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Theorem pntrlog2bndlem2 25509
Description: Lemma for pntrlog2bnd 25515. Bound on the difference between the Selberg function and its approximation, inside a sum. (Contributed by Mario Carneiro, 31-May-2016.)
Hypotheses
Ref Expression
pntsval.1 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))))
pntrlog2bnd.r 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
pntrlog2bndlem2.1 (𝜑𝐴 ∈ ℝ+)
pntrlog2bndlem2.2 (𝜑 → ∀𝑦 ∈ ℝ+ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
Assertion
Ref Expression
pntrlog2bndlem2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
Distinct variable groups:   𝑖,𝑎,𝑛,𝑥,𝑦,𝐴   𝜑,𝑛,𝑥   𝑆,𝑛,𝑥,𝑦   𝑅,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑖,𝑎)   𝑅(𝑖,𝑎)   𝑆(𝑖,𝑎)

Proof of Theorem pntrlog2bndlem2
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 1red 10278 . 2 (𝜑 → 1 ∈ ℝ)
2 elioore 12429 . . . . . . . 8 (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ)
32adantl 468 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ)
4 chpcl 25092 . . . . . . 7 (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
53, 4syl 17 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℝ)
65recnd 10291 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (ψ‘𝑥) ∈ ℂ)
7 fzfid 13002 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) ∈ Fin)
83adantr 467 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
9 elfznn 12599 . . . . . . . . . . . 12 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
109adantl 468 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
1110peano2nnd 11260 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℕ)
128, 11nndivred 11292 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ)
13 chpcl 25092 . . . . . . . . 9 ((𝑥 / (𝑛 + 1)) ∈ ℝ → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
1412, 13syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
1514, 12readdcld 10292 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
167, 15fsumrecl 14695 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℝ)
1716recnd 10291 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
183recnd 10291 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℂ)
19 eliooord 12457 . . . . . . . . . 10 (𝑥 ∈ (1(,)+∞) → (1 < 𝑥𝑥 < +∞))
2019adantl 468 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 < 𝑥𝑥 < +∞))
2120simpld 483 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 𝑥)
223, 21rplogcld 24617 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ+)
2322rpcnd 12094 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℂ)
2418, 23mulcld 10283 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℂ)
25 1rp 12056 . . . . . . . . 9 1 ∈ ℝ+
2625a1i 11 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ+)
27 1red 10278 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℝ)
2827, 3, 21ltled 10408 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ≤ 𝑥)
293, 26, 28rpgecld 12131 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ∈ ℝ+)
3029rpne0d 12097 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 ≠ 0)
3122rpne0d 12097 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ≠ 0)
3218, 23, 30, 31mulne0d 10902 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ≠ 0)
336, 17, 24, 32divdird 11062 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))))
3433mpteq2dva 4891 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))))
3529, 22rpmulcld 12108 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 · (log‘𝑥)) ∈ ℝ+)
365, 35rerpdivcld 12123 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) ∈ ℝ)
3716, 35rerpdivcld 12123 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
386, 18, 23, 30, 31divdiv1d 11055 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = ((ψ‘𝑥) / (𝑥 · (log‘𝑥))))
395, 29rerpdivcld 12123 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
4039recnd 10291 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
4140, 23, 31divrecd 11027 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) / 𝑥) / (log‘𝑥)) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
4238, 41eqtr3d 2810 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) / (𝑥 · (log‘𝑥))) = (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥))))
4342mpteq2dva 4891 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) = (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))))
4422rprecred 12103 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 / (log‘𝑥)) ∈ ℝ)
4529ex 398 . . . . . . . 8 (𝜑 → (𝑥 ∈ (1(,)+∞) → 𝑥 ∈ ℝ+))
4645ssrdv 3764 . . . . . . 7 (𝜑 → (1(,)+∞) ⊆ ℝ+)
47 chpo1ub 25411 . . . . . . . 8 (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
4847a1i 11 . . . . . . 7 (𝜑 → (𝑥 ∈ ℝ+ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
