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Theorem selberg2lem 24984
Description: Lemma for selberg2 24985. 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 11674 . . . . . . . . 9 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2 chpcl 24595 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
31, 2syl 17 . . . . . . . 8 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
43recnd 9925 . . . . . . 7 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5 rprege0 11682 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6 flge0nn0 12441 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8 nn0p1nn 11182 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
97, 8syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
109nnrpd 11705 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
1110relogcld 24118 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
1211recnd 9925 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13 relogcl 24071 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1413recnd 9925 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1512, 14subcld 10244 . . . . . . 7 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
164, 15mulcld 9917 . . . . . 6 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17 fzfid 12592 . . . . . . 7 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18 elfznn 12199 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1918adantl 480 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2019nnrpd 11705 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21 1rp 11671 . . . . . . . . . . . . 13 1 ∈ ℝ+
22 rpaddcl 11689 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
2321, 22mpan2 702 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
2423relogcld 24118 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25 relogcl 24071 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
2624, 25resubcld 10310 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27 rpre 11674 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
28 chpcl 24595 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
3026, 29remulcld 9927 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
3130recnd 9925 . . . . . . . 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 14260 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34 rpcnne0 11685 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35 divsubdir 10573 . . . . . 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 1317 . . . . 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 9917 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
384, 14mulcld 9917 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
3937, 38, 33sub32d 10276 . . . . . . 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 10338 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
4140oveq1d 6542 . . . . . . 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 6088 . . . . . . . . . . 11 (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43 oveq1 6534 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
4443fveq2d 6092 . . . . . . . . . . 11 (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
4542, 44jca 552 . . . . . . . . . 10 (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
46 fveq2 6088 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
47 oveq1 6534 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
4847fveq2d 6092 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
4946, 48jca 552 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
50 fveq2 6088 . . . . . . . . . . . 12 (𝑚 = 1 → (log‘𝑚) = (log‘1))
51 log1 24081 . . . . . . . . . . . 12 (log‘1) = 0
5250, 51syl6eq 2659 . . . . . . . . . . 11 (𝑚 = 1 → (log‘𝑚) = 0)
53 oveq1 6534 . . . . . . . . . . . . . 14 (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
54 1m1e0 10939 . . . . . . . . . . . . . 14 (1 − 1) = 0
5553, 54syl6eq 2659 . . . . . . . . . . . . 13 (𝑚 = 1 → (𝑚 − 1) = 0)
5655fveq2d 6092 . . . . . . . . . . . 12 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
57 2pos 10962 . . . . . . . . . . . . 13 0 < 2
58 0re 9897 . . . . . . . . . . . . . 14 0 ∈ ℝ
59 chpeq0 24678 . . . . . . . . . . . . . 14 (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
6058, 59ax-mp 5 . . . . . . . . . . . . 13 ((ψ‘0) = 0 ↔ 0 < 2)
6157, 60mpbir 219 . . . . . . . . . . . 12 (ψ‘0) = 0
6256, 61syl6eq 2659 . . . . . . . . . . 11 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
6352, 62jca 552 . . . . . . . . . 10 (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
64 fveq2 6088 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
65 oveq1 6534 . . . . . . . . . . . 12 (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
6665fveq2d 6092 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
6764, 66jca 552 . . . . . . . . . 10 (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
68 nnuz 11558 . . . . . . . . . . 11 ℕ = (ℤ‘1)
699, 68syl6eleq 2697 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
70 elfznn 12199 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
7170adantl 480 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
7271nnrpd 11705 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
7372relogcld 24118 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
7473recnd 9925 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
7571nnred 10885 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
76 peano2rem 10200 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
7775, 76syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
78 chpcl 24595 . . . . . . . . . . . 12 ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
7977, 78syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
8079recnd 9925 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
8145, 49, 63, 67, 69, 74, 80fsumparts 14328 . . . . . . . . 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)))))
827nn0zd 11315 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
83 fzval3 12362 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8482, 83syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8584eqcomd 2615 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
8619nncnd 10886 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
87 ax-1cn 9851 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
88 pncan 10139 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
8986, 87, 88sylancl 692 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
90 npcan 10142 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
9186, 87, 90sylancl 692 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
9289, 91eqtr4d 2646 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
9392fveq2d 6092 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
94 nnm1nn0 11184 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
9519, 94syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
