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Theorem selberg2lem 27594
Description: Lemma for selberg2 27595. Equation 10.4.12 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)
Assertion
Ref Expression
selberg2lem (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
Distinct variable group:   𝑥,𝑛

Proof of Theorem selberg2lem
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 rpre 13043 . . . . . . . . 9 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2 chpcl 27167 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
31, 2syl 17 . . . . . . . 8 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
43recnd 11289 . . . . . . 7 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5 rprege0 13050 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6 flge0nn0 13860 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8 nn0p1nn 12565 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
97, 8syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
109nnrpd 13075 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
1110relogcld 26665 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
1211recnd 11289 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13 relogcl 26617 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1413recnd 11289 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1512, 14subcld 11620 . . . . . . 7 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
164, 15mulcld 11281 . . . . . 6 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17 fzfid 14014 . . . . . . 7 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18 elfznn 13593 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1918adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2019nnrpd 13075 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21 1rp 13038 . . . . . . . . . . . . 13 1 ∈ ℝ+
22 rpaddcl 13057 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
2321, 22mpan2 691 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
2423relogcld 26665 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25 relogcl 26617 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
2624, 25resubcld 11691 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27 rpre 13043 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
28 chpcl 27167 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
3026, 29remulcld 11291 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
3130recnd 11289 . . . . . . . 8 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
3220, 31syl 17 . . . . . . 7 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
3317, 32fsumcl 15769 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34 rpcnne0 13053 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35 divsubdir 11961 . . . . . 6 ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ ∧ Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
3616, 33, 34, 35syl3anc 1373 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
374, 12mulcld 11281 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
384, 14mulcld 11281 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
3937, 38, 33sub32d 11652 . . . . . . 7 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
404, 12, 14subdid 11719 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
4140oveq1d 7446 . . . . . . 7 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
42 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43 fvoveq1 7454 . . . . . . . . . . 11 (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
4442, 43jca 511 . . . . . . . . . 10 (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
45 fveq2 6906 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
46 fvoveq1 7454 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
4745, 46jca 511 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
48 fveq2 6906 . . . . . . . . . . . 12 (𝑚 = 1 → (log‘𝑚) = (log‘1))
49 log1 26627 . . . . . . . . . . . 12 (log‘1) = 0
5048, 49eqtrdi 2793 . . . . . . . . . . 11 (𝑚 = 1 → (log‘𝑚) = 0)
51 oveq1 7438 . . . . . . . . . . . . . 14 (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52 1m1e0 12338 . . . . . . . . . . . . . 14 (1 − 1) = 0
5351, 52eqtrdi 2793 . . . . . . . . . . . . 13 (𝑚 = 1 → (𝑚 − 1) = 0)
5453fveq2d 6910 . . . . . . . . . . . 12 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55 2pos 12369 . . . . . . . . . . . . 13 0 < 2
56 0re 11263 . . . . . . . . . . . . . 14 0 ∈ ℝ
57 chpeq0 27252 . . . . . . . . . . . . . 14 (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
5856, 57ax-mp 5 . . . . . . . . . . . . 13 ((ψ‘0) = 0 ↔ 0 < 2)
5955, 58mpbir 231 . . . . . . . . . . . 12 (ψ‘0) = 0
6054, 59eqtrdi 2793 . . . . . . . . . . 11 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
6150, 60jca 511 . . . . . . . . . 10 (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
62 fveq2 6906 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
63 fvoveq1 7454 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
6462, 63jca 511 . . . . . . . . . 10 (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
65 nnuz 12921 . . . . . . . . . . 11 ℕ = (ℤ‘1)
669, 65eleqtrdi 2851 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
67 elfznn 13593 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
6867adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
6968nnrpd 13075 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
7069relogcld 26665 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
7170recnd 11289 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
7268nnred 12281 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73 peano2rem 11576 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
7472, 73syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75 chpcl 27167 . . . . . . . . . . . 12 ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
7674, 75syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
7776recnd 11289 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
7844, 47, 61, 64, 66, 71, 77fsumparts 15842 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))))
797nn0zd 12639 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