4946, 48o1res2 14524 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
50 divlogrlim 24623 . . . . . . 7 (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0
51 rlimo1 14577 . . . . . . 7 ((𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ⇝𝑟 0 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
5250, 51mp1i 13 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 / (log‘𝑥))) ∈ 𝑂(1))
5339, 44, 49, 52o1mul2 14585 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / 𝑥) · (1 / (log‘𝑥)))) ∈ 𝑂(1))
5443, 53eqeltrd 2853 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((ψ‘𝑥) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
55 pntrlog2bndlem2.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ+)
5655rpred 12092 . . . . . . . 8 (𝜑𝐴 ∈ ℝ)
5756, 1readdcld 10292 . . . . . . 7 (𝜑 → (𝐴 + 1) ∈ ℝ)
5857adantr 467 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℝ)
5927, 44readdcld 10292 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) ∈ ℝ)
60 ioossre 12459 . . . . . . 7 (1(,)+∞) ⊆ ℝ
6157recnd 10291 . . . . . . 7 (𝜑 → (𝐴 + 1) ∈ ℂ)
62 o1const 14580 . . . . . . 7 (((1(,)+∞) ⊆ ℝ ∧ (𝐴 + 1) ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
6360, 61, 62sylancr 576 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (𝐴 + 1)) ∈ 𝑂(1))
64 1cnd 10279 . . . . . . . 8 (𝜑 → 1 ∈ ℂ)
65 o1const 14580 . . . . . . . 8 (((1(,)+∞) ⊆ ℝ ∧ 1 ∈ ℂ) → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
6660, 64, 65sylancr 576 . . . . . . 7 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ 1) ∈ 𝑂(1))
6727, 44, 66, 52o1add2 14584 . . . . . 6 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (1 + (1 / (log‘𝑥)))) ∈ 𝑂(1))
6858, 59, 63, 67o1mul2 14585 . . . . 5 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ ((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ 𝑂(1))
6958, 59remulcld 10293 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℝ)
7037recnd 10291 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
71 chpge0 25094 . . . . . . . . . . . 12 ((𝑥 / (𝑛 + 1)) ∈ ℝ → 0 ≤ (ψ‘(𝑥 / (𝑛 + 1))))
7212, 71syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (ψ‘(𝑥 / (𝑛 + 1))))
7310nnrpd 12090 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
7425a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ+)
7573, 74rpaddcld 12107 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℝ+)
7629adantr 467 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
7776rpge0d 12096 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑥)
788, 75, 77divge0d 12132 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑥 / (𝑛 + 1)))
7914, 12, 72, 78addge0d 10826 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
807, 15, 79fsumge0 14756 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
8116, 35, 80divge0d 12132 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
8237, 81absidd 14391 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
8369recnd 10291 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ∈ ℂ)
8483abscld 14405 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))) ∈ ℝ)
8516, 29rerpdivcld 12123 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ∈ ℝ)
8629relogcld 24611 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (log‘𝑥) ∈ ℝ)
8786, 27readdcld 10292 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℝ)
8858, 87remulcld 10293 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · ((log‘𝑥) + 1)) ∈ ℝ)
8958, 3remulcld 10293 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
9010nnrecred 11289 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℝ)
917, 90fsumrecl 14695 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ∈ ℝ)
9289, 91remulcld 10293 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ∈ ℝ)
9389, 87remulcld 10293 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) ∈ ℝ)
9456ad2antrr 706 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℝ)
95 1red 10278 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
9694, 95readdcld 10292 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℝ)
9796, 8remulcld 10293 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℝ)
9897, 90remulcld 10293 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) · (1 / 𝑛)) ∈ ℝ)
9997, 11nndivred 11292 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ∈ ℝ)
10097, 10nndivred 11292 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) ∈ ℝ)
10194, 12remulcld 10293 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 · (𝑥 / (𝑛 + 1))) ∈ ℝ)
102 fveq2 6348 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / (𝑛 + 1)) → (ψ‘𝑦) = (ψ‘(𝑥 / (𝑛 + 1))))
103 oveq2 6820 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (𝑥 / (𝑛 + 1)) → (𝐴 · 𝑦) = (𝐴 · (𝑥 / (𝑛 + 1))))
