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Theorem selberg2lem 25284
Description: Lemma for selberg2 25285. 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 11877 . . . . . . . . 9 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2 chpcl 24895 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
31, 2syl 17 . . . . . . . 8 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
43recnd 10106 . . . . . . 7 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5 rprege0 11885 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6 flge0nn0 12661 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8 nn0p1nn 11370 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
97, 8syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
109nnrpd 11908 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
1110relogcld 24414 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
1211recnd 10106 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13 relogcl 24367 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1413recnd 10106 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1512, 14subcld 10430 . . . . . . 7 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
164, 15mulcld 10098 . . . . . 6 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17 fzfid 12812 . . . . . . 7 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18 elfznn 12408 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1918adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2019nnrpd 11908 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21 1rp 11874 . . . . . . . . . . . . 13 1 ∈ ℝ+
22 rpaddcl 11892 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
2321, 22mpan2 707 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
2423relogcld 24414 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25 relogcl 24367 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
2624, 25resubcld 10496 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27 rpre 11877 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
28 chpcl 24895 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
3026, 29remulcld 10108 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
3130recnd 10106 . . . . . . . 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 14508 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34 rpcnne0 11888 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35 divsubdir 10759 . . . . . 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 1366 . . . . 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 10098 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
384, 14mulcld 10098 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
3937, 38, 33sub32d 10462 . . . . . . 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 10524 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
4140oveq1d 6705 . . . . . . 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 6229 . . . . . . . . . . 11 (𝑚 = 𝑛 → (log‘𝑚) = (log‘𝑛))
43 oveq1 6697 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 − 1) = (𝑛 − 1))
4443fveq2d 6233 . . . . . . . . . . 11 (𝑚 = 𝑛 → (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1)))
4542, 44jca 553 . . . . . . . . . 10 (𝑚 = 𝑛 → ((log‘𝑚) = (log‘𝑛) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(𝑛 − 1))))
46 fveq2 6229 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (log‘𝑚) = (log‘(𝑛 + 1)))
47 oveq1 6697 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑚 − 1) = ((𝑛 + 1) − 1))
4847fveq2d 6233 . . . . . . . . . . 11 (𝑚 = (𝑛 + 1) → (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1)))
4946, 48jca 553 . . . . . . . . . 10 (𝑚 = (𝑛 + 1) → ((log‘𝑚) = (log‘(𝑛 + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘((𝑛 + 1) − 1))))
50 fveq2 6229 . . . . . . . . . . . 12 (𝑚 = 1 → (log‘𝑚) = (log‘1))
51 log1 24377 . . . . . . . . . . . 12 (log‘1) = 0
5250, 51syl6eq 2701 . . . . . . . . . . 11 (𝑚 = 1 → (log‘𝑚) = 0)
53 oveq1 6697 . . . . . . . . . . . . . 14 (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
54 1m1e0 11127 . . . . . . . . . . . . . 14 (1 − 1) = 0
5553, 54syl6eq 2701 . . . . . . . . . . . . 13 (𝑚 = 1 → (𝑚 − 1) = 0)
5655fveq2d 6233 . . . . . . . . . . . 12 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
57 2pos 11150 . . . . . . . . . . . . 13 0 < 2
58 0re 10078 . . . . . . . . . . . . . 14 0 ∈ ℝ
59 chpeq0 24978 . . . . . . . . . . . . . 14 (0 ∈ ℝ → ((ψ‘0) = 0 ↔ 0 < 2))
6058, 59ax-mp 5 . . . . . . . . . . . . 13 ((ψ‘0) = 0 ↔ 0 < 2)
6157, 60mpbir 221 . . . . . . . . . . . 12 (ψ‘0) = 0
6256, 61syl6eq 2701 . . . . . . . . . . 11 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = 0)
6352, 62jca 553 . . . . . . . . . 10 (𝑚 = 1 → ((log‘𝑚) = 0 ∧ (ψ‘(𝑚 − 1)) = 0))
64 fveq2 6229 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (log‘𝑚) = (log‘((⌊‘𝑥) + 1)))
65 oveq1 6697 . . . . . . . . . . . 12 (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 − 1) = (((⌊‘𝑥) + 1) − 1))
6665fveq2d 6233 . . . . . . . . . . 11 (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1)))
6764, 66jca 553 . . . . . . . . . 10 (𝑚 = ((⌊‘𝑥) + 1) → ((log‘𝑚) = (log‘((⌊‘𝑥) + 1)) ∧ (ψ‘(𝑚 − 1)) = (ψ‘(((⌊‘𝑥) + 1) − 1))))
68 nnuz 11761 . . . . . . . . . . 11 ℕ = (ℤ‘1)
699, 68syl6eleq 2740 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
70 elfznn 12408 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
7170adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
7271nnrpd 11908 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
7372relogcld 24414 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
7473recnd 10106 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
7571nnred 11073 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
76 peano2rem 10386 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
7775, 76syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
78 chpcl 24895 . . . . . . . . . . . 12 ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
7977, 78syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
8079recnd 10106 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
8145, 49, 63, 67, 69, 74, 80fsumparts 14582 . . . . . . . . 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 11518 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
83 fzval3 12576 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8482, 83syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8584eqcomd 2657 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
8619nncnd 11074 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