96 chpp1 24626 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℕ0 → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
9795, 96syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 − 1) + 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))))
9891fveq2d 6092 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
9998oveq2d 6543 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10093, 97, 993eqtrd 2647 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
101100oveq1d 6542 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))))
10295nn0red 11202 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
103 chpcl 24595 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
104102, 103syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
105104recnd 9925 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
106 vmacl 24589 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
10719, 106syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
108107recnd 9925 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
109105, 108pncan2d 10246 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
110101, 109eqtrd 2643 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
111110oveq2d 6543 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
11220relogcld 24118 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
113112recnd 9925 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
114108, 113mulcomd 9918 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
115111, 114eqtr4d 2646 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
11685, 115sumeq12rdv 14234 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
1177nn0cnd 11203 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
118 pncan 10139 . . . . . . . . . . . . . . . . 17 (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
119117, 87, 118sylancl 692 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
120119fveq2d 6092 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
121 chpfl 24621 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
1221, 121syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
123120, 122eqtrd 2643 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
124123oveq2d 6543 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
12512, 4mulcomd 9918 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
126124, 125eqtrd 2643 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127 0cn 9889 . . . . . . . . . . . . . 14 0 ∈ ℂ
128127mul01i 10078 . . . . . . . . . . . . 13 (0 · 0) = 0
129128a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (0 · 0) = 0)
130126, 129oveq12d 6545 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
13137subid1d 10233 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
132130, 131eqtrd 2643 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
13389fveq2d 6092 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
134133oveq2d 6543 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
13585, 134sumeq12rdv 14234 . . . . . . . . . 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)))
136132, 135oveq12d 6545 . . . . . . . . 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))))
13781, 116, 1363eqtr3d 2651 . . . . . . . 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))))
138137oveq1d 6542 . . . . . . 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))))
13939, 41, 1383eqtr4d 2653 . . . . . 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))))
140139oveq1d 6542 . . . . 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))
141 div23 10556 . . . . . . 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))))
1424, 15, 34, 141syl3anc 1317 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
143142oveq1d 6542 . . . . 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)))
14436, 140, 1433eqtr3rd 2652 . . . 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))
145144mpteq2ia 4662 . . 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))
146 ovex 6555 . . . . 5 (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V
147146a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
148 ovex 6555 . . . . 5 𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V
149148a1i 11 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
150 reex 9884 . . . . . . . 8 ℝ ∈ V
151 rpssre 11678 . . . . . . . 8 + ⊆ ℝ
152150, 151ssexi 4726 . . . . . . 7 + ∈ V
153152a1i 11 . . . . . 6 (⊤ → ℝ+ ∈ V)
154 ovex 6555 . . . . . . 7 ((ψ‘𝑥) / 𝑥) ∈ V
155154a1i 11 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
15615adantl 480 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
157 eqidd 2610 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
158 eqidd 2610 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
159153, 155, 156, 157, 158offval2 6790 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
160 chpo1ub 24914 . . . . . 6 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
161 0red 9898 . . . . . . . 8 (⊤ → 0 ∈ ℝ)
162 1red 9912 . . . . . . . 8 (⊤ → 1 ∈ ℝ)
163 divrcnv 14372 . . . . . . . . 9 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
16487, 163mp1i 13 . . . . . . . 8 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
165 rpreccl 11692 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
166165rpred 11707 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
167166adantl 480 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
16811, 13resubcld 10310 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
169168adantl 480 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
170 rpaddcl 11689 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
17121, 170mpan2 702 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
172171relogcld 24118 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
173172, 13resubcld 10310 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
1747nn0red 11202 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
175 1red 9912 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
176 flle 12420 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
1771, 176syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
178174, 1, 175, 177leadd1dd 10493 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
17910, 171logled 24122 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
180178, 179mpbid 220 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
18111, 172, 13, 180lesub1dd 10495 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
182 logdifbnd 24465 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
183168, 173, 166, 181, 182letrd 10046 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
184183ad2antrl 759 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
185 fllep1 12422 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
1861, 185syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ≤ ((⌊‘𝑥) + 1))
187 logleb 24098 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ ((⌊‘𝑥) + 1) ∈ ℝ+) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