104102, 103breq12d 4810 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (𝑥 / (𝑛 + 1)) → ((ψ‘𝑦) ≤ (𝐴 · 𝑦) ↔ (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1)))))
105 pntrlog2bndlem2.2 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑦 ∈ ℝ+ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
106105ad2antrr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ∀𝑦 ∈ ℝ+ (ψ‘𝑦) ≤ (𝐴 · 𝑦))
10776, 75rpdivcld 12109 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℝ+)
108104, 106, 107rspcdva 3471 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (𝐴 · (𝑥 / (𝑛 + 1))))
10914, 101, 12, 108leadd1dd 10864 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
11061ad2antrr 706 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝐴 + 1) ∈ ℂ)
11118adantr 467 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
11210nncnd 11259 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
113 1cnd 10279 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℂ)
114112, 113addcld 10282 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ∈ ℂ)
11511nnne0d 11288 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 + 1) ≠ 0)
116110, 111, 114, 115divassd 11059 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 + 1) · (𝑥 / (𝑛 + 1))))
11794recnd 10291 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝐴 ∈ ℂ)
118111, 114, 115divcld 11024 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ∈ ℂ)
119117, 113, 118adddird 10288 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · (𝑥 / (𝑛 + 1))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))))
120118mulid2d 10281 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (𝑥 / (𝑛 + 1))) = (𝑥 / (𝑛 + 1)))
121120oveq2d 6828 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 · (𝑥 / (𝑛 + 1))) + (1 · (𝑥 / (𝑛 + 1)))) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
122116, 119, 1213eqtrd 2812 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) = ((𝐴 · (𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
123109, 122breqtrrd 4825 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / (𝑛 + 1)))
12456adantr 467 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ)
12555adantr 467 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝐴 ∈ ℝ+)
126125rpge0d 12096 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝐴)
12726rpge0d 12096 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 1)
128124, 27, 126, 127addge0d 10826 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (𝐴 + 1))
12929rpge0d 12096 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ 𝑥)
13058, 3, 128, 129mulge0d 10827 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ ((𝐴 + 1) · 𝑥))
131130adantr 467 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝐴 + 1) · 𝑥))
13210nnred 11258 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ)
133132lep1d 11178 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ (𝑛 + 1))
13473, 75, 97, 131, 133lediv2ad 12114 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / (𝑛 + 1)) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
13515, 99, 100, 123, 134letrd 10417 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) / 𝑛))
13697recnd 10291 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
13710nnne0d 11288 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≠ 0)
138136, 112, 137divrecd 11027 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝐴 + 1) · 𝑥) / 𝑛) = (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
139135, 138breqtrd 4823 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
1407, 15, 98, 139fsumle 14760 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
14189recnd 10291 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · 𝑥) ∈ ℂ)
142112, 137reccld 11017 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 / 𝑛) ∈ ℂ)
1437, 141, 142fsummulc2 14745 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((𝐴 + 1) · 𝑥) · (1 / 𝑛)))
144140, 143breqtrrd 4825 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)))
145 harmonicubnd 24978 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ ∧ 1 ≤ 𝑥) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
1463, 28, 145syl2anc 574 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛) ≤ ((log‘𝑥) + 1))
14791, 87, 89, 130, 146lemul2ad 11187 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · Σ𝑛 ∈ (1...(⌊‘𝑥))(1 / 𝑛)) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
14816, 92, 93, 144, 147letrd 10417 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)))
14961adantr 467 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝐴 + 1) ∈ ℂ)
15087recnd 10291 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
