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Theorem selberg2lem 27608
Description: Lemma for selberg2 27609. 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 13040 . . . . . . . . 9 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
2 chpcl 27181 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
31, 2syl 17 . . . . . . . 8 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℝ)
43recnd 11286 . . . . . . 7 (𝑥 ∈ ℝ+ → (ψ‘𝑥) ∈ ℂ)
5 rprege0 13047 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
6 flge0nn0 13856 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
75, 6syl 17 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℕ0)
8 nn0p1nn 12562 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
97, 8syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℕ)
109nnrpd 13072 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ ℝ+)
1110relogcld 26679 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℝ)
1211recnd 11286 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘((⌊‘𝑥) + 1)) ∈ ℂ)
13 relogcl 26631 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
1413recnd 11286 . . . . . . . 8 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℂ)
1512, 14subcld 11617 . . . . . . 7 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
164, 15mulcld 11278 . . . . . 6 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ ℂ)
17 fzfid 14010 . . . . . . 7 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) ∈ Fin)
18 elfznn 13589 . . . . . . . . . 10 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1918adantl 481 . . . . . . . . 9 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
2019nnrpd 13072 . . . . . . . 8 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
21 1rp 13035 . . . . . . . . . . . . 13 1 ∈ ℝ+
22 rpaddcl 13054 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑛 + 1) ∈ ℝ+)
2321, 22mpan2 691 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 + 1) ∈ ℝ+)
2423relogcld 26679 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘(𝑛 + 1)) ∈ ℝ)
25 relogcl 26631 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (log‘𝑛) ∈ ℝ)
2624, 25resubcld 11688 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ∈ ℝ)
27 rpre 13040 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℝ)
28 chpcl 27181 . . . . . . . . . . 11 (𝑛 ∈ ℝ → (ψ‘𝑛) ∈ ℝ)
2927, 28syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℝ)
3026, 29remulcld 11288 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℝ)
3130recnd 11286 . . . . . . . 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 15765 . . . . . 6 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
34 rpcnne0 13050 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
35 divsubdir 11958 . . . . . 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 1370 . . . . 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 11278 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) ∈ ℂ)
384, 14mulcld 11278 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
3937, 38, 33sub32d 11649 . . . . . . 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 11716 . . . . . . . 8 (𝑥 ∈ ℝ+ → ((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − ((ψ‘𝑥) · (log‘𝑥))))
4140oveq1d 7445 . . . . . . 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 7453 . . . . . . . . . . 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 7453 . . . . . . . . . . 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 26641 . . . . . . . . . . . 12 (log‘1) = 0
5048, 49eqtrdi 2790 . . . . . . . . . . 11 (𝑚 = 1 → (log‘𝑚) = 0)
51 oveq1 7437 . . . . . . . . . . . . . 14 (𝑚 = 1 → (𝑚 − 1) = (1 − 1))
52 1m1e0 12335 . . . . . . . . . . . . . 14 (1 − 1) = 0
5351, 52eqtrdi 2790 . . . . . . . . . . . . 13 (𝑚 = 1 → (𝑚 − 1) = 0)
5453fveq2d 6910 . . . . . . . . . . . 12 (𝑚 = 1 → (ψ‘(𝑚 − 1)) = (ψ‘0))
55 2pos 12366 . . . . . . . . . . . . 13 0 < 2
56 0re 11260 . . . . . . . . . . . . . 14 0 ∈ ℝ
57 chpeq0 27266 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . 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 7453 . . . . . . . . . . 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 12918 . . . . . . . . . . 11 ℕ = (ℤ‘1)
669, 65eleqtrdi 2848 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
67 elfznn 13589 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