80 fzval3 13773 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8179, 80syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8281eqcomd 2743 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83 nnm1nn0 12567 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
8419, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
8584nn0red 12588 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86 chpcl 27167 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
8785, 86syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
8887recnd 11289 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89 vmacl 27161 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
9019, 89syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
9190recnd 11289 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
9219nncnd 12282 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93 ax-1cn 11213 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
94 pncan 11514 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
9592, 93, 94sylancl 586 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
96 npcan 11517 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
9792, 93, 96sylancl 586 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
9895, 97eqtr4d 2780 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
9998fveq2d 6910 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
100 chpp1 27198 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ ℕ0 → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
10184, 100syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
10297fveq2d 6910 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
103102oveq2d 7447 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10499, 101, 1033eqtrd 2781 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10588, 91, 104mvrladdd 11676 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106105oveq2d 7447 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
10720relogcld 26665 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108107recnd 11289 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
10991, 108mulcomd 11282 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110106, 109eqtr4d 2780 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
11182, 110sumeq12rdv 15743 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
1127nn0cnd 12589 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
113 pncan 11514 . . . . . . . . . . . . . . . . 17 (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
114112, 93, 113sylancl 586 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
115114fveq2d 6910 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
116 chpfl 27193 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
1171, 116syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118115, 117eqtrd 2777 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119118oveq2d 7447 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
12012, 4mulcomd 11282 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121119, 120eqtrd 2777 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122 0cn 11253 . . . . . . . . . . . . . 14 0 ∈ ℂ
123122mul01i 11451 . . . . . . . . . . . . 13 (0 · 0) = 0
124123a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (0 · 0) = 0)
125121, 124oveq12d 7449 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
12637subid1d 11609 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127125, 126eqtrd 2777 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
12895fveq2d 6910 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
129128oveq2d 7447 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
13082, 129sumeq12rdv 15743 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
131127, 130oveq12d 7449 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1)))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
13278, 111, 1313eqtr3d 2785 . . . . . . . 8 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))))
133132oveq1d 7446 . . . . . . 7 (𝑥 ∈ ℝ+ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) − ((ψ‘𝑥) · (log‘𝑥))))
13439, 41, 1333eqtr4d 2787 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))))
135134oveq1d 7446 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
136 div23 11941 . . . . . . 7 (((ψ‘𝑥) ∈ ℂ ∧ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
1374, 15, 34, 136syl3anc 1373 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138137oveq1d 7446 . . . . 5 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)))
13936, 135, 1383eqtr3rd 2786 . . . 4 (𝑥 ∈ ℝ+ → ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
140139mpteq2ia 5245 . . 3 (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))
141 ovexd 7466 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142 ovexd 7466 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143 reex 11246 . . . . . . . 8 ℝ ∈ V
144 rpssre 13042 . . . . . . . 8 + ⊆ ℝ
145143, 144ssexi 5322 . . . . . . 7 + ∈ V
146145a1i 11 . . . . . 6 (⊤ → ℝ+ ∈ V)
147 ovexd 7466 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
14815adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149 eqidd 2738 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
150 eqidd 2738 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151146, 147, 148, 149, 150offval2 7717 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
152 chpo1ub 27524 . . . . . 6 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153 0red 11264 . . . . . . . 8 (⊤ → 0 ∈ ℝ)
154 1red 11262 . . . . . . . 8 (⊤ → 1 ∈ ℝ)
155 divrcnv 15888 . . . . . . . . 9 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
15693, 155mp1i 13 . . . . . . . 8 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
157 rpreccl 13061 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
158157rpred 13077 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
159158adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
16011, 13resubcld 11691 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161160adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162 rpaddcl 13057 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
16321, 162mpan2 691 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
164163relogcld 26665 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
165164, 13resubcld 11691 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