151149, 18, 150mul32d 10469 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · 𝑥) · ((log‘𝑥) + 1)) = (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
152148, 151breqtrd 4823 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥))
15316, 88, 29ledivmul2d 12146 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ≤ ((𝐴 + 1) · ((log‘𝑥) + 1)) ↔ Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) · 𝑥)))
154152, 153mpbird 248 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) ≤ ((𝐴 + 1) · ((log‘𝑥) + 1)))
15585, 88, 22, 154lediv1dd 12150 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) / (log‘𝑥)) ≤ (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)))
15617, 18, 23, 30, 31divdiv1d 11055 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / 𝑥) / (log‘𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))
157 1cnd 10279 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 ∈ ℂ)
15823, 157addcld 10282 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) + 1) ∈ ℂ)
159149, 158, 23, 31divassd 11059 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
16023, 157, 23, 31divdird 11062 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) + 1) / (log‘𝑥)) = (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))))
16123, 31dividd 11022 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → ((log‘𝑥) / (log‘𝑥)) = 1)
162161oveq1d 6827 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (((log‘𝑥) / (log‘𝑥)) + (1 / (log‘𝑥))) = (1 + (1 / (log‘𝑥))))
163160, 162eqtr2d 2809 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 + (1 / (log‘𝑥))) = (((log‘𝑥) + 1) / (log‘𝑥)))
164163oveq2d 6828 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) = ((𝐴 + 1) · (((log‘𝑥) + 1) / (log‘𝑥))))
165159, 164eqtr4d 2811 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → (((𝐴 + 1) · ((log‘𝑥) + 1)) / (log‘𝑥)) = ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
166155, 156, 1653brtr3d 4828 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ≤ ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))))
16769leabsd 14383 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝐴 + 1) · (1 + (1 / (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
16837, 69, 84, 166, 167letrd 10417 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
16982, 168eqbrtrd 4819 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
170169adantrr 697 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘((𝐴 + 1) · (1 + (1 / (log‘𝑥))))))
1711, 68, 69, 70, 170o1le 14613 . . . 4 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
17236, 37, 54, 171o1add2 14584 . . 3 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) / (𝑥 · (log‘𝑥))) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) / (𝑥 · (log‘𝑥))))) ∈ 𝑂(1))
17334, 172eqeltrd 2853 . 2 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
1745, 16readdcld 10292 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
175174, 35rerpdivcld 12123 . 2 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
176 pntrlog2bnd.r . . . . . . . . . . . 12 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎))
177176pntrf 25494 . . . . . . . . . . 11 𝑅:ℝ+⟶ℝ
178177ffvelrni 6518 . . . . . . . . . 10 ((𝑥 / (𝑛 + 1)) ∈ ℝ+ → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
179107, 178syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℝ)
180179recnd 10291 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
18176, 73rpdivcld 12109 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
182177ffvelrni 6518 . . . . . . . . . 10 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
183181, 182syl 17 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℝ)
184183recnd 10291 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) ∈ ℂ)
185180, 184subcld 10615 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) ∈ ℂ)
186185abscld 14405 . . . . . 6 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ∈ ℝ)
187132, 186remulcld 10293 . . . . 5 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
1887, 187fsumrecl 14695 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ∈ ℝ)
189188, 35rerpdivcld 12123 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℝ)
190189recnd 10291 . 2 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
19173rpge0d 12096 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ 𝑛)
192185absge0d 14413 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))))
193132, 186, 191, 192mulge0d 10827 . . . . . . 7 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
1947, 187, 193fsumge0 14756 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))))
195188, 35, 194divge0d 12132 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → 0 ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
196189, 195absidd 14391 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))))
