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Mirrors  >  Home  >  MPE Home  >  Th. List  >  vmalogdivsum2 Structured version   Visualization version   GIF version

Theorem vmalogdivsum2 25272
Description: The sum Σ𝑛𝑥, Λ(𝑛)log(𝑥 / 𝑛) / 𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). Exercise 9.1.7 of [Shapiro], p. 336. (Contributed by Mario Carneiro, 30-May-2016.)
Assertion
Ref Expression
vmalogdivsum2 (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem vmalogdivsum2
Dummy variables 𝑘 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fzfid 12812 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2 elfznn 12408 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
32adantl 481 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
43nnrpd 11908 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
54relogcld 24414 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (log‘𝑘) ∈ ℝ)
65, 3nndivred 11107 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) ∈ ℝ)
71, 6fsumrecl 14509 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℝ)
87recnd 10106 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℂ)
9 elioore 12243 . . . . . . . . . . . . 13 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
109adantl 481 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
11 1rp 11874 . . . . . . . . . . . . 13 1 ∈ ℝ+
1211a1i 11 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
13 1red 10093 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
14 eliooord 12271 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
1514adantl 481 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
1615simpld 474 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
1713, 10, 16ltled 10223 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
1810, 12, 17rpgecld 11949 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
1918relogcld 24414 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
2019resqcld 13075 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℝ)
2120rehalfcld 11317 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℝ)
2221recnd 10106 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℂ)
2319recnd 10106 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2410, 16rplogcld 24420 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2524rpne0d 11915 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
268, 22, 23, 25divsubdird 10878 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((((log‘𝑥)↑2) / 2) / (log‘𝑥))))
277, 21resubcld 10496 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℝ)
2827recnd 10106 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℂ)
2928, 23, 25divrecd 10842 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥))))
3020recnd 10106 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℂ)
31 2cnd 11131 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
32 2ne0 11151 . . . . . . . . . 10 2 ≠ 0
3332a1i 11 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
3430, 31, 23, 33, 25divdiv32d 10864 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((((log‘𝑥)↑2) / (log‘𝑥)) / 2))
3523sqvald 13045 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) = ((log‘𝑥) · (log‘𝑥)))
3635oveq1d 6705 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)))
3723, 23, 25divcan3d 10844 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)) = (log‘𝑥))
3836, 37eqtrd 2685 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (log‘𝑥))
3938oveq1d 6705 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / (log‘𝑥)) / 2) = ((log‘𝑥) / 2))
4034, 39eqtrd 2685 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((log‘𝑥) / 2))
4140oveq2d 6706 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((((log‘𝑥)↑2) / 2) / (log‘𝑥))) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)))
4226, 29, 413eqtr3rd 2694 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥))))
4342mpteq2dva 4777 . . . 4 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))))
4424rprecred 11921 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
4518ex 449 . . . . . . 7 (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
4645ssrdv 3642 . . . . . 6 (⊤ → (1(,)+∞) ⊆ ℝ+)
47 eqid 2651 . . . . . . . . 9 (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)))
4847logdivsum 25267 . . . . . . . 8 ((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))):ℝ+⟶ℝ ∧ (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟 ∧ (((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ⇝𝑟 1 ∧ 1 ∈ ℝ+ ∧ e ≤ 1) → (abs‘(((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)))‘1) − 1)) ≤ ((log‘1) / 1)))
4948simp2i 1091 . . . . . . 7 (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟
50 rlimdmo1 14392 . . . . . . 7 ((𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟 → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
5149, 50mp1i 13 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
5246, 51o1res2 14338 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
53 divlogrlim 24426 . . . . . 6 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
54 rlimo1 14391 . . . . . 6 ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
5553, 54mp1i 13 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
5627, 44, 52, 55o1mul2 14399 . . . 4 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
5743, 56eqeltrd 2730 . . 3 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
588, 23, 25divcld 10839 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) ∈ ℂ)
5923halfcld 11315 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
6058, 59subcld 10430 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
61 elfznn 12408 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
6261adantl 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
63 vmacl 24889 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
6462, 63syl 17 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
6564, 62nndivred 11107 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
6618adantr 480 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
6762nnrpd 11908 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
6866, 67rpdivcld 11927 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
6968relogcld 24414 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
7065, 69remulcld 10108 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
