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Theorem vmalogdivsum2 25144
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 12720 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
2 elfznn 12320 . . . . . . . . . . . . 13 (𝑘 ∈ (1...(⌊‘𝑥)) → 𝑘 ∈ ℕ)
32adantl 482 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℕ)
43nnrpd 11822 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℝ+)
54relogcld 24290 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (log‘𝑘) ∈ ℝ)
65, 3nndivred 11021 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) ∈ ℝ)
71, 6fsumrecl 14406 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℝ)
87recnd 10020 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) ∈ ℂ)
9 elioore 12155 . . . . . . . . . . . . 13 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
109adantl 482 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
11 1rp 11788 . . . . . . . . . . . . 13 1 ∈ ℝ+
1211a1i 11 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
13 1red 10007 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
14 eliooord 12183 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
1514adantl 482 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
1615simpld 475 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
1713, 10, 16ltled 10137 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
1810, 12, 17rpgecld 11863 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
1918relogcld 24290 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
2019resqcld 12983 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℝ)
2120rehalfcld 11231 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℝ)
2221recnd 10020 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / 2) ∈ ℂ)
2319recnd 10020 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2410, 16rplogcld 24296 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2524rpne0d 11829 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
268, 22, 23, 25divsubdird 10792 . . . . . 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 10410 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℝ)
2827recnd 10020 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) ∈ ℂ)
2928, 23, 25divrecd 10756 . . . . . 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 10020 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) ∈ ℂ)
31 2cnd 11045 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ∈ ℂ)
32 2ne0 11065 . . . . . . . . . 10 2 ≠ 0
3332a1i 11 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 2 ≠ 0)
3430, 31, 23, 33, 25divdiv32d 10778 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((((log‘𝑥)↑2) / (log‘𝑥)) / 2))
3523sqvald 12953 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥)↑2) = ((log‘𝑥) · (log‘𝑥)))
3635oveq1d 6625 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)))
3723, 23, 25divcan3d 10758 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) · (log‘𝑥)) / (log‘𝑥)) = (log‘𝑥))
3836, 37eqtrd 2655 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥)↑2) / (log‘𝑥)) = (log‘𝑥))
3938oveq1d 6625 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / (log‘𝑥)) / 2) = ((log‘𝑥) / 2))
4034, 39eqtrd 2655 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((((log‘𝑥)↑2) / 2) / (log‘𝑥)) = ((log‘𝑥) / 2))
4140oveq2d 6626 . . . . . 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 2664 . . . . 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 4709 . . . 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 11835 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
4518ex 450 . . . . . . 7 (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
4645ssrdv 3593 . . . . . 6 (⊤ → (1(,)+∞) ⊆ ℝ+)
47 eqid 2621 . . . . . . . . 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 25139 . . . . . . . 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 1069 . . . . . . 7 (𝑥 ∈ ℝ+ ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ dom ⇝𝑟
50 rlimdmo1 14290 . . . . . . 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 14236 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2))) ∈ 𝑂(1))
53 divlogrlim 24298 . . . . . 6 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
54 rlimo1 14289 . . . . . 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 14297 . . . 4 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) − (((log‘𝑥)↑2) / 2)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
5743, 56eqeltrd 2698 . . 3 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
588, 23, 25divcld 10753 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) ∈ ℂ)
5923halfcld 11229 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
6058, 59subcld 10344 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
61 elfznn 12320 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
6261adantl 482 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
63 vmacl 24761 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
6462, 63syl 17 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
6564, 62nndivred 11021 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
6618adantr 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
6762nnrpd 11822 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
6866, 67rpdivcld 11841 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
6968relogcld 24290 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
7065, 69remulcld 10022 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
711, 70fsumrecl 14406 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℝ)
7271recnd 10020 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
7324rpcnd 11826 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
7472, 73, 25divcld 10753 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) ∈ ℂ)
7573halfcld 11229 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
7674, 75subcld 10344 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
7758, 74, 59nnncan2d 10379 . . . . . . 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 10792 . . . . . . 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 12720 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
8064adantr 481 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℝ)
8162adantr 481 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℕ)
82 elfznn 12320 . . . . . . . . . . . . . . 15 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
8382adantl 482 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
8481, 83nnmulcld 11020 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (𝑛 · 𝑚) ∈ ℕ)
