridd 11279 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · 1) = (log‘𝑥)) | 
| 30 | 25, 29 | eqtrd 2776 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = (log‘𝑥)) | 
| 31 | 30 | oveq2d 7448 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})))) = (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) | 
| 32 | 31 | mpteq2dva 5241 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(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)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(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)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝐴 = (1r‘𝑍)) | 
| 38 | 1, 2, 3, 5, 6, 7, 33, 34, 35, 36, 37 | rpvmasum2 27557 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (♯‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))))) ∈ 𝑂(1)) | 
| 39 | 32, 38 | eqeltrrd 2841 | 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 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∩ cin 3949 ∅c0 4332 {csn 4625 ↦ cmpt 5224 ◡ccnv 5683 “ cima 5687 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 · cmul 11161 − cmin 11493 / cdiv 11921 ℕcn 12267 ℝ+crp 13035 ...cfz 13548 ⌊cfl 13831 ♯chash 14370 𝑂(1)co1 15523 Σcsu 15723 ϕcphi 16802 Basecbs 17248 0gc0g 17485 1rcur 20179 Unitcui 20356 ℤRHomczrh 21511 ℤ/nℤczn 21514 logclog 26597 Λcvma 27136 DChrcdchr 27277 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-disj 5110 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-rpss 7744 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-o1 15527 df-lo1 15528 df-sum 15724 df-ef 16104 df-e 16105 df-sin 16106 df-cos 16107 df-tan 16108 df-pi 16109 df-dvds 16292 df-gcd 16533 df-prm 16710 df-numer 16773 df-denom 16774 df-phi 16804 df-pc 16876 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-qus 17555 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-gim 19278 df-ga 19309 df-cntz 19336 df-oppg 19365 df-od 19547 df-gex 19548 df-pgp 19549 df-lsm 19655 df-pj1 19656 df-cmn 19801 df-abl 19802 df-cyg 19897 df-dprd 20016 df-dpj 20017 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-dvr 20402 df-rhm 20473 df-subrng 20547 df-subrg 20571 df-drng 20732 df-lmod 20861 df-lss 20931 df-lsp 20971 df-sra 21173 df-rgmod 21174 df-lidl 21219 df-rsp 21220 df-2idl 21261 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-zring 21459 df-zrh 21515 df-zn 21518 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-0p 25706 df-limc 25902 df-dv 25903 df-ply 26228 df-idp 26229 df-coe 26230 df-dgr 26231 df-quot 26334 df-ulm 26421 df-log 26599 df-cxp 26600 df-atan 26911 df-em 27037 df-cht 27141 df-vma 27142 df-chp 27143 df-ppi 27144 df-mu 27145 df-dchr 27278 | 
| This theorem is referenced by: rplogsum 27572 | 
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