1667nn0red 12588 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
167 1red 11262 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
168 flle 13839 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
1691, 168syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
170166, 1, 167, 169leadd1dd 11877 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
17110, 163logled 26669 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
172170, 171mpbid 232 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
17311, 164, 13, 172lesub1dd 11879 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174 logdifbnd 27037 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175160, 165, 158, 173, 174letrd 11418 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
176175ad2antrl 728 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
177 fllep1 13841 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
1781, 177syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ≤ ((⌊‘𝑥) + 1))
179 logleb 26645 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ ((⌊‘𝑥) + 1) ∈ ℝ+) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
18010, 179mpdan 687 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
181178, 180mpbid 232 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
18211, 13subge0d 11853 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
183181, 182mpbird 257 . . . . . . . . 9 (𝑥 ∈ ℝ+ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
184183ad2antrl 728 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
185153, 154, 156, 159, 161, 176, 184rlimsqz2 15687 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
186 rlimo1 15653 . . . . . . 7 ((𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
187185, 186syl 17 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
188 o1mul 15651 . . . . . 6 (((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
189152, 187, 188sylancr 587 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘f · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
190151, 189eqeltrrd 2842 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191 nnrp 13046 . . . . . . . . 9 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
192191ssriv 3987 . . . . . . . 8 ℕ ⊆ ℝ+
193192a1i 11 . . . . . . 7 (⊤ → ℕ ⊆ ℝ+)
194193sselda 3983 . . . . . 6 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
195194, 31syl 17 . . . . 5 ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196 chpo1ub 27524 . . . . . . . 8 (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197196a1i 11 . . . . . . 7 (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198 rerpdivcl 13065 . . . . . . . . 9 (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
19929, 198mpancom 688 . . . . . . . 8 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
200199adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20131adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
202 rpreccl 13061 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
203202rpred 13077 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
204 chpge0 27169 . . . . . . . . . . 11 (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
20527, 204syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
206 logdifbnd 27037 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
20726, 203, 29, 205, 206lemul1ad 12207 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
20827lep1d 12199 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+𝑛 ≤ (𝑛 + 1))
209 logleb 26645 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
21023, 209mpdan 687 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
211208, 210mpbid 232 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
21224, 25subge0d 11853 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
213211, 212mpbird 257 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
21426, 29, 213, 205mulge0d 11840 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
21530, 214absidd 15461 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216 rpregt0 13049 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217 divge0 12137 . . . . . . . . . . . 12 ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
21829, 205, 216, 217syl21anc 838 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((ψ‘𝑛) / 𝑛))
219199, 218absidd 15461 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
22029recnd 11289 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
221 rpcn 13045 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℂ)
222 rpne0 13051 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ≠ 0)
223220, 221, 222divrec2d 12047 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224219, 223eqtrd 2777 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225207, 215, 2243brtr4d 5175 . . . . . . . 8 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
226225ad2antrl 728 . . . . . . 7 ((⊤ ∧ (𝑛 ∈ ℝ+ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
227154, 197, 200, 201, 226o1le 15689 . . . . . 6 (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228193, 227o1res2 15599 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229195, 228o1fsum 15849 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230141, 142, 190, 229o1sub2 15662 . . 3 (⊤ → (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
231140, 230eqeltrrid 2846 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232231mptru 1547 1 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wtru 1541  wcel 2108  wne 2940  Vcvv 3480  wss 3951   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  f cof 7695  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  +crp 13034  ...cfz 13547  ..^cfzo 13694  cfl 13830  abscabs 15273  𝑟 crli 15521  𝑂(1)co1 15522  Σcsu 15722  logclog 26596  Λcvma 27135  ψcchp 27136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-o1 15526  df-lo1 15527  df-sum 15723  df-ef 16103  df-e 16104  df-sin 16105  df-cos 16106  df-pi 16108  df-dvds 16291  df-gcd 16532  df-prm 16709  df-pc 16875  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-log 26598  df-cxp 26599  df-cht 27140  df-vma 27141  df-chp 27142  df-ppi 27143
This theorem is referenced by:  selberg2  27595  selberg3lem2  27602
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