8580, 84nndivred 11021 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
8679, 85fsumrecl 14406 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℝ)
8786recnd 10020 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) ∈ ℂ)
8870recnd 10020 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
891, 87, 88fsumsub 14459 . . . . . . . . 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 10020 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
9162nncnd 10988 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
9262nnne0d 11017 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
9390, 91, 92divcld 10753 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
9483nnrecred 11018 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℝ)
9579, 94fsumrecl 14406 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℝ)
9695recnd 10020 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ∈ ℂ)
9769recnd 10020 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
9893, 96, 97subdid 10438 . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (Λ‘𝑛) ∈ ℂ)
10091adantr 481 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ∈ ℂ)
10183nncnd 10988 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℂ)
10292adantr 481 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑛 ≠ 0)
10383nnne0d 11017 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ≠ 0)
10499, 100, 101, 102, 103divdiv1d 10784 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
10599, 100, 102divcld 10753 . . . . . . . . . . . . . . . 16 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
106105, 101, 103divrecd 10756 . . . . . . . . . . . . . . 15 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (((Λ‘𝑛) / 𝑛) / 𝑚) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
107104, 106eqtr3d 2657 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · (1 / 𝑚)))
108107sumeq2dv 14375 . . . . . . . . . . . . 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 10746 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (1 / 𝑚) ∈ ℂ)
11079, 93, 109fsummulc2 14455 . . . . . . . . . . . . 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 2658 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)) = (((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚)))
112111oveq1d 6625 . . . . . . . . . . 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 2658 . . . . . . . . . 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 14375 . . . . . . . . 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 24858 . . . . . . . . . . . . . . 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 6625 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) / 𝑘) = ((log‘𝑘) / 𝑘))
118 fzfid 12720 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (1...𝑘) ∈ Fin)
119 dvdsssfz1 14975 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
1203, 119syl 17 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘))
121 ssfi 8132 . . . . . . . . . . . . . . 15 (((1...𝑘) ∈ Fin ∧ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ (1...𝑘)) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ∈ Fin)
122118, 120, 121syl2anc 692 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦𝑘} ∈ Fin)
1233nncnd 10988 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ∈ ℂ)
124 ssrab2 3671 . . . . . . . . . . . . . . . . . 18 {𝑦 ∈ ℕ ∣ 𝑦𝑘} ⊆ ℕ
125 simprr 795 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})
126124, 125sseldi 3585 . . . . . . . . . . . . . . . . 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 10020 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → (Λ‘𝑛) ∈ ℂ)
129128anassrs 679 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘}) → (Λ‘𝑛) ∈ ℂ)
1303nnne0d 11017 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → 𝑘 ≠ 0)
131122, 123, 129, 130fsumdivc 14457 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → (Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} (Λ‘𝑛) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
132117, 131eqtr3d 2657 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑥))) → ((log‘𝑘) / 𝑘) = Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
133132sumeq2dv 14375 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑘 ∈ (1...(⌊‘𝑥))Σ𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘} ((Λ‘𝑛) / 𝑘))
134 oveq2 6618 . . . . . . . . . . . 12 (𝑘 = (𝑛 · 𝑚) → ((Λ‘𝑛) / 𝑘) = ((Λ‘𝑛) / (𝑛 · 𝑚)))
1352ad2antrl 763 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ∈ ℕ)
136135nncnd 10988 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ∈ ℂ)
137135nnne0d 11017 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → 𝑘 ≠ 0)
138128, 136, 137divcld 10753 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ (𝑘 ∈ (1...(⌊‘𝑥)) ∧ 𝑛 ∈ {𝑦 ∈ ℕ ∣ 𝑦𝑘})) → ((Λ‘𝑛) / 𝑘) ∈ ℂ)
139134, 10, 138dvdsflsumcom 24831 . . . . . . . . . . 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 2655 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑥))((log‘𝑘) / 𝑘) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑛) / (𝑛 · 𝑚)))
141140oveq1d 6625 . . . . . . . . 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 2666 . . . . . . . 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 6625 . . . . . . 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 2661 . . . . . 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 4709 . . . . 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 10007 . . . . . . 7 (⊤ → 1 ∈ ℝ)
1471, 65fsumrecl 14406 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
148147, 24rerpdivcld 11855 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℝ)
149 ioossre 12185 . . . . . . . . . . 11 (1(,)+∞) ⊆ ℝ
150 ax-1cn 9946 . . . . . . . . . . 11 1 ∈ ℂ
151 o1const 14292 . . . . . . . . . . 11 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
152149, 150, 151mp2an 707 . . . . . . . . . 10 (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1)
153152a1i 11 . . . . . . . . 9 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
154148recnd 10020 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) ∈ ℂ)
15512rpcnd 11826 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
156147recnd 10020 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
157156, 23, 23, 25divsubdird 10792 . . . . . . . . . . . . 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 10344 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
159158, 23, 25divrecd 10756 . . . . . . . . . . . . 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 10751 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