6867adantl 481 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
6968nnrpd 13072 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ+)
7069relogcld 26679 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℝ)
7170recnd 11286 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (log‘𝑚) ∈ ℂ)
7268nnred 12278 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℝ)
73 peano2rem 11573 . . . . . . . . . . . . 13 (𝑚 ∈ ℝ → (𝑚 − 1) ∈ ℝ)
7472, 73syl 17 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑚 − 1) ∈ ℝ)
75 chpcl 27181 . . . . . . . . . . . 12 ((𝑚 − 1) ∈ ℝ → (ψ‘(𝑚 − 1)) ∈ ℝ)
7674, 75syl 17 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℝ)
7776recnd 11286 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑚 − 1)) ∈ ℂ)
7844, 47, 61, 64, 66, 71, 77fsumparts 15838 . . . . . . . . 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 12636 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℤ)
80 fzval3 13769 . . . . . . . . . . . 12 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8179, 80syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
8281eqcomd 2740 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
83 nnm1nn0 12564 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ0)
8419, 83syl 17 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℕ0)
8584nn0red 12585 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 − 1) ∈ ℝ)
86 chpcl 27181 . . . . . . . . . . . . . . 15 ((𝑛 − 1) ∈ ℝ → (ψ‘(𝑛 − 1)) ∈ ℝ)
8785, 86syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℝ)
8887recnd 11286 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑛 − 1)) ∈ ℂ)
89 vmacl 27175 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
9019, 89syl 17 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
9190recnd 11286 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
9219nncnd 12279 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
93 ax-1cn 11210 . . . . . . . . . . . . . . . . 17 1 ∈ ℂ
94 pncan 11511 . . . . . . . . . . . . . . . . 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 11514 . . . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . . . 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 27212 . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑛 − 1)) + (Λ‘((𝑛 − 1) + 1))) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10499, 101, 1033eqtrd 2778 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘((𝑛 + 1) − 1)) = ((ψ‘(𝑛 − 1)) + (Λ‘𝑛)))
10588, 91, 104mvrladdd 11673 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1))) = (Λ‘𝑛))
106105oveq2d 7446 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((log‘𝑛) · (Λ‘𝑛)))
10720relogcld 26679 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
108107recnd 11286 . . . . . . . . . . . 12 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
10991, 108mulcomd 11279 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) = ((log‘𝑛) · (Λ‘𝑛)))
110106, 109eqtr4d 2777 . . . . . . . . . 10 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = ((Λ‘𝑛) · (log‘𝑛)))
11182, 110sumeq12rdv 15739 . . . . . . . . 9 (𝑥 ∈ ℝ+ → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))((log‘𝑛) · ((ψ‘((𝑛 + 1) − 1)) − (ψ‘(𝑛 − 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)))
1127nn0cnd 12586 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℂ)
113 pncan 11511 . . . . . . . . . . . . . . . . 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 27207 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
1171, 116syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (ψ‘(⌊‘𝑥)) = (ψ‘𝑥))
118115, 117eqtrd 2774 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (ψ‘(((⌊‘𝑥) + 1) − 1)) = (ψ‘𝑥))
119118oveq2d 7446 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)))
12012, 4mulcomd 11279 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘𝑥)) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
121119, 120eqtrd 2774 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
122 0cn 11250 . . . . . . . . . . . . . 14 0 ∈ ℂ
123122mul01i 11448 . . . . . . . . . . . . 13 (0 · 0) = 0
124123a1i 11 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (0 · 0) = 0)
125121, 124oveq12d 7448 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((log‘((⌊‘𝑥) + 1)) · (ψ‘(((⌊‘𝑥) + 1) − 1))) − (0 · 0)) = (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0))
