MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  selberglem2 Structured version   Visualization version   GIF version

Theorem selberglem2 26294
Description: Lemma for selberg 26296. (Contributed by Mario Carneiro, 23-May-2016.)
Hypothesis
Ref Expression
selberglem1.t 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)
Assertion
Ref Expression
selberglem2 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Distinct variable group:   𝑚,𝑛,𝑥
Allowed substitution hints:   𝑇(𝑥,𝑚,𝑛)

Proof of Theorem selberglem2
StepHypRef Expression
1 reex 10718 . . . . . . 7 ℝ ∈ V
2 rpssre 12491 . . . . . . 7 + ⊆ ℝ
31, 2ssexi 5200 . . . . . 6 + ∈ V
43a1i 11 . . . . 5 (⊤ → ℝ+ ∈ V)
5 fzfid 13444 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
6 elfznn 13039 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
76adantl 485 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8 mucl 25890 . . . . . . . . . 10 (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
97, 8syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
109zred 12180 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
1110recnd 10759 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
12 fzfid 13444 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
13 elfznn 13039 . . . . . . . . . . . . . . 15 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
1413adantl 485 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
1514nnrpd 12524 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ+)
1615relogcld 25378 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
1716resqcld 13715 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℝ)
1812, 17fsumrecl 15196 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℝ)
19 simplr 769 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
2018, 19rerpdivcld 12557 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) ∈ ℝ)
2120recnd 10759 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) ∈ ℂ)
22 selberglem1.t . . . . . . . . . 10 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)
23 simpr 488 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
246nnrpd 12524 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
25 rpdivcl 12509 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
2623, 24, 25syl2an 599 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
2726relogcld 25378 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
2827resqcld 13715 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
29 2re 11802 . . . . . . . . . . . . 13 2 ∈ ℝ
30 remulcl 10712 . . . . . . . . . . . . . 14 ((2 ∈ ℝ ∧ (log‘(𝑥 / 𝑛)) ∈ ℝ) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
3129, 27, 30sylancr 590 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
32 resubcl 11040 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
3329, 31, 32sylancr 590 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
3428, 33readdcld 10760 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
3534, 7nndivred 11782 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛) ∈ ℝ)
3622, 35eqeltrid 2838 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℝ)
3736recnd 10759 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
3821, 37subcld 11087 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) ∈ ℂ)
3911, 38mulcld 10751 . . . . . 6 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
405, 39fsumcl 15195 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
4111, 37mulcld 10751 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · 𝑇) ∈ ℂ)
425, 41fsumcl 15195 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) ∈ ℂ)
43 2cn 11803 . . . . . . 7 2 ∈ ℂ
44 relogcl 25331 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
4544adantl 485 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
4645recnd 10759 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
47 mulcl 10711 . . . . . . 7 ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
4843, 46, 47sylancr 590 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
4942, 48subcld 11087 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) ∈ ℂ)
50 eqidd 2740 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
51 eqidd 2740 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))))
524, 40, 49, 50, 51offval2 7456 . . . 4 (⊤ → ((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∘f + (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))))
5340, 42, 48addsubassd 11107 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇)) − (2 · (log‘𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))))
54 rpcnne0 12502 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
5554adantl 485 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
5655simpld 498 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
5710adantr 484 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (μ‘𝑛) ∈ ℝ)
5857, 17remulcld 10761 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
5912, 58fsumrecl 15196 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
6059recnd 10759 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℂ)
6155simprd 499 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
625, 56, 60, 61fsumdivc 15246 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥))
6317recnd 10759 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℂ)
6412, 63fsumcl 15195 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℂ)
6555adantr 484 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66 divass 11406 . . . . . . . . . . 11 (((μ‘𝑛) ∈ ℂ ∧ Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((μ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2)) / 𝑥) = ((μ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)))
6711, 64, 65, 66syl3anc 1372 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2)) / 𝑥) = ((μ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)))
6812, 11, 63fsummulc2 15244 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)))
6968oveq1d 7197 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2)) / 𝑥) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥))
7021, 37npcand 11091 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) + 𝑇) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥))
7170oveq2d 7198 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) + 𝑇)) = ((μ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)))
7211, 38, 37adddid 10755 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) + 𝑇)) = (((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + ((μ‘𝑛) · 𝑇)))
7371, 72eqtr3d 2776 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) = (((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + ((μ‘𝑛) · 𝑇)))
7467, 69, 733eqtr3d 2782 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) = (((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + ((μ‘𝑛) · 𝑇)))
7574sumeq2dv 15165 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + ((μ‘𝑛) · 𝑇)))
765, 39, 41fsumadd 15201 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + ((μ‘𝑛) · 𝑇)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇)))
7762, 75, 763eqtrrd 2779 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥))
7877oveq1d 7197 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇)) − (2 · (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
7953, 78eqtr3d 2776 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
8079mpteq2dva 5135 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) + (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))))
8152, 80eqtrd 2774 . . 3 (⊤ → ((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∘f + (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))) = (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))))
82 1red 10732 . . . . 5 (⊤ → 1 ∈ ℝ)
835, 28fsumrecl 15196 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
8483, 23rerpdivcld 12557 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ)
8584recnd 10759 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℂ)
86 2cnd 11806 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
87 2nn0 12005 . . . . . . . 8 2 ∈ ℕ0
88 logexprlim 25973 . . . . . . . 8 (2 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥)) ⇝𝑟 (!‘2))
8987, 88mp1i 13 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥)) ⇝𝑟 (!‘2))
90 2cnd 11806 . . . . . . . 8 (⊤ → 2 ∈ ℂ)
91 rlimconst 15003 . . . . . . . 8 ((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 2) ⇝𝑟 2)
922, 90, 91sylancr 590 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ 2) ⇝𝑟 2)
9385, 86, 89, 92rlimadd 15102 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ⇝𝑟 ((!‘2) + 2))
94 rlimo1 15076 . . . . . 6 ((𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ⇝𝑟 ((!‘2) + 2) → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ 𝑂(1))