87 ax-1cn 10032 . . . . . . . . . . . . . . . . . 18 1 ∈ ℂ
88 pncan 10325 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 + 1) − 1) = 𝑛)
8986, 87, 88sylancl 695 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = 𝑛)
90 npcan 10328 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑛 − 1) + 1) = 𝑛)
9186, 87, 90sylancl 695 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 − 1) + 1) = 𝑛)
9289, 91eqtr4d 2688 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 1) = ((𝑛 − 1) + 1))
9392fveq2d 6233 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘((𝑛 − 1) + 1)))
94 nnm1nn0 11372 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
9519, 94syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
96 chpp1 24926 . . . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘((𝑛 − 1) + 1)) = (Λ‘𝑛))
9998oveq2d 6706 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10093, 97, 993eqtrd 2689 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
101100oveq1d 6705 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))))
10295nn0red 11390 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
103 chpcl 24895 . . . . . . . . . . . . . . . 16 ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
104102, 103syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
105104recnd 10106 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
106 vmacl 24889 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
10719, 106syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
108107recnd 10106 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
109105, 108pncan2d 10432 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑛 − 1)) + (Λ‘𝑛)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
110101, 109eqtrd 2685 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
111110oveq2d 6706 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
11220relogcld 24414 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
113112recnd 10106 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
114108, 113mulcomd 10099 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
115111, 114eqtr4d 2688 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
11685, 115sumeq12rdv 14482 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
1177nn0cnd 11391 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
118 pncan 10325 . . . . . . . . . . . . . . . . 17 (((⌊‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
119117, 87, 118sylancl 695 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) − 1) = (⌊‘𝑥))
120119fveq2d 6233 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘(⌊‘𝑥)))
121 chpfl 24921 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
1221, 121syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
123120, 122eqtrd 2685 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
124123oveq2d 6706 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
12512, 4mulcomd 10099 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
126124, 125eqtrd 2685 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127 0cn 10070 . . . . . . . . . . . . . 14 0 ∈ ℂ
128127mul01i 10264 . . . . . . . . . . . . 13 (0 · 0) = 0
129128a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (0 · 0) = 0)
130126, 129oveq12d 6708 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
13137subid1d 10419 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
132130, 131eqtrd 2685 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
13389fveq2d 6233 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = (ψ‘𝑛))
134133oveq2d 6706 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
13585, 134sumeq12rdv 14482 . . . . . . . . . 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 6708 . . . . . . . . 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 2693 . . . . . . . 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 6705 . . . . . . 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 2695 . . . . . 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 6705 . . . . 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 10742 . . . . . . 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 1366 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
143142oveq1d 6705 . . . . 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 2694 . . . 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 4773 . . 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 ovexd 6720 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
147 ovexd 6720 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
148 reex 10065 . . . . . . . 8 ℝ ∈ V
149 rpssre 11881 . . . . . . . 8 + ⊆ ℝ
150148, 149ssexi 4836 . . . . . . 7 + ∈ V
151150a1i 11 . . . . . 6 (⊤ → ℝ+ ∈ V)
152 ovexd 6720 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
15315adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
154 eqidd 2652 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
155 eqidd 2652 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
156151, 152, 153, 154, 155offval2 6956 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))))
157 chpo1ub 25214 . . . . . 6 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
158 0red 10079 . . . . . . . 8 (⊤ → 0 ∈ ℝ)
159 1red 10093 . . . . . . . 8 (⊤ → 1 ∈ ℝ)
160 divrcnv 14628 . . . . . . . . 9 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
16187, 160mp1i 13 . . . . . . . 8 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
162 rpreccl 11895 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
163162rpred 11910 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
164163adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
16511, 13resubcld 10496 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
166165adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
167 rpaddcl 11892 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
16821, 167mpan2 707 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
169168relogcld 24414 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
170169, 13resubcld 10496 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
1717nn0red 11390 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
172 1red 10093 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
173 flle 12640 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
1741, 173syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
175171, 1, 172, 174leadd1dd 10679 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