1976, 17addcld 10282 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
198197, 24, 32divcld 11024 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ∈ ℂ)
199198abscld 14405 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))) ∈ ℝ)
2008, 10nndivred 11292 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
201 chpcl 25092 . . . . . . . . . . . 12 ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
202200, 201syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
203202, 200readdcld 10292 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℝ)
204203, 15resubcld 10681 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℝ)
205132, 204remulcld 10293 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℝ)
206176pntrval 25493 . . . . . . . . . . . . . . 15 ((𝑥 / (𝑛 + 1)) ∈ ℝ+ → (𝑅‘(𝑥 / (𝑛 + 1))) = ((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))))
207107, 206syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / (𝑛 + 1))) = ((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))))
208176pntrval 25493 . . . . . . . . . . . . . . 15 ((𝑥 / 𝑛) ∈ ℝ+ → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛)))
209181, 208syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑛)) = ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛)))
210207, 209oveq12d 6830 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))))
21114recnd 10291 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ∈ ℂ)
212202recnd 10291 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
213111, 112, 137divcld 11024 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
214211, 118, 212, 213sub4d 10664 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / (𝑛 + 1))) − (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) − (𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))))
215210, 214eqtrd 2808 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))) = (((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))))
216215fveq2d 6352 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) = (abs‘(((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
217211, 212subcld 10615 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
218118, 213subcld 10615 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)) ∈ ℂ)
219217, 218abs2dif2d 14427 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛))) − ((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) ≤ ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
220216, 219eqbrtrd 4819 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ≤ ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))))
22173, 75, 8, 77, 133lediv2ad 12114 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛))
222 chpwordi 25125 . . . . . . . . . . . . . 14 (((𝑥 / (𝑛 + 1)) ∈ ℝ ∧ (𝑥 / 𝑛) ∈ ℝ ∧ (𝑥 / (𝑛 + 1)) ≤ (𝑥 / 𝑛)) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
22312, 200, 221, 222syl3anc 1480 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / (𝑛 + 1))) ≤ (ψ‘(𝑥 / 𝑛)))
22414, 202, 223abssuble0d 14401 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) = ((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))))
22512, 200, 221abssuble0d 14401 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛))) = ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1))))
226224, 225oveq12d 6830 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) = (((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))) + ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1)))))
227212, 213, 211, 118addsub4d 10662 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (((ψ‘(𝑥 / 𝑛)) − (ψ‘(𝑥 / (𝑛 + 1)))) + ((𝑥 / 𝑛) − (𝑥 / (𝑛 + 1)))))
228226, 227eqtr4d 2811 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘((ψ‘(𝑥 / (𝑛 + 1))) − (ψ‘(𝑥 / 𝑛)))) + (abs‘((𝑥 / (𝑛 + 1)) − (𝑥 / 𝑛)))) = (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
229220, 228breqtrd 4823 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛)))) ≤ (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
230186, 204, 132, 191, 229lemul2ad 11187 . . . . . . . 8 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
2317, 187, 205, 230fsumle 14760 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
232204recnd 10291 . . . . . . . . . 10 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) ∈ ℂ)
233112, 232mulcld 10283 . . . . . . . . 9 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℂ)
2347, 233fsumcl 14694 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) ∈ ℂ)
2356, 17negdi2d 10629 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (-(ψ‘𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
23629rprege0d 12099 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
237 flge0nn0 12851 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