711, 70fsumrecl 14509 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
7271recnd 10106 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
7324rpcnd 11912 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
7472, 73, 25divcld 10839 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
7573halfcld 11315 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
7674, 75subcld 10430 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
7758, 74, 59nnncan2d 10465 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
788, 72, 23, 25divsubdird 10878 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥))))
79 fzfid 12812 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
8064adantr 480 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℝ)
8162adantr 480 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
82 elfznn 12408 . . . . . . . . . . . . . . 15 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
8382adantl 481 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
8481, 83nnmulcld 11106 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℕ)
8580, 84nndivred 11107 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
8679, 85fsumrecl 14509 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
8786recnd 10106 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℂ)
8870recnd 10106 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
891, 87, 88fsumsub 14564 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
9064recnd 10106 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
9162nncnd 11074 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
9262nnne0d 11103 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
9390, 91, 92divcld 10839 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
9483nnrecred 11104 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
9579, 94fsumrecl 14509 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
9695recnd 10106 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
9769recnd 10106 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
9893, 96, 97subdid 10524 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
9990adantr 480 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℂ)
10091adantr 480 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℂ)
10183nncnd 11074 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
10292adantr 480 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ≠ 0)
10383nnne0d 11103 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
10499, 100, 101, 102, 103divdiv1d 10870 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
10599, 100, 102divcld 10839 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
106105, 101, 103divrecd 10842 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
107104, 106eqtr3d 2687 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
108107sumeq2dv 14477 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
109101, 103reccld 10832 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
11079, 93, 109fsummulc2 14560 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
111108, 110eqtr4d 2688 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
112111oveq1d 6705 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
11398, 112eqtr4d 2688 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
114113sumeq2dv 14477 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
115 vmasum 24986 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) = (log‘𝑘))
1163, 115syl 17 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) = (log‘𝑘))
117116oveq1d 6705 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) / 𝑘) = ((log‘𝑘) / 𝑘))
118 fzfid 12812 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
119 dvdsssfz1 15087 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
1203, 119syl 17 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
121 ssfi 8221 . . . . . . . . . . . . . . 15 (((1...𝑘) ∈ Fin ∧ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘)) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ∈ Fin)
122118, 120, 121syl2anc 694 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ∈ Fin)
1233nncnd 11074 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℂ)
124 ssrab2 3720 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ ℕ
125 simprr 811 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
126124, 125sseldi 3634 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ ℕ)
127126, 63syl 17 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (Λ‘𝑛) ∈ ℝ)
128127recnd 10106 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (Λ‘𝑛) ∈ ℂ)
129128anassrs 681 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (Λ‘𝑛) ∈ ℂ)
1303nnne0d 11103 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≠ 0)
131122, 123, 129, 130fsumdivc 14562 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
132117, 131eqtr3d 2687 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
133132sumeq2dv 14477 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
134 oveq2 6698 . . . . . . . . . . . 12 (𝑘 = (𝑛 · 𝑚) → ((Λ‘𝑛) / 𝑘) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
1352ad2antrl 764 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ∈ ℕ)
136135nncnd 11074 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ∈ ℂ)
137135nnne0d 11103 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ≠ 0)
138128, 136, 137divcld 10839 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → ((Λ‘𝑛) / 𝑘) ∈ ℂ)
139134, 10, 138dvdsflsumcom 24959 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
140133, 139eqtrd 2685 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
141140oveq1d 6705 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
14289, 114, 1413eqtr4rd 2696 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
143142oveq1d 6705 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
14477, 78, 1433eqtr2d 2691 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
145144mpteq2dva 4777 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))))
146 1red 10093 . . . . . . 7 (⊤ → 1 ∈ ℝ)
1471, 65fsumrecl 14509 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
148147, 24rerpdivcld 11941 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
149 ioossre 12273 . . . . . . . . . . 11 (1(,)+∞) ⊆ ℝ
150 ax-1cn 10032 . . . . . . . . . . 11 1 ∈ ℂ