17610, 168logled 24418 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (((⌊‘𝑥) + 1) ≤ (𝑥 + 1) ↔ (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1))))
177175, 176mpbid 222 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ≤ (log‘(𝑥 + 1)))
17811, 169, 13, 177lesub1dd 10681 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
179 logdifbnd 24765 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
180165, 170, 163, 178, 179letrd 10232 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
181180ad2antrl 764 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
182 fllep1 12642 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
1831, 182syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ≤ ((⌊‘𝑥) + 1))
184 logleb 24394 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+ ∧ ((⌊‘𝑥) + 1) ∈ ℝ+) → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
18510, 184mpdan 703 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ≤ ((⌊‘𝑥) + 1) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
186183, 185mpbid 222 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1)))
18711, 13subge0d 10655 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ↔ (log‘𝑥) ≤ (log‘((⌊‘𝑥) + 1))))
188186, 187mpbird 247 . . . . . . . . 9 (𝑥 ∈ ℝ+ → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
189188ad2antrl 764 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → 0 ≤ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))
190158, 159, 161, 164, 166, 181, 189rlimsqz2 14425 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
191 rlimo1 14391 . . . . . . 7 ((𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
192190, 191syl 17 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ 𝑂(1))
193 o1mul 14389 . . . . . 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))
194157, 192, 193sylancr 696 . . . . 5 (⊤ → ((𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∘𝑓 · (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
195156, 194eqeltrrd 2731 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
196 nnrp 11880 . . . . . . . . 9 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
197196ssriv 3640 . . . . . . . 8 ℕ ⊆ ℝ+
198197a1i 11 . . . . . . 7 (⊤ → ℕ ⊆ ℝ+)
199198sselda 3636 . . . . . 6 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
200199, 31syl 17 . . . . 5 ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
201 chpo1ub 25214 . . . . . . . 8 (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
202201a1i 11 . . . . . . 7 (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
203 rerpdivcl 11899 . . . . . . . . 9 (((ψ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20429, 203mpancom 704 . . . . . . . 8 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
205204adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → ((ψ‘𝑛) / 𝑛) ∈ ℝ)
20631adantl 481 . . . . . . 7 ((⊤ ∧ 𝑛 ∈ ℝ+) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
207 rpreccl 11895 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
208207rpred 11910 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
209 chpge0 24897 . . . . . . . . . . 11 (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
21027, 209syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
211 logdifbnd 24765 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
21226, 208, 29, 210, 211lemul1ad 11001 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
21327lep1d 10993 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+𝑛 ≤ (𝑛 + 1))
214 logleb 24394 . . . . . . . . . . . . . 14 ((𝑛 ∈ ℝ+ ∧ (𝑛 + 1) ∈ ℝ+) → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
21523, 214mpdan 703 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+ → (𝑛 ≤ (𝑛 + 1) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
216213, 215mpbid 222 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (log‘𝑛) ≤ (log‘(𝑛 + 1)))
21724, 25subge0d 10655 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)) ↔ (log‘𝑛) ≤ (log‘(𝑛 + 1))))
218216, 217mpbird 247 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((log‘(𝑛 + 1)) − (log‘𝑛)))
21926, 29, 218, 210mulge0d 10642 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
22030, 219absidd 14205 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
221 rpregt0 11884 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
222 divge0 10930 . . . . . . . . . . . 12 ((((ψ‘𝑛) ∈ ℝ ∧ 0 ≤ (ψ‘𝑛)) ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → 0 ≤ ((ψ‘𝑛) / 𝑛))
22329, 210, 221, 222syl21anc 1365 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → 0 ≤ ((ψ‘𝑛) / 𝑛))
224204, 223absidd 14205 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
22529recnd 10106 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
226 rpcn 11879 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℂ)
227 rpne0 11886 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ≠ 0)
228225, 226, 227divrec2d 10843 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
229224, 228eqtrd 2685 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
230212, 220, 2293brtr4d 4717 . . . . . . . 8 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
231230ad2antrl 764 . . . . . . 7 ((⊤ ∧ (𝑛 ∈ ℝ+ ∧ 1 ≤ 𝑛)) → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ≤ (abs‘((ψ‘𝑛) / 𝑛)))
232159, 202, 205, 206, 231o1le 14427 . . . . . 6 (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
233198, 232o1res2 14338 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
234200, 233o1fsum 14589 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
235146, 147, 195, 234o1sub2 14400 . . 3 (⊤ → (𝑥 ∈ ℝ+ ↦ ((((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥))) ∈ 𝑂(1))
236145, 235syl5eqelr 2735 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
237236trud 1533 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 196  wa 383   = wceq 1523  wtru 1524  wcel 2030  wne 2823  Vcvv 3231  wss 3607   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  𝑓 cof 6937  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  +crp 11870  ...cfz 12364  ..^cfzo 12504  cfl 12631  abscabs 14018  𝑟 crli 14260  𝑂(1)co1 14261  Σcsu 14460  logclog 24346  Λcvma 24863  ψcchp 24864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-o1 14265  df-lo1 14266  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-dvds 15028  df-gcd 15264  df-prm 15433  df-pc 15589  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-cht 24868  df-vma 24869  df-chp 24870  df-ppi 24871
This theorem is referenced by:  selberg2  25285  selberg3lem2  25292
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