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Mirrors > Home > MPE Home > Th. List > selberglem3 | Structured version Visualization version GIF version |
Description: Lemma for selberg 27526. Estimation of the left-hand side of logsqvma2 27521. (Contributed by Mario Carneiro, 23-May-2016.) |
Ref | Expression |
---|---|
selberglem3 | ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 7442 | . . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑))) | |
2 | 1 | oveq1d 7434 | . . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) |
3 | 2 | oveq2d 7435 | . . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2))) |
4 | rpre 13017 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ) | |
5 | ssrab2 4073 | . . . . . . . . . . 11 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ | |
6 | simprr 771 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) | |
7 | 5, 6 | sselid 3974 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ) |
8 | mucl 27118 | . . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ) | |
9 | 7, 8 | syl 17 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ) |
10 | 9 | zcnd 12700 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ) |
11 | elfznn 13565 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ) | |
12 | 11 | nnrpd 13049 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+) |
13 | 12 | ad2antrl 726 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ+) |
14 | 7 | nnrpd 13049 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ+) |
15 | 13, 14 | rpdivcld 13068 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ+) |
16 | relogcl 26554 | . . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℝ) | |
17 | 16 | recnd 11274 | . . . . . . . . . 10 ⊢ ((𝑛 / 𝑑) ∈ ℝ+ → (log‘(𝑛 / 𝑑)) ∈ ℂ) |
18 | 15, 17 | syl 17 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (log‘(𝑛 / 𝑑)) ∈ ℂ) |
19 | 18 | sqcld 14144 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ) |
20 | 10, 19 | mulcld 11266 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ) |
21 | 3, 4, 20 | dvdsflsumcom 27165 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2))) |
22 | elfznn 13565 | . . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ) | |
23 | 22 | 3ad2ant3 1132 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ) |
24 | 23 | nncnd 12261 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ) |
25 | elfznn 13565 | . . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ) | |
26 | 25 | 3ad2ant2 1131 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ) |
27 | 26 | nncnd 12261 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ) |
28 | 26 | nnne0d 12295 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0) |
29 | 24, 27, 28 | divcan3d 12028 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚) |
30 | 29 | fveq2d 6900 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚)) |
31 | 30 | oveq1d 7434 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2)) |
32 | 31 | oveq2d 7435 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ+ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘𝑚)↑2))) |
33 | 32 | 2sumeq2dv 15687 | . . . . . 6 ⊢ (𝑥 ∈ ℝ+ → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2))) |
34 | 21, 33 | eqtrd 2765 | . . . . 5 ⊢ (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2))) |
35 | 34 | oveq1d 7434 | . . . 4 ⊢ (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) = (Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥)) |
36 | 35 | oveq1d 7434 | . . 3 ⊢ (𝑥 ∈ ℝ+ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥))) = ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) |
37 | 36 | mpteq2ia 5252 | . 2 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) |
38 | eqid 2725 | . . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) | |
39 | 38 | selberglem2 27524 | . 2 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) |
40 | 37, 39 | eqeltri 2821 | 1 ⊢ (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 {crab 3418 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 1c1 11141 + caddc 11143 · cmul 11145 − cmin 11476 / cdiv 11903 ℕcn 12245 2c2 12300 ℤcz 12591 ℝ+crp 13009 ...cfz 13519 ⌊cfl 13791 ↑cexp 14062 𝑂(1)co1 15466 Σcsu 15668 ∥ cdvds 16234 logclog 26533 μcmu 27072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 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 12506 df-xnn0 12578 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-fac 14269 df-bc 14298 df-hash 14326 df-shft 15050 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-limsup 15451 df-clim 15468 df-rlim 15469 df-o1 15470 df-lo1 15471 df-sum 15669 df-ef 16047 df-e 16048 df-sin 16049 df-cos 16050 df-tan 16051 df-pi 16052 df-dvds 16235 df-gcd 16473 df-prm 16646 df-pc 16809 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840 df-ulm 26358 df-log 26535 df-cxp 26536 df-atan 26844 df-em 26970 df-mu 27078 |
This theorem is referenced by: (None) |
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