selberglem3 - Metamath Proof Explorer

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Theorem selberglem3 26131

Description: Lemma for selberg 26132. Estimation of the left-hand side of logsqvma2 26127. (Contributed by Mario Carneiro, 23-May-2016.)

**Assertion**
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)

Distinct variable group: 𝑛,𝑑,𝑥,𝑦

Proof of Theorem selberglem3 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 7158	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
2	1	oveq1d 7150	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
3	2	oveq2d 7151	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
4		rpre 12385	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
5		ssrab2 4007	. . . . . . . . . . 11 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
6		simprr 772	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
7	5, 6	sseldi 3913	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
8		mucl 25726	. . . . . . . . . 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 12076	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
11		elfznn 12931	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
12	11	nnrpd 12417	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
13	12	ad2antrl 727	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ⁺)
14	7	nnrpd 12417	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ⁺)
15	13, 14	rpdivcld 12436	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ⁺)
16		relogcl 25167	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ⁺ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
17	16	recnd 10658	. . . . . . . . . 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 13504	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
20	10, 19	mulcld 10650	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
21	3, 4, 20	dvdsflsumcom 25773	. . . . . 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 12931	. . . . . . . . . . . . 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 11641	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
25		elfznn 12931	. . . . . . . . . . . . 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 11641	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 11675	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	24, 27, 28	divcan3d 11410	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6649	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
31	30	oveq1d 7150	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥)) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
32	31	oveq2d 7151	. . . . . . 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 15054	. . . . . 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 2833	. . . . 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 7150	. . . 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 7150	. . 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 5121	. 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 2798	. . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑)
39	38	selberglem2 26130	. 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 2886	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ℤcz 11969 ℝ⁺crp 12377 ...cfz 12885 ⌊cfl 13155 ↑cexp 13425 𝑂(1)co1 14835 Σcsu 15034 ∥ cdvds 15599 logclog 25146 μcmu 25680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 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 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-o1 14839 df-lo1 14840 df-sum 15035 df-ef 15413 df-e 15414 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-dvds 15600 df-gcd 15834 df-prm 16006 df-pc 16164 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-ulm 24972 df-log 25148 df-cxp 25149 df-atan 25453 df-em 25578 df-mu 25686

This theorem is referenced by: (None)

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