238 nn0p1nn 11556 . . . . . . . . . . . . . . . . . . . . 21 ((⌊‘𝑥) ∈ ℕ0 → ((⌊‘𝑥) + 1) ∈ ℕ)
239236, 237, 2383syl 18 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℕ)
2403, 239nndivred 11292 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ)
241 2re 11313 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℝ
242241a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (1(,)+∞)) → 2 ∈ ℝ)
243 flltp1 12831 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 ∈ ℝ → 𝑥 < ((⌊‘𝑥) + 1))
2443, 243syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 < ((⌊‘𝑥) + 1))
245239nncnd 11259 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℂ)
246245mulid1d 10280 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · 1) = ((⌊‘𝑥) + 1))
247244, 246breqtrrd 4825 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → 𝑥 < (((⌊‘𝑥) + 1) · 1))
248239nnrpd 12090 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ ℝ+)
2493, 27, 248ltdivmuld 12143 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑥 ∈ (1(,)+∞)) → ((𝑥 / ((⌊‘𝑥) + 1)) < 1 ↔ 𝑥 < (((⌊‘𝑥) + 1) · 1)))
250247, 249mpbird 248 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 1)
251 1lt2 11418 . . . . . . . . . . . . . . . . . . . 20 1 < 2
252251a1i 11 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥 ∈ (1(,)+∞)) → 1 < 2)
253240, 27, 242, 250, 252lttrd 10421 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) < 2)
254 chpeq0 25175 . . . . . . . . . . . . . . . . . . 19 ((𝑥 / ((⌊‘𝑥) + 1)) ∈ ℝ → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0 ↔ (𝑥 / ((⌊‘𝑥) + 1)) < 2))
255240, 254syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0 ↔ (𝑥 / ((⌊‘𝑥) + 1)) < 2))
256253, 255mpbird 248 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / ((⌊‘𝑥) + 1))) = 0)
257256oveq1d 6827 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (0 + (𝑥 / ((⌊‘𝑥) + 1))))
258240recnd 10291 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / ((⌊‘𝑥) + 1)) ∈ ℂ)
259258addid2d 10460 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → (0 + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
260257, 259eqtrd 2808 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))) = (𝑥 / ((⌊‘𝑥) + 1)))
261260oveq2d 6828 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))))
262239nnne0d 11288 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ≠ 0)
26318, 245, 262divcan2d 11026 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · (𝑥 / ((⌊‘𝑥) + 1))) = 𝑥)
264261, 263eqtrd 2808 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) = 𝑥)
26518div1d 11016 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥 ∈ (1(,)+∞)) → (𝑥 / 1) = 𝑥)
266265fveq2d 6352 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → (ψ‘(𝑥 / 1)) = (ψ‘𝑥))
267266, 265oveq12d 6830 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘(𝑥 / 1)) + (𝑥 / 1)) = ((ψ‘𝑥) + 𝑥))
268267oveq2d 6828 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = (1 · ((ψ‘𝑥) + 𝑥)))
2695, 3readdcld 10292 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℝ)
270269recnd 10291 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1(,)+∞)) → ((ψ‘𝑥) + 𝑥) ∈ ℂ)
271270mulid2d 10281 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘𝑥) + 𝑥)) = ((ψ‘𝑥) + 𝑥))
272268, 271eqtrd 2808 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1))) = ((ψ‘𝑥) + 𝑥))
273264, 272oveq12d 6830 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
274270, 18negsubdi2d 10631 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = (𝑥 − ((ψ‘𝑥) + 𝑥)))
2756, 18pncand 10616 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + 𝑥) − 𝑥) = (ψ‘𝑥))
276275negeqd 10498 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → -(((ψ‘𝑥) + 𝑥) − 𝑥) = -(ψ‘𝑥))
277273, 274, 2763eqtr2d 2814 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → ((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) = -(ψ‘𝑥))
2783flcld 12829 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1(,)+∞)) → (⌊‘𝑥) ∈ ℤ)
279 fzval3 12767 . . . . . . . . . . . . . 14 ((⌊‘𝑥) ∈ ℤ → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
280278, 279syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1(,)+∞)) → (1...(⌊‘𝑥)) = (1..^((⌊‘𝑥) + 1)))
281280eqcomd 2780 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → (1..^((⌊‘𝑥) + 1)) = (1...(⌊‘𝑥)))
282112, 113pncan2d 10617 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑛 + 1) − 𝑛) = 1)
283282oveq1d 6827 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (1 · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
28415recnd 10291 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
285284mulid2d 10281 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
286283, 285eqtrd 2808 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
