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Theorem vmalogdivsum 26687
Description: The sum Σ𝑛𝑥, Λ(𝑛)log𝑛 / 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
vmalogdivsum (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem vmalogdivsum
StepHypRef Expression
1 elioore 13109 . . . . . . . 8 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
21adantl 482 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
3 1rp 12734 . . . . . . . 8 1 ∈ ℝ+
43a1i 11 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
5 1red 10976 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
6 eliooord 13138 . . . . . . . . . 10 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
76adantl 482 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
87simpld 495 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
95, 2, 8ltled 11123 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
102, 4, 9rpgecld 12811 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
1110ex 413 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
1211ssrdv 3927 . . . 4 (⊤ → (1(,)+∞) ⊆ ℝ+)
13 vmadivsum 26630 . . . . 5 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)
1413a1i 11 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
1512, 14o1res2 15272 . . 3 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1))
16 fzfid 13693 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17 elfznn 13285 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1817adantl 482 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19 vmacl 26267 . . . . . . . . 9 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
2018, 19syl 17 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
2120, 18nndivred 12027 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
2221recnd 11003 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
2316, 22fsumcl 15445 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
2410relogcld 25778 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
2524recnd 11003 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2623, 25subcld 11332 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) ∈ ℂ)
2718nnrpd 12770 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
2827relogcld 25778 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
2921, 28remulcld 11005 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ)
3016, 29fsumrecl 15446 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℝ)
312, 8rplogcld 25784 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
3230, 31rerpdivcld 12803 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℝ)
3324rehalfcld 12220 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℝ)
3432, 33resubcld 11403 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℝ)
3534recnd 11003 . . . 4 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)) ∈ ℂ)
3633recnd 11003 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / 2) ∈ ℂ)
3723, 36subcld 11332 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) ∈ ℂ)
3832recnd 11003 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) ∈ ℂ)
3937, 38, 36nnncan2d 11367 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
4023, 36, 36subsub4d 11363 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2))))
41252halvesd 12219 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / 2) + ((log‘𝑥) / 2)) = (log‘𝑥))
4241oveq2d 7291 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (((log‘𝑥) / 2) + ((log‘𝑥) / 2))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
4340, 42eqtrd 2778 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
4443oveq1d 7290 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − ((log‘𝑥) / 2)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))))
4523, 36, 38sub32d 11364 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2)))
4610adantr 481 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
4746relogcld 25778 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℝ)
4821, 47remulcld 11005 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℝ)
4948recnd 11003 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℂ)
5029recnd 11003 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ)
5116, 49, 50fsumsub 15500 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
5246, 27relogdivd 25781 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) = ((log‘𝑥) − (log‘𝑛)))
5352oveq2d 7291 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = (((Λ‘𝑛) / 𝑛) · ((log‘𝑥) − (log‘𝑛))))
5425adantr 481 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑥) ∈ ℂ)
5528recnd 11003 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
5622, 54, 55subdid 11431 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · ((log‘𝑥) − (log‘𝑛))) = ((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
5753, 56eqtrd 2778 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
5857sumeq2dv 15415 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − (((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
5920recnd 11003 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
6018nncnd 11989 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
6118nnne0d 12023 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
6259, 60, 61divcld 11751 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
6316, 25, 62fsummulc1 15497 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)))
6463oveq1d 7290 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
6551, 58, 643eqtr4d 2788 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))))
6665oveq1d 7290 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥)))
6723, 25mulcld 10995 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) ∈ ℂ)
6830recnd 11003 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) ∈ ℂ)
6931rpne0d 12777 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
7067, 68, 25, 69divsubdird 11790 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛))) / (log‘𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
7123, 25, 69divcan4d 11757 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
7271oveq1d 7290 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · (log‘𝑥)) / (log‘𝑥)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
7366, 70, 723eqtrd 2782 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))))
7473oveq1d 7290 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) − ((log‘𝑥) / 2)))
7545, 74eqtr4d 2781 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘𝑥) / 2)) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))
7639, 44, 753eqtr3d 2786 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2)))
7776mpteq2dva 5174 . . . . 5 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) = (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))))
78 vmalogdivsum2 26686 . . . . 5 (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
7977, 78eqeltrdi 2847 . . . 4 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2)))) ∈ 𝑂(1))
8026, 35, 79o1dif 15339 . . 3 (⊤ → ((𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1) ↔ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)))
8115, 80mpbid 231 . 2 (⊤ → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
8281mptru 1546 1 (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · (log‘𝑛)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wtru 1540  wcel 2106   class class class wbr 5074  cmpt 5157  cfv 6433  (class class class)co 7275  cc 10869  cr 10870  1c1 10872   + caddc 10874   · cmul 10876  +∞cpnf 11006   < clt 11009  cmin 11205   / cdiv 11632  cn 11973  2c2 12028  +crp 12730  (,)cioo 13079  ...cfz 13239  cfl 13510  𝑂(1)co1 15195  Σcsu 15397  logclog 25710  Λcvma 26241
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-o1 15199  df-lo1 15200  df-sum 15398  df-ef 15777  df-e 15778  df-sin 15779  df-cos 15780  df-tan 15781  df-pi 15782  df-dvds 15964  df-gcd 16202  df-prm 16377  df-pc 16538  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-ulm 25536  df-log 25712  df-cxp 25713  df-atan 26017  df-em 26142  df-cht 26246  df-vma 26247  df-chp 26248  df-ppi 26249
This theorem is referenced by:  selberg3r  26717
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