12637subid1d 11606 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))) − 0) = ((ψ‘𝑥) · (log‘((⌊‘𝑥) + 1))))
127125, 126eqtrd 2774 . . . . . . . . . 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 7446 . . . . . . . . . . 11 ((𝑥 ∈ ℝ+𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘((𝑛 + 1) − 1))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
13082, 129sumeq12rdv 15739 . . . . . . . . . 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 7448 . . . . . . . . 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 2782 . . . . . . . 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 7445 . . . . . . 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 2784 . . . . . 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 7445 . . . . 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 11938 . . . . . . 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 1370 . . . . . 6 (𝑥 ∈ ℝ+ → (((ψ‘𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) / 𝑥) = (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
138137oveq1d 7445 . . . . 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 2783 . . . 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 5250 . . 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 7465 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ∈ V)
142 ovexd 7465 . . . 4 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥) ∈ V)
143 reex 11243 . . . . . . . 8 ℝ ∈ V
144 rpssre 13039 . . . . . . . 8 + ⊆ ℝ
145143, 144ssexi 5327 . . . . . . 7 + ∈ V
146145a1i 11 . . . . . 6 (⊤ → ℝ+ ∈ V)
147 ovexd 7465 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((ψ‘𝑥) / 𝑥) ∈ V)
14815adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℂ)
149 eqidd 2735 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) = (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)))
150 eqidd 2735 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) = (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))))
151146, 147, 148, 149, 150offval2 7716 . . . . 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 27538 . . . . . 6 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
153 0red 11261 . . . . . . . 8 (⊤ → 0 ∈ ℝ)
154 1red 11259 . . . . . . . 8 (⊤ → 1 ∈ ℝ)
155 divrcnv 15884 . . . . . . . . 9 (1 ∈ ℂ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
15693, 155mp1i 13 . . . . . . . 8 (⊤ → (𝑥 ∈ ℝ+ ↦ (1 / 𝑥)) ⇝𝑟 0)
157 rpreccl 13058 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ+)
158157rpred 13074 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (1 / 𝑥) ∈ ℝ)
159158adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1 / 𝑥) ∈ ℝ)
16011, 13resubcld 11688 . . . . . . . . 9 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
161160adantl 481 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ∈ ℝ)
162 rpaddcl 13054 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝑥 + 1) ∈ ℝ+)
16321, 162mpan2 691 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → (𝑥 + 1) ∈ ℝ+)
164163relogcld 26679 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (log‘(𝑥 + 1)) ∈ ℝ)
165164, 13resubcld 11688 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ∈ ℝ)
1667nn0red 12585 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ∈ ℝ)
167 1red 11259 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → 1 ∈ ℝ)
168 flle 13835 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
1691, 168syl 17 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → (⌊‘𝑥) ≤ 𝑥)
170166, 1, 167, 169leadd1dd 11874 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+ → ((⌊‘𝑥) + 1) ≤ (𝑥 + 1))
17110, 163logled 26683 . . . . . . . . . . . 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 11876 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)) ≤ ((log‘(𝑥 + 1)) − (log‘𝑥)))
174 logdifbnd 27051 . . . . . . . . . 10 (𝑥 ∈ ℝ+ → ((log‘(𝑥 + 1)) − (log‘𝑥)) ≤ (1 / 𝑥))
175160, 165, 158, 173, 174letrd 11415 . . . . . . . . 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 13837 . . . . . . . . . . . 12 (𝑥 ∈ ℝ → 𝑥 ≤ ((⌊‘𝑥) + 1))
1781, 177syl 17 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ≤ ((⌊‘𝑥) + 1))