18810, 187mpdan 698 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
189186, 188mpbid 220 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
19011, 13subge0d 10469 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
191189, 190mpbird 245 . . . . . . . . 9 (𝑥 ∈ ℝ+ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
192191ad2antrl 759 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
193161, 162, 164, 167, 169, 184, 192rlimsqz2 14178 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
194 rlimo1 14144 . . . . . . 7 ((𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
195193, 194syl 17 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
196 o1mul 14142 . . . . . 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))
197160, 195, 196sylancr 693 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
198159, 197eqeltrrd 2688 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
199 nnrp 11677 . . . . . . . . 9 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
200199ssriv 3571 . . . . . . . 8 ℕ ⊆ ℝ+
201200a1i 11 . . . . . . 7 (⊤ → ℕ ⊆ ℝ+)
202201sselda 3567 . . . . . 6 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
203202, 31syl 17 . . . . 5 ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
204 chpo1ub 24914 . . . . . . . 8 (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
205204a1i 11 . . . . . . 7 (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
206 rerpdivcl 11696 . . . . . . . . 9 (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20729, 206mpancom 699 . . . . . . . 8 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
208207adantl 480 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20931adantl 480 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
210 rpreccl 11692 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
211210rpred 11707 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
212 chpge0 24597 . . . . . . . . . . 11 (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
21327, 212syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
214 logdifbnd 24465 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
21526, 211, 29, 213, 214lemul1ad 10815 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
21627lep1d 10807 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+𝑛 ≤ (𝑛 + 1))
217 logleb 24098 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
21823, 217mpdan 698 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
219216, 218mpbid 220 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
22024, 25subge0d 10469 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
221219, 220mpbird 245 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
22226, 29, 221, 213mulge0d 10456 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
22330, 222absidd 13958 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
224 rpregt0 11681 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
225 divge0 10744 . . . . . . . . . . . 12 ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
22629, 213, 224, 225syl21anc 1316 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((ψ‘𝑛) / 𝑛))
227207, 226absidd 13958 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
22829recnd 9925 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
229 rpcn 11676 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℂ)
230 rpne0 11683 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ≠ 0)
231228, 229, 230divrec2d 10657 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
232227, 231eqtrd 2643 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
233215, 223, 2323brtr4d 4609 . . . . . . . 8 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
234233ad2antrl 759 . . . . . . 7 ((⊤ ∧ (𝑛 ∈ ℝ+ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
235162, 205, 208, 209, 234o1le 14180 . . . . . 6 (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
236201, 235o1res2 14091 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
237203, 236o1fsum 14335 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
238147, 149, 198, 237o1sub2 14153 . . 3 (⊤ → (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
239145, 238syl5eqelr 2692 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
240239trud 1483 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 194  wa 382   = wceq 1474  wtru 1475  wcel 1976  wne 2779  Vcvv 3172  wss 3539   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  𝑓 cof 6771  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796   · cmul 9798   < clt 9931  cle 9932  cmin 10118   / cdiv 10536  cn 10870  2c2 10920  0cn0 11142  cz 11213  cuz 11522  +crp 11667  ...cfz 12155  ..^cfzo 12292  cfl 12411  abscabs 13771  𝑟 crli 14013  𝑂(1)co1 14014  Σcsu 14213  logclog 24050  Λcvma 24563  ψcchp 24564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871  ax-addf 9872  ax-mulf 9873
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6773  df-om 6936  df-1st 7037  df-2nd 7038  df-supp 7161  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-2o 7426  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-ixp 7773  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fsupp 8137  df-fi 8178  df-sup 8209  df-inf 8210  df-oi 8276  df-card 8626  df-cda 8851  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-ioo 12009  df-ioc 12010  df-ico 12011  df-icc 12012  df-fz 12156  df-fzo 12293  df-fl 12413  df-mod 12489  df-seq 12622  df-exp 12681  df-fac 12881  df-bc 12910  df-hash 12938  df-shft 13604  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-limsup 13999  df-clim 14016  df-rlim 14017  df-o1 14018  df-lo1 14019  df-sum 14214  df-ef 14586  df-e 14587  df-sin 14588  df-cos 14589  df-pi 14591  df-dvds 14771  df-gcd 15004  df-prm 15173  df-pc 15329  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-sets 15650  df-ress 15651  df-plusg 15730  df-mulr 15731  df-starv 15732  df-sca 15733  df-vsca 15734  df-ip 15735  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-hom 15742  df-cco 15743  df-rest 15855  df-topn 15856  df-0g 15874  df-gsum 15875  df-topgen 15876  df-pt 15877  df-prds 15880  df-xrs 15934  df-qtop 15939  df-imas 15940  df-xps 15942  df-mre 16018  df-mrc 16019  df-acs 16021  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-submnd 17108  df-mulg 17313  df-cntz 17522  df-cmn 17967  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-fbas 19513  df-fg 19514  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cld 20581  df-ntr 20582  df-cls 20583  df-nei 20660  df-lp 20698  df-perf 20699  df-cn 20789  df-cnp 20790  df-haus 20877  df-tx 21123  df-hmeo 21316  df-fil 21408  df-fm 21500  df-flim 21501  df-flf 21502  df-xms 21883  df-ms 21884  df-tms 21885  df-cncf 22437  df-limc 23381  df-dv 23382  df-log 24052  df-cxp 24053  df-cht 24568  df-vma 24569  df-chp 24570  df-ppi 24571
This theorem is referenced by:  selberg2  24985  selberg3lem2  24992
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