161160oveq2d 6626 . . . . . . . . . . . . 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 2664 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥))))
163162mpteq2dva 4709 . . . . . . . . . . 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 10410 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℝ)
165 vmadivsum 25088 . . . . . . . . . . . . . 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 14236 . . . . . . . . . . . 12 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
168164, 44, 167, 55o1mul2 14297 . . . . . . . . . . 11 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
169163, 168eqeltrd 2698 . . . . . . . . . 10 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥)) − 1)) ∈ 𝑂(1))
170154, 155, 169o1dif 14302 . . . . . . . . 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 14212 . . . . . . 7 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) / (log‘𝑥))) ∈ ≤𝑂(1))
17395, 69resubcld 10410 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛))) ∈ ℝ)
17465, 173remulcld 10022 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
1751, 174fsumrecl 14406 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ∈ ℝ)
176175, 24rerpdivcld 11855 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)) ∈ ℝ)
177 1red 10007 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
178 vmage0 24764 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
17962, 178syl 17 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
18064, 67, 179divge0d 11864 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) / 𝑛))
18168rpred 11824 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
18291mulid2d 10010 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
183 fznnfl 12609 . . . . . . . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
18710adantr 481 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
188177, 187, 67lemuldivd 11873 . . . . . . . . . . . . . . 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 24653 . . . . . . . . . . . . . 14 (((𝑥 / 𝑛) ∈ ℝ ∧ 1 ≤ (𝑥 / 𝑛)) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
191181, 189, 190syl2anc 692 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) ≤ ((log‘(𝑥 / 𝑛)) + 1))
19295, 69, 177lesubadd2d 10578 . . . . . . . . . . . . 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 10916 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) ≤ (((Λ‘𝑛) / 𝑛) · 1))
19593mulid1d 10009 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 1) = ((Λ‘𝑛) / 𝑛))
196194, 195breqtrd 4644 . . . . . . . . . 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 14469 . . . . . . . . 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 11882 . . . . . . . 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 752 . . . . . . 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 14324 . . . . . 6 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥))) ∈ ≤𝑂(1))
201 0red 9993 . . . . . . 7 (⊤ → 0 ∈ ℝ)
202 harmoniclbnd 24652 . . . . . . . . . . . 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 10569 . . . . . . . . . . 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 10556 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
2071, 174, 206fsumge0 14465 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))))
208175, 24, 207divge0d 11864 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))(1 / 𝑚) − (log‘(𝑥 / 𝑛)))) / (log‘𝑥)))
209176, 201, 208o1lo12 14211 . . . . . 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 2698 . . . 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 14302 . . 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 1490 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 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  wne 2790  {crab 2911  wss 3559   class class class wbr 4618  cmpt 4678  dom cdm 5079  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7907  cc 9886  cr 9887  0cc0 9888  1c1 9889   + caddc 9891   · cmul 9893  +∞cpnf 10023   < clt 10026  cle 10027  cmin 10218   / cdiv 10636  cn 10972  2c2 11022  +crp 11784  (,)cioo 12125  ...cfz 12276  cfl 12539  cexp 12808  abscabs 13916  𝑟 crli 14158  𝑂(1)co1 14159  ≤𝑂(1)clo1 14160  Σcsu 14358  eceu 14729  cdvds 14918  logclog 24222  Λcvma 24735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965  ax-pre-sup 9966  ax-addf 9967  ax-mulf 9968
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-supp 7248  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-ixp 7861  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-fsupp 8228  df-fi 8269  df-sup 8300  df-inf 8301  df-oi 8367  df-card 8717  df-cda 8942  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-xnn0 11316  df-z 11330  df-dec 11446  df-uz 11640  df-q 11741  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-ioo 12129  df-ioc 12130  df-ico 12131  df-icc 12132  df-fz 12277  df-fzo 12415  df-fl 12541  df-mod 12617  df-seq 12750  df-exp 12809  df-fac 13009  df-bc 13038  df-hash 13066  df-shft 13749  df-cj 13781  df-re 13782  df-im 13783  df-sqrt 13917  df-abs 13918  df-limsup 14144  df-clim 14161  df-rlim 14162  df-o1 14163  df-lo1 14164  df-sum 14359  df-ef 14734  df-e 14735  df-sin 14736  df-cos 14737  df-pi 14739  df-dvds 14919  df-gcd 15152  df-prm 15321  df-pc 15477  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-mulr 15887  df-starv 15888  df-sca 15889  df-vsca 15890  df-ip 15891  df-tset 15892  df-ple 15893  df-ds 15896  df-unif 15897  df-hom 15898  df-cco 15899  df-rest 16015  df-topn 16016  df-0g 16034  df-gsum 16035  df-topgen 16036  df-pt 16037  df-prds 16040  df-xrs 16094  df-qtop 16099  df-imas 16100  df-xps 16102  df-mre 16178  df-mrc 16179  df-acs 16181  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-mulg 17473  df-cntz 17682  df-cmn 18127  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-mopn 19674  df-fbas 19675  df-fg 19676  df-cnfld 19679  df-top 20631  df-topon 20648  df-topsp 20661  df-bases 20674  df-cld 20746  df-ntr 20747  df-cls 20748  df-nei 20825  df-lp 20863  df-perf 20864  df-cn 20954  df-cnp 20955  df-haus 21042  df-cmp 21113  df-tx 21288  df-hmeo 21481  df-fil 21573  df-fm 21665  df-flim 21666  df-flf 21667  df-xms 22048  df-ms 22049  df-tms 22050  df-cncf 22604  df-limc 23553  df-dv 23554  df-log 24224  df-cxp 24225  df-em 24636  df-cht 24740  df-vma 24741  df-chp 24742  df-ppi 24743
This theorem is referenced by:  vmalogdivsum  25145  2vmadivsumlem  25146  selberg4lem1  25166
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