287281, 286sumeq12rdv 14668 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
288277, 287oveq12d 6830 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = (-(ψ‘𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
289 oveq2 6820 . . . . . . . . . . . . . . 15 (𝑚 = 𝑛 → (𝑥 / 𝑚) = (𝑥 / 𝑛))
290289fveq2d 6352 . . . . . . . . . . . . . 14 (𝑚 = 𝑛 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 𝑛)))
291290, 289oveq12d 6830 . . . . . . . . . . . . 13 (𝑚 = 𝑛 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))
292291ancli 539 . . . . . . . . . . . 12 (𝑚 = 𝑛 → (𝑚 = 𝑛 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
293 oveq2 6820 . . . . . . . . . . . . . . 15 (𝑚 = (𝑛 + 1) → (𝑥 / 𝑚) = (𝑥 / (𝑛 + 1)))
294293fveq2d 6352 . . . . . . . . . . . . . 14 (𝑚 = (𝑛 + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / (𝑛 + 1))))
295294, 293oveq12d 6830 . . . . . . . . . . . . 13 (𝑚 = (𝑛 + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))
296295ancli 539 . . . . . . . . . . . 12 (𝑚 = (𝑛 + 1) → (𝑚 = (𝑛 + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
297 oveq2 6820 . . . . . . . . . . . . . . 15 (𝑚 = 1 → (𝑥 / 𝑚) = (𝑥 / 1))
298297fveq2d 6352 . . . . . . . . . . . . . 14 (𝑚 = 1 → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / 1)))
299298, 297oveq12d 6830 . . . . . . . . . . . . 13 (𝑚 = 1 → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))
300299ancli 539 . . . . . . . . . . . 12 (𝑚 = 1 → (𝑚 = 1 ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / 1)) + (𝑥 / 1))))
301 oveq2 6820 . . . . . . . . . . . . . . 15 (𝑚 = ((⌊‘𝑥) + 1) → (𝑥 / 𝑚) = (𝑥 / ((⌊‘𝑥) + 1)))
302301fveq2d 6352 . . . . . . . . . . . . . 14 (𝑚 = ((⌊‘𝑥) + 1) → (ψ‘(𝑥 / 𝑚)) = (ψ‘(𝑥 / ((⌊‘𝑥) + 1))))
303302, 301oveq12d 6830 . . . . . . . . . . . . 13 (𝑚 = ((⌊‘𝑥) + 1) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1))))
304303ancli 539 . . . . . . . . . . . 12 (𝑚 = ((⌊‘𝑥) + 1) → (𝑚 = ((⌊‘𝑥) + 1) ∧ ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) = ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))))
305 nnuz 11947 . . . . . . . . . . . . 13 ℕ = (ℤ‘1)
306239, 305syl6eleq 2863 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1(,)+∞)) → ((⌊‘𝑥) + 1) ∈ (ℤ‘1))
307 elfznn 12599 . . . . . . . . . . . . . 14 (𝑚 ∈ (1...((⌊‘𝑥) + 1)) → 𝑚 ∈ ℕ)
308307adantl 468 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℕ)
309308nncnd 11259 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑚 ∈ ℂ)
3103adantr 467 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → 𝑥 ∈ ℝ)
311310, 308nndivred 11292 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (𝑥 / 𝑚) ∈ ℝ)
312 chpcl 25092 . . . . . . . . . . . . . . 15 ((𝑥 / 𝑚) ∈ ℝ → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
313311, 312syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → (ψ‘(𝑥 / 𝑚)) ∈ ℝ)
314313, 311readdcld 10292 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℝ)
315314recnd 10291 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑚 ∈ (1...((⌊‘𝑥) + 1))) → ((ψ‘(𝑥 / 𝑚)) + (𝑥 / 𝑚)) ∈ ℂ)
316292, 296, 300, 304, 306, 309, 315fsumparts 14767 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
317212, 213addcld 10282 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) ∈ ℂ)
318211, 118addcld 10282 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) ∈ ℂ)
319317, 318negsubdi2d 10631 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛))))
320319oveq2d 6828 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = (𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))))
321112, 232mulneg2d 10707 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · -(((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
322320, 321eqtr3d 2810 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (1(,)+∞)) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = -(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
323281, 322sumeq12rdv 14668 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(𝑛 · (((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))) − ((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
324316, 323eqtr3d 2810 . . . . . . . . . 10 ((𝜑𝑥 ∈ (1(,)+∞)) → (((((⌊‘𝑥) + 1) · ((ψ‘(𝑥 / ((⌊‘𝑥) + 1))) + (𝑥 / ((⌊‘𝑥) + 1)))) − (1 · ((ψ‘(𝑥 / 1)) + (𝑥 / 1)))) − Σ𝑛 ∈ (1..^((⌊‘𝑥) + 1))(((𝑛 + 1) − 𝑛) · ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
325235, 288, 3243eqtr2d 2814 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
3267, 233fsumneg 14748 . . . . . . . . 9 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))-(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))))
327325, 326eqtr2d 2809 . . . . . . . 8 ((𝜑𝑥 ∈ (1(,)+∞)) → -Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = -((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
