MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  selberg2b Structured version   Visualization version   GIF version

Theorem selberg2b 25286
Description: Convert eventual boundedness in selberg2 25285 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 10077 . . . . . . 7 1 ∈ ℝ
2 elicopnf 12307 . . . . . . 7 (1 ∈ ℝ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
31, 2mp1i 13 . . . . . 6 (⊤ → (𝑥 ∈ (1[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 1 ≤ 𝑥)))
43simprbda 652 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ)
54ex 449 . . . 4 (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ))
65ssrdv 3642 . . 3 (⊤ → (1[,)+∞) ⊆ ℝ)
71a1i 11 . . 3 (⊤ → 1 ∈ ℝ)
8 chpcl 24895 . . . . . . . . 9 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
94, 8syl 17 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (ψ‘𝑥) ∈ ℝ)
10 1rp 11874 . . . . . . . . . . 11 1 ∈ ℝ+
1110a1i 11 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ∈ ℝ+)
123simplbda 653 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 1 ≤ 𝑥)
134, 11, 12rpgecld 11949 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 𝑥 ∈ ℝ+)
1413relogcld 24414 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (log‘𝑥) ∈ ℝ)
159, 14remulcld 10108 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
16 fzfid 12812 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
17 elfznn 12408 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
1817adantl 481 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
19 vmacl 24889 . . . . . . . . . 10 (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
2018, 19syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
214adantr 480 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
2221, 18nndivred 11107 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
23 chpcl 24895 . . . . . . . . . 10 ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2422, 23syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
2520, 24remulcld 10108 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
2616, 25fsumrecl 14509 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
2715, 26readdcld 10107 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
2827, 13rerpdivcld 11941 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
29 2re 11128 . . . . . . 7 2 ∈ ℝ
3029a1i 11 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 2 ∈ ℝ)
3130, 14remulcld 10108 . . . . 5 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (2 · (log‘𝑥)) ∈ ℝ)
3228, 31resubcld 10496 . . . 4 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℝ)
3332recnd 10106 . . 3 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ ℂ)
3413ex 449 . . . . 5 (⊤ → (𝑥 ∈ (1[,)+∞) → 𝑥 ∈ ℝ+))
3534ssrdv 3642 . . . 4 (⊤ → (1[,)+∞) ⊆ ℝ+)
36 selberg2 25285 . . . . 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 14338 . . 3 (⊤ → (𝑥 ∈ (1[,)+∞) ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
39 chpcl 24895 . . . . . . 7 (𝑦 ∈ ℝ → (ψ‘𝑦) ∈ ℝ)
4039ad2antrl 764 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (ψ‘𝑦) ∈ ℝ)
41 simprl 809 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ)
4210a1i 11 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ∈ ℝ+)
43 simprr 811 . . . . . . . 8 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 1 ≤ 𝑦)
4441, 42, 43rpgecld 11949 . . . . . . 7 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → 𝑦 ∈ ℝ+)
4544relogcld 24414 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (log‘𝑦) ∈ ℝ)
4640, 45remulcld 10108 . . . . 5 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
47 fzfid 12812 . . . . . 6 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
48 elfznn 12408 . . . . . . . . 9 (𝑛 ∈ (1...(⌊‘𝑦)) → 𝑛 ∈ ℕ)
4948adantl 481 . . . . . . . 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 480 . . . . . . . . 9 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
5251, 49nndivred 11107 . . . . . . . 8 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑦 / 𝑛) ∈ ℝ)
53 chpcl 24895 . . . . . . . 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 10108 . . . . . 6 (((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
5647, 55fsumrecl 14509 . . . . 5 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
5746, 56readdcld 10107 . . . 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 10108 . . . 4 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → (2 · (log‘𝑦)) ∈ ℝ)
6057, 59readdcld 10107 . . 3 ((⊤ ∧ (𝑦 ∈ ℝ ∧ 1 ≤ 𝑦)) → ((((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) + (2 · (log‘𝑦))) ∈ ℝ)
6132adantr 480 . . . . . 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 10106 . . . . 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 14219 . . . 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 480 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℝ)
6564recnd 10106 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
6665abscld 14219 . . . . 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 480 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℝ)
6867recnd 10106 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ∈ ℂ)
6968abscld 14219 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ∈ ℝ)
7066, 69readdcld 10107 . . . 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 798 . . . 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 14241 . . . 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 819 . . . . . . . 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 480 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ+)
764adantr 480 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 ∈ ℝ)
77 simprr 811 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥 < 𝑦)
7876, 73, 77ltled 10223 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑥𝑦)
7973, 75, 78rpgecld 11949 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 𝑦 ∈ ℝ+)
8079relogcld 24414 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑦) ∈ ℝ)
8174, 80remulcld 10108 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑦) · (log‘𝑦)) ∈ ℝ)
8256ad2ant2r 798 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