151 o1const 14394 . . . . . . . . . . 11 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
152149, 150, 151mp2an 708 . . . . . . . . . 10 (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)
153152a1i 11 . . . . . . . . 9 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
154148recnd 10106 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
15512rpcnd 11912 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
156147recnd 10106 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
157156, 23, 23, 25divsubdird 10878 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))))
158156, 23subcld 10430 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
159158, 23, 25divrecd 10842 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) / (log‘𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
16023, 25dividd 10837 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
161160oveq2d 6706 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − ((log‘𝑥) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1))
162157, 159, 1613eqtr3rd 2694 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
163162mpteq2dva 4777 . . . . . . . . . . 11 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))))
164147, 19resubcld 10496 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
165 vmadivsum 25216 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
166165a1i 11 . . . . . . . . . . . . 13 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
16746, 166o1res2 14338 . . . . . . . . . . . 12 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
168164, 44, 167, 55o1mul2 14399 . . . . . . . . . . 11 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
169163, 168eqeltrd 2730 . . . . . . . . . 10 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
170154, 155, 169o1dif 14404 . . . . . . . . 9 (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)))
171153, 170mpbird 247 . . . . . . . 8 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ 𝑂(1))
172148, 171o1lo1d 14314 . . . . . . 7 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ≤𝑂(1))
17395, 69resubcld 10496 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
17465, 173remulcld 10108 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
1751, 174fsumrecl 14509 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
176175, 24rerpdivcld 11941 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ∈ ℝ)
177 1red 10093 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
178 vmage0 24892 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
17962, 178syl 17 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
18064, 67, 179divge0d 11950 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
18168rpred 11910 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
18291mulid2d 10096 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
183 fznnfl 12701 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
18410, 183syl 17 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
185184simplbda 653 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
186182, 185eqbrtrd 4707 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
18710adantr 480 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
188177, 187, 67lemuldivd 11959 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
189186, 188mpbid 222 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
190 harmonicubnd 24781 . . . . . . . . . . . . . 14 (((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
191181, 189, 190syl2anc 694 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
19295, 69, 177lesubadd2d 10664 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ≤ 1 ↔ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1)))
193191, 192mpbird 247 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ≤ 1)
194173, 177, 65, 180, 193lemul2ad 11002 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ (((Λ‘𝑛) / 𝑛) · 1))
19593mulid1d 10095 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 1) = ((Λ‘𝑛) / 𝑛))
196194, 195breqtrd 4711 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) / 𝑛))
1971, 174, 65, 196fsumle 14575 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
198175, 147, 24, 197lediv1dd 11968 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
199198adantrr 753 . . . . . . 7 ((⊤ ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)))
200146, 172, 148, 176, 199lo1le 14426 . . . . . 6 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1))
201 0red 10079 . . . . . . 7 (⊤ → 0 ∈ ℝ)
202 harmoniclbnd 24780 . . . . . . . . . . . 12 ((𝑥 / 𝑛) ∈ ℝ+ → (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))
20368, 202syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚))
20495, 69subge0d 10655 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ↔ (log‘(𝑥 / 𝑛)) ≤ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
205203, 204mpbird 247 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))))
20665, 173, 180, 205mulge0d 10642 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
2071, 174, 206fsumge0 14571 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
208175, 24, 207divge0d 11950 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
209176, 201, 208o1lo12 14313 . . . . . 6 (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1)))
210200, 209mpbird 247 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ 𝑂(1))
211145, 210eqeltrd 2730 . . . 4 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
21260, 76, 211o1dif 14404 . . 3 (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)))
21357, 212mpbid 222 . 2 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
214213trud 1533 1 (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wtru 1524  wcel 2030  wne 2823  {crab 2945  wss 3607   class class class wbr 4685  cmpt 4762  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  +crp 11870  (,)cioo 12213  ...cfz 12364  cfl 12631  cexp 12900  abscabs 14018  𝑟 crli 14260  𝑂(1)co1 14261  ≤𝑂(1)clo1 14262  Σcsu 14460  eceu 14837  cdvds 15027  logclog 24346  Λcvma 24863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-o1 14265  df-lo1 14266  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-em 24764  df-cht 24868  df-vma 24869  df-chp 24870  df-ppi 24871
This theorem is referenced by:  vmalogdivsum  25273  2vmadivsumlem  25274  selberg4lem1  25294
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