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Theorem selberg2b 26055
Description: Convert eventual boundedness in selberg2 26054 to boundedness on any interval [𝐴, +∞). (We have to bound away from zero because the log terms diverge at zero.) (Contributed by Mario Carneiro, 25-May-2016.)
Assertion
Ref Expression
selberg2b 𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐
Distinct variable group:   𝑛,𝑐,𝑥

Proof of Theorem selberg2b
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 1re 10629 . . . . . . 7 1 ∈ ℝ
2 elicopnf 12821 . . . . . . 7 (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
31, 2mp1i 13 . . . . . 6 (⊤ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
43simprbda 499 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ)
54ex 413 . . . 4 (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ))
65ssrdv 3970 . . 3 (⊤ → (1[,)+∞) ⊆ ℝ)
71a1i 11 . . 3 (⊤ → 1 ∈ ℝ)
8 chpcl 25628 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
94, 8syl 17 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (ψ‘𝑥) ∈ ℝ)
10 1rp 12381 . . . . . . . . . . 11 1 ∈ ℝ+
1110a1i 11 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
123simplbda 500 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
134, 11, 12rpgecld 12458 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ+)
1413relogcld 25133 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (log‘𝑥) ∈ ℝ)
159, 14remulcld 10659 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
16 fzfid 13329 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17 elfznn 12924 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1817adantl 482 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19 vmacl 25622 . . . . . . . . . 10 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
214adantr 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
2221, 18nndivred 11679 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
23 chpcl 25628 . . . . . . . . . 10 ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2422, 23syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2520, 24remulcld 10659 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
2616, 25fsumrecl 15079 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
2715, 26readdcld 10658 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
2827, 13rerpdivcld 12450 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
29 2re 11699 . . . . . . 7 2 ∈ ℝ
3029a1i 11 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 2 ∈ ℝ)
3130, 14remulcld 10659 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
3228, 31resubcld 11056 . . . 4 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
3332recnd 10657 . . 3 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
3413ex 413 . . . . 5 (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ+))
3534ssrdv 3970 . . . 4 (⊤ → (1[,)+∞) ⊆ ℝ+)
36 selberg2 26054 . . . . 5 (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
3736a1i 11 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
3835, 37o1res2 14908 . . 3 (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
39 chpcl 25628 . . . . . . 7 (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
4039ad2antrl 724 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (ψ‘𝑦) ∈ ℝ)
41 simprl 767 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ)
4210a1i 11 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ∈ ℝ+)
43 simprr 769 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ 𝑦)
4441, 42, 43rpgecld 12458 . . . . . . 7 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ+)
4544relogcld 25133 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (log‘𝑦) ∈ ℝ)
4640, 45remulcld 10659 . . . . 5 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
47 fzfid 13329 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
48 elfznn 12924 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑦)) → 𝑛 ∈ ℕ)
4948adantl 482 . . . . . . . 8 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
5049, 19syl 17 . . . . . . 7 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
5141adantr 481 . . . . . . . . 9 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
5251, 49nndivred 11679 . . . . . . . 8 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
53 chpcl 25628 . . . . . . . 8 ((𝑦 / 𝑛) ∈ ℝ → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
5452, 53syl 17 . . . . . . 7 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
5550, 54remulcld 10659 . . . . . 6 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
5647, 55fsumrecl 15079 . . . . 5 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
5746, 56readdcld 10658 . . . 4 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
5829a1i 11 . . . . 5 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 2 ∈ ℝ)
5958, 45remulcld 10659 . . . 4 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
6057, 59readdcld 10658 . . 3 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
6132adantr 481 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
6261recnd 10657 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
6362abscld 14784 . . . 4 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ ℝ)
6428adantr 481 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
6564recnd 10657 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
6665abscld 14784 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ∈ ℝ)
6731adantr 481 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℝ)
6867recnd 10657 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℂ)
6968abscld 14784 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ∈ ℝ)
7066, 69readdcld 10658 . . . 4 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))) ∈ ℝ)
7160ad2ant2r 743 . . . 4 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
7265, 68abs2dif2d 14806 . . . 4 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))))
73 simprll 775 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ)
7473, 39syl 17 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑦) ∈ ℝ)
7513adantr 481 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ+)
764adantr 481 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ)
77 simprr 769 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
7876, 73, 77ltled 10776 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥𝑦)
7973, 75, 78rpgecld 12458 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ+)
8079relogcld 25133 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑦) ∈ ℝ)
8174, 80remulcld 10659 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
8256ad2ant2r 743 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
8381, 82readdcld 10658 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℝ)
8429a1i 11 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 2 ∈ ℝ)
8584, 80remulcld 10659 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
8676, 8syl 17 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ∈ ℝ)
8775relogcld 25133 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ∈ ℝ)
8886, 87remulcld 10659 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
8926adantr 481 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