179 logleb 26659 . . . . . . . . . . . 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 11850 . . . . . . . . . 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 15683 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥))) ⇝𝑟 0)
186 rlimo1 15649 . . . . . . 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 15647 . . . . . 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 2839 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((ψ‘𝑥) / 𝑥) · ((log‘((⌊‘𝑥) + 1)) − (log‘𝑥)))) ∈ 𝑂(1))
191 nnrp 13043 . . . . . . . . 9 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ+)
192191ssriv 3998 . . . . . . . 8 ℕ ⊆ ℝ+
193192a1i 11 . . . . . . 7 (⊤ → ℕ ⊆ ℝ+)
194193sselda 3994 . . . . . 6 ((⊤ ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℝ+)
195194, 31syl 17 . . . . 5 ((⊤ ∧ 𝑛 ∈ ℕ) → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ∈ ℂ)
196 chpo1ub 27538 . . . . . . . 8 (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1)
197196a1i 11 . . . . . . 7 (⊤ → (𝑛 ∈ ℝ+ ↦ ((ψ‘𝑛) / 𝑛)) ∈ 𝑂(1))
198 rerpdivcl 13062 . . . . . . . . 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 13058 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ+)
203202rpred 13074 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (1 / 𝑛) ∈ ℝ)
204 chpge0 27183 . . . . . . . . . . 11 (𝑛 ∈ ℝ → 0 ≤ (ψ‘𝑛))
20527, 204syl 17 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (ψ‘𝑛))
206 logdifbnd 27051 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((log‘(𝑛 + 1)) − (log‘𝑛)) ≤ (1 / 𝑛))
20726, 203, 29, 205, 206lemul1ad 12204 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) ≤ ((1 / 𝑛) · (ψ‘𝑛)))
20827lep1d 12196 . . . . . . . . . . . . 13 (𝑛 ∈ ℝ+𝑛 ≤ (𝑛 + 1))
209 logleb 26659 . . . . . . . . . . . . . 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 11850 . . . . . . . . . . . 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 11837 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → 0 ≤ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
21530, 214absidd 15457 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) = (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)))
216 rpregt0 13046 . . . . . . . . . . . 12 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℝ ∧ 0 < 𝑛))
217 divge0 12134 . . . . . . . . . . . 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 15457 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((ψ‘𝑛) / 𝑛))
22029recnd 11286 . . . . . . . . . . 11 (𝑛 ∈ ℝ+ → (ψ‘𝑛) ∈ ℂ)
221 rpcn 13042 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ∈ ℂ)
222 rpne0 13048 . . . . . . . . . . 11 (𝑛 ∈ ℝ+𝑛 ≠ 0)
223220, 221, 222divrec2d 12044 . . . . . . . . . 10 (𝑛 ∈ ℝ+ → ((ψ‘𝑛) / 𝑛) = ((1 / 𝑛) · (ψ‘𝑛)))
224219, 223eqtrd 2774 . . . . . . . . 9 (𝑛 ∈ ℝ+ → (abs‘((ψ‘𝑛) / 𝑛)) = ((1 / 𝑛) · (ψ‘𝑛)))
225207, 215, 2243brtr4d 5179 . . . . . . . 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 15685 . . . . . 6 (⊤ → (𝑛 ∈ ℝ+ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
228193, 227o1res2 15595 . . . . 5 (⊤ → (𝑛 ∈ ℕ ↦ (((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛))) ∈ 𝑂(1))
229195, 228o1fsum 15845 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑛 + 1)) − (log‘𝑛)) · (ψ‘𝑛)) / 𝑥)) ∈ 𝑂(1))
230141, 142, 190, 229o1sub2 15658 . . 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 2843 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1))
232231mptru 1543 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 1536  wtru 1537  wcel 2105  wne 2937  Vcvv 3477  wss 3962   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  f cof 7694  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  0cn0 12523  cz 12610  cuz 12875  +crp 13031  ...cfz 13543  ..^cfzo 13690  cfl 13826  abscabs 15269  𝑟 crli 15517  𝑂(1)co1 15518  Σcsu 15718  logclog 26610  Λcvma 27149  ψcchp 27150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-o1 15522  df-lo1 15523  df-sum 15719  df-ef 16099  df-e 16100  df-sin 16101  df-cos 16102  df-pi 16104  df-dvds 16287  df-gcd 16528  df-prm 16705  df-pc 16870  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-mulg 19098  df-cntz 19347  df-cmn 19814  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-log 26612  df-cxp 26613  df-cht 27154  df-vma 27155  df-chp 27156  df-ppi 27157
This theorem is referenced by:  selberg2  27609  selberg3lem2  27616
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