328234, 197, 327neg11d 10627 . . . . . . 7 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (((ψ‘(𝑥 / 𝑛)) + (𝑥 / 𝑛)) − ((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1))))) = ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
329231, 328breqtrd 4823 . . . . . 6 ((𝜑𝑥 ∈ (1(,)+∞)) → Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) ≤ ((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))))
330188, 174, 35, 329lediv1dd 12150 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ≤ (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))))
331175leabsd 14383 . . . . 5 ((𝜑𝑥 ∈ (1(,)+∞)) → (((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
332189, 175, 199, 330, 331letrd 10417 . . . 4 ((𝜑𝑥 ∈ (1(,)+∞)) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
333196, 332eqbrtrd 4819 . . 3 ((𝜑𝑥 ∈ (1(,)+∞)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
334333adantrr 697 . 2 ((𝜑 ∧ (𝑥 ∈ (1(,)+∞) ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ≤ (abs‘(((ψ‘𝑥) + Σ𝑛 ∈ (1...(⌊‘𝑥))((ψ‘(𝑥 / (𝑛 + 1))) + (𝑥 / (𝑛 + 1)))) / (𝑥 · (log‘𝑥)))))
3351, 173, 175, 190, 334o1le 14613 1 (𝜑 → (𝑥 ∈ (1(,)+∞) ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(𝑛 · (abs‘((𝑅‘(𝑥 / (𝑛 + 1))) − (𝑅‘(𝑥 / 𝑛))))) / (𝑥 · (log‘𝑥)))) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383   = wceq 1634  wcel 2148  wral 3064  wss 3729   class class class wbr 4797  cmpt 4876  cfv 6042  (class class class)co 6812  cc 10157  cr 10158  0cc0 10159  1c1 10160   + caddc 10162   · cmul 10164  +∞cpnf 10294   < clt 10297  cle 10298  cmin 10489  -cneg 10490   / cdiv 10907  cn 11243  2c2 11293  0cn0 11516  cz 11601  cuz 11910  +crp 12052  (,)cioo 12399  ...cfz 12555  ..^cfzo 12695  cfl 12821  abscabs 14204  𝑟 crli 14446  𝑂(1)co1 14447  Σcsu 14646  logclog 24543  Λcvma 25060  ψcchp 25061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-inf2 8723  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236  ax-pre-sup 10237  ax-addf 10238  ax-mulf 10239
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-fal 1640  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rmo 3072  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-iin 4668  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-se 5223  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-isom 6051  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-of 7065  df-om 7234  df-1st 7336  df-2nd 7337  df-supp 7468  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-oadd 7738  df-er 7917  df-map 8032  df-pm 8033  df-ixp 8084  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-fsupp 8453  df-fi 8494  df-sup 8525  df-inf 8526  df-oi 8592  df-card 8986  df-cda 9213  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-div 10908  df-nn 11244  df-2 11302  df-3 11303  df-4 11304  df-5 11305  df-6 11306  df-7 11307  df-8 11308  df-9 11309  df-n0 11517  df-xnn0 11588  df-z 11602  df-dec 11718  df-uz 11911  df-q 12014  df-rp 12053  df-xneg 12168  df-xadd 12169  df-xmul 12170  df-ioo 12403  df-ioc 12404  df-ico 12405  df-icc 12406  df-fz 12556  df-fzo 12696  df-fl 12823  df-mod 12899  df-seq 13031  df-exp 13090  df-fac 13287  df-bc 13316  df-hash 13344  df-shft 14037  df-cj 14069  df-re 14070  df-im 14071  df-sqrt 14205  df-abs 14206  df-limsup 14432  df-clim 14449  df-rlim 14450  df-o1 14451  df-lo1 14452  df-sum 14647  df-ef 15026  df-e 15027  df-sin 15028  df-cos 15029  df-pi 15031  df-dvds 15212  df-gcd 15446  df-prm 15614  df-pc 15769  df-struct 16086  df-ndx 16087  df-slot 16088  df-base 16090  df-sets 16091  df-ress 16092  df-plusg 16182  df-mulr 16183  df-starv 16184  df-sca 16185  df-vsca 16186  df-ip 16187  df-tset 16188  df-ple 16189  df-ds 16192  df-unif 16193  df-hom 16194  df-cco 16195  df-rest 16311  df-topn 16312  df-0g 16330  df-gsum 16331  df-topgen 16332  df-pt 16333  df-prds 16336  df-xrs 16390  df-qtop 16395  df-imas 16396  df-xps 16398  df-mre 16474  df-mrc 16475  df-acs 16477  df-mgm 17470  df-sgrp 17512  df-mnd 17523  df-submnd 17564  df-mulg 17769  df-cntz 17977  df-cmn 18422  df-psmet 19973  df-xmet 19974  df-met 19975  df-bl 19976  df-mopn 19977  df-fbas 19978  df-fg 19979  df-cnfld 19982  df-top 20939  df-topon 20956  df-topsp 20978  df-bases 20991  df-cld 21064  df-ntr 21065  df-cls 21066  df-nei 21143  df-lp 21181  df-perf 21182  df-cn 21272  df-cnp 21273  df-haus 21360  df-tx 21606  df-hmeo 21799  df-fil 21890  df-fm 21982  df-flim 21983  df-flf 21984  df-xms 22365  df-ms 22366  df-tms 22367  df-cncf 22921  df-limc 23871  df-dv 23872  df-log 24545  df-cxp 24546  df-em 24961  df-cht 25065  df-vma 25066  df-chp 25067  df-ppi 25068
This theorem is referenced by:  pntrlog2bndlem3  25510
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