9088, 89readdcld 10658 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
91 chpge0 25630 . . . . . . . . . . 11 (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
9276, 91syl 17 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (ψ‘𝑥))
9312adantr 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ≤ 𝑥)
9476, 93logge0d 25140 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (log‘𝑥))
9586, 87, 92, 94mulge0d 11205 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((ψ‘𝑥) · (log‘𝑥)))
96 vmage0 25625 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
9718, 96syl 17 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
98 chpge0 25630 . . . . . . . . . . . . 13 ((𝑥 / 𝑛) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / 𝑛)))
9922, 98syl 17 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
10020, 24, 97, 99mulge0d 11205 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
10116, 25, 100fsumge0 15138 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
102101adantr 481 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
10388, 89, 95, 102addge0d 11204 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
10490, 75, 103divge0d 12459 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
10564, 104absidd 14770 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
10610a1i 11 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ∈ ℝ+)
107 chpwordi 25661 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
10876, 73, 78, 107syl3anc 1363 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ≤ (ψ‘𝑦))
10975, 79logled 25137 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (𝑥𝑦 ↔ (log‘𝑥) ≤ (log‘𝑦)))
11078, 109mpbid 233 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ≤ (log‘𝑦))
11186, 74, 87, 80, 92, 94, 108, 110lemul12ad 11570 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ≤ ((ψ‘𝑦) · (log‘𝑦)))
112 fzfid 13329 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
11348adantl 482 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℕ)
114113, 19syl 17 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (Λ‘𝑛) ∈ ℝ)
11576adantr 481 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ∈ ℝ)
116115, 113nndivred 11679 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ∈ ℝ)
117116, 23syl 17 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
118114, 117remulcld 10659 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
119112, 118fsumrecl 15079 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
120113, 96syl 17 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (Λ‘𝑛))
121116, 98syl 17 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ (ψ‘(𝑥 / 𝑛)))
122114, 117, 120, 121mulge0d 11205 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
123 flword2 13171 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥𝑦) → (⌊‘𝑦) ∈ (ℤ‘(⌊‘𝑥)))
12476, 73, 78, 123syl3anc 1363 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (⌊‘𝑦) ∈ (ℤ‘(⌊‘𝑥)))
125 fzss2 12935 . . . . . . . . . . . 12 ((⌊‘𝑦) ∈ (ℤ‘(⌊‘𝑥)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
126124, 125syl 17 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑥)) ⊆ (1...(⌊‘𝑦)))
127112, 118, 122, 126fsumless 15139 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
12873adantr 481 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
129128, 113nndivred 11679 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
130129, 53syl 17 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑦 / 𝑛)) ∈ ℝ)
131114, 130remulcld 10659 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
132113nnrpd 12417 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
13378adantr 481 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥𝑦)
134115, 128, 132, 133lediv1dd 12477 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ≤ (𝑦 / 𝑛))
135 chpwordi 25661 . . . . . . . . . . . . 13 (((𝑥 / 𝑛) ∈ ℝ ∧ (𝑦 / 𝑛) ∈ ℝ ∧ (𝑥 / 𝑛) ≤ (𝑦 / 𝑛)) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
136116, 129, 134, 135syl3anc 1363 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
137117, 130, 114, 120, 136lemul2ad 11568 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
138112, 118, 131, 137fsumle 15142 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
13989, 119, 82, 127, 138letrd 10785 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))))
14088, 89, 81, 82, 111, 139le2addd 11247 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
14190, 83, 106, 76, 103, 140, 93lediv12ad 12478 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) / 1))
14283recnd 10657 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℂ)
143142div1d 11396 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) / 1) = (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
144141, 143breqtrd 5083 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
145105, 144eqbrtrd 5079 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) ≤ (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))))
146 2rp 12382 . . . . . . . . 9 2 ∈ ℝ+
147 rpge0 12390 . . . . . . . . 9 (2 ∈ ℝ+ → 0 ≤ 2)
148146, 147mp1i 13 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ 2)
14984, 87, 148, 94mulge0d 11205 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (2 · (log‘𝑥)))
15067, 149absidd 14770 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) = (2 · (log‘𝑥)))
15187, 80, 84, 148, 110lemul2ad 11568 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ≤ (2 · (log‘𝑦)))
152150, 151eqbrtrd 5079 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ≤ (2 · (log‘𝑦)))
15366, 69, 83, 85, 145, 152le2addd 11247 . . . 4 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((abs‘((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥)) + (abs‘(2 · (log‘𝑥)))) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
15463, 70, 71, 72, 153letrd 10785 . . 3 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))))
1556, 7, 33, 38, 60, 154o1bddrp 14887 . 2 (⊤ → ∃𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐)
156155mptru 1535 1 𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wtru 1529  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057  cmpt 5137  cfv 6348  (class class class)co 7145  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  +∞cpnf 10660   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  cuz 12231  +crp 12377  [,)cico 12728  ...cfz 12880  cfl 13148  abscabs 14581  𝑂(1)co1 14831  Σcsu 15030  logclog 25065  Λcvma 25596  ψcchp 25597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-dju 9318  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-o1 14835  df-lo1 14836  df-sum 15031  df-ef 15409  df-e 15410  df-sin 15411  df-cos 15412  df-tan 15413  df-pi 15414  df-dvds 15596  df-gcd 15832  df-prm 16004  df-pc 16162  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-mulg 18163  df-cntz 18385  df-cmn 18837  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-limc 24391  df-dv 24392  df-ulm 24892  df-log 25067  df-cxp 25068  df-atan 25372  df-em 25497  df-cht 25601  df-vma 25602  df-chp 25603  df-ppi 25604  df-mu 25605
This theorem is referenced by:  chpdifbnd  26058
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