8381, 82readdcld 10107 . . . . 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 10108 . . . . 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 24414 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ∈ ℝ)
8886, 87remulcld 10108 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℝ)
8926adantr 480 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
9088, 89readdcld 10107 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) ∈ ℝ)
91 chpge0 24897 . . . . . . . . . . 11 (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
9276, 91syl 17 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (ψ‘𝑥))
9312adantr 480 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 1 ≤ 𝑥)
9476, 93logge0d 24421 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (log‘𝑥))
9586, 87, 92, 94mulge0d 10642 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ ((ψ‘𝑥) · (log‘𝑥)))
96 vmage0 24892 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
9718, 96syl 17 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
98 chpge0 24897 . . . . . . . . . . . . 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 10642 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
10116, 25, 100fsumge0 14571 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ (1[,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
102101adantr 480 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
10388, 89, 95, 102addge0d 10641 . . . . . . . 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 11950 . . . . . . 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 14205 . . . . . 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 24928 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥𝑦) → (ψ‘𝑥) ≤ (ψ‘𝑦))
10876, 73, 78, 107syl3anc 1366 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (ψ‘𝑥) ≤ (ψ‘𝑦))
10975, 79logled 24418 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (𝑥𝑦 ↔ (log‘𝑥) ≤ (log‘𝑦)))
11078, 109mpbid 222 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (log‘𝑥) ≤ (log‘𝑦))
11186, 74, 87, 80, 92, 94, 108, 110lemul12ad 11004 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → ((ψ‘𝑥) · (log‘𝑥)) ≤ ((ψ‘𝑦) · (log‘𝑦)))
112 fzfid 12812 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (1...(⌊‘𝑦)) ∈ Fin)
11348adantl 481 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥 ∈ ℝ)
116115, 113nndivred 11107 . . . . . . . . . . . . 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 10108 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℝ)
119112, 118fsumrecl 14509 . . . . . . . . . 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 10642 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 0 ≤ ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))))
123 flword2 12654 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥𝑦) → (⌊‘𝑦) ∈ (ℤ‘(⌊‘𝑥)))
12476, 73, 78, 123syl3anc 1366 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (⌊‘𝑦) ∈ (ℤ‘(⌊‘𝑥)))
125 fzss2 12419 . . . . . . . . . . . 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 14572 . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑦 ∈ ℝ)
129128, 113nndivred 11107 . . . . . . . . . . . . 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 10108 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → ((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛))) ∈ ℝ)
132113nnrpd 11908 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑛 ∈ ℝ+)
13378adantr 480 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → 𝑥𝑦)
134115, 128, 132, 133lediv1dd 11968 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (𝑥 / 𝑛) ≤ (𝑦 / 𝑛))
135 chpwordi 24928 . . . . . . . . . . . . 13 (((𝑥 / 𝑛) ∈ ℝ ∧ (𝑦 / 𝑛) ∈ ℝ ∧ (𝑥 / 𝑛) ≤ (𝑦 / 𝑛)) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
136116, 129, 134, 135syl3anc 1366 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) ∧ 𝑛 ∈ (1...(⌊‘𝑦))) → (ψ‘(𝑥 / 𝑛)) ≤ (ψ‘(𝑦 / 𝑛)))
137117, 130, 114, 120, 136lemul2ad 11002 . . . . . . . . . . 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 14575 . . . . . . . . . 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 10232 . . . . . . . . 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 10684 . . . . . . . 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 11969 . . . . . . 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 10106 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (((ψ‘𝑦) · (log‘𝑦)) + Σ𝑛 ∈ (1...(⌊‘𝑦))((Λ‘𝑛) · (ψ‘(𝑦 / 𝑛)))) ∈ ℂ)
143142div1d 10831 . . . . . . 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 4711 . . . . . 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 4707 . . . . 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 11875 . . . . . . . . 9 2 ∈ ℝ+
147 rpge0 11883 . . . . . . . . 9 (2 ∈ ℝ+ → 0 ≤ 2)
148146, 147mp1i 13 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ 2)
14984, 87, 148, 94mulge0d 10642 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → 0 ≤ (2 · (log‘𝑥)))
15067, 149absidd 14205 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) = (2 · (log‘𝑥)))
15187, 80, 84, 148, 110lemul2ad 11002 . . . . . 6 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (2 · (log‘𝑥)) ≤ (2 · (log‘𝑦)))
152150, 151eqbrtrd 4707 . . . . 5 (((⊤ ∧ 𝑥 ∈ (1[,)+∞)) ∧ ((𝑦 ∈ ℝ ∧ 1 ≤ 𝑦) ∧ 𝑥 < 𝑦)) → (abs‘(2 · (log‘𝑥))) ≤ (2 · (log‘𝑦)))
15366, 69, 83, 85, 145, 152le2addd 10684 . . . 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 10232 . . 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 14317 . 2 (⊤ → ∃𝑐 ∈ ℝ+𝑥 ∈ (1[,)+∞)(abs‘(((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ≤ 𝑐)
156155trud 1533 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 196  wa 383  wtru 1524  wcel 2030  wral 2941  wrex 2942  wss 3607   class class class wbr 4685  cmpt 4762  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  cuz 11725  +crp 11870  [,)cico 12215  ...cfz 12364  cfl 12631  abscabs 14018  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-disj 4653  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-cmp 21238  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-em 24764  df-cht 24868  df-vma 24869  df-chp 24870  df-ppi 24871  df-mu 24872
This theorem is referenced by:  chpdifbnd  25289
  Copyright terms: Public domain W3C validator