9593, 94syl 17 . . . . 5 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ 𝑂(1))
96 readdcl 10710 . . . . . 6 (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ ∧ 2 ∈ ℝ) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
9784, 29, 96sylancl 589 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
9840abscld 14898 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
9997recnd 10759 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℂ)
10099abscld 14898 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ ℝ)
10139abscld 14898 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
1025, 101fsumrecl 15196 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
1035, 39fsumabs 15261 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
104 readdcl 10710 . . . . . . . . . . . 12 ((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 2 ∈ ℝ) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
10528, 29, 104sylancl 589 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
106105, 19rerpdivcld 12557 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
1075, 106fsumrecl 15196 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
10838abscld 14898 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℝ)
10911, 38absmuld 14916 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = ((abs‘(μ‘𝑛)) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
11011abscld 14898 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
111 1red 10732 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
11238absge0d 14906 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
113 mule1 25897 . . . . . . . . . . . . . . 15 (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
1147, 113syl 17 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
115110, 111, 108, 112, 114lemul1ad 11669 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (1 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
116108recnd 10759 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
117116mulid2d 10749 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
118115, 117breqtrd 5066 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘(μ‘𝑛)) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
119109, 118eqbrtrd 5062 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
12065simpld 498 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
121120, 38absmuld 14916 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = ((abs‘𝑥) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
122120, 21, 37subdid 11186 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) = ((𝑥 · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) − (𝑥 · 𝑇)))
12365simprd 499 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
12464, 120, 123divcan2d 11508 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2))
12534recnd 10759 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
1267nnrpd 12524 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
127 rpcnne0 12502 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
128126, 127syl 17 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
129 divass 11406 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) / 𝑛) = (𝑥 · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)))
13022oveq2i 7193 . . . . . . . . . . . . . . . . . . . 20 (𝑥 · 𝑇) = (𝑥 · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
131129, 130eqtr4di 2792 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) / 𝑛) = (𝑥 · 𝑇))
132 div23 11407 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((𝑥 · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) / 𝑛) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
133131, 132eqtr3d 2776 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (𝑥 · 𝑇) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
134120, 125, 128, 133syl3anc 1372 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · 𝑇) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
135124, 134oveq12d 7200 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) − (𝑥 · 𝑇)) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))))))
136122, 135eqtrd 2774 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))))))
137136fveq2d 6690 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(𝑥 · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))))
138 rprege0 12499 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
139 absid 14758 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
14019, 138, 1393syl 18 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑥) = 𝑥)
141140oveq1d 7197 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((abs‘𝑥) · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) = (𝑥 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
142121, 137, 1413eqtr3d 2782 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))) = (𝑥 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))))
1437nncnd 11744 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
144143mulid2d 10749 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
145 rpre 12492 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
146145adantl 485 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
147 fznnfl 13333 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
148146, 147syl 17 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
149148simplbda 503 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
150144, 149eqbrtrd 5062 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
15119rpred 12526 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
152111, 151, 126lemuldivd 12575 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
153150, 152mpbid 235 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
154 log2sumbnd 26292 . . . . . . . . . . . . . 14 (((𝑥 / 𝑛) ∈ ℝ+ ∧ 1 ≤ (𝑥 / 𝑛)) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))) ≤ (((log‘(𝑥 / 𝑛))↑2) + 2))
15526, 153, 154syl2anc 587 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) − ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))) ≤ (((log‘(𝑥 / 𝑛))↑2) + 2))
156142, 155eqbrtrrd 5064 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (((log‘(𝑥 / 𝑛))↑2) + 2))
157108, 105, 19lemuldiv2d 12576 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (((log‘(𝑥 / 𝑛))↑2) + 2) ↔ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ≤ ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥)))
158156, 157mpbid 235 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ≤ ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
159101, 108, 106, 119, 158letrd 10887 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
1605, 101, 106, 159fsumle 15259 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
1615, 105fsumrecl 15196 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
162 remulcl 10712 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑥 · 2) ∈ ℝ)
163146, 29, 162sylancl 589 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 · 2) ∈ ℝ)
16483, 163readdcld 10760 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) ∈ ℝ)
16528recnd 10759 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
166 2cnd 11806 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
1675, 165, 166fsumadd 15201 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2))
168 fsumconst 15250 . . . . . . . . . . . . . . . 16 (((1...(⌊‘𝑥)) ∈ Fin ∧ 2 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
1695, 43, 168sylancl 589 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
170138adantl 485 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
171 flge0nn0 13293 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
172 hashfz1 13810 . . . . . . . . . . . . . . . . 17 ((⌊‘𝑥) ∈ ℕ0 → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
173170, 171, 1723syl 18 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑥 ∈ ℝ+) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
174173oveq1d 7197 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((♯‘(1...(⌊‘𝑥))) · 2) = ((⌊‘𝑥) · 2))
175169, 174eqtrd 2774 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((⌊‘𝑥) · 2))
176175oveq2d 7198 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
177167, 176eqtrd 2774 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
178 reflcl 13269 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
179146, 178syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → (⌊‘𝑥) ∈ ℝ)
18029a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℝ)
181179, 180remulcld 10761 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) · 2) ∈ ℝ)
182 flle 13272 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
183146, 182syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → (⌊‘𝑥) ≤ 𝑥)
184 2pos 11831 . . . . . . . . . . . . . . . . 17 0 < 2
18529, 184pm3.2i 474 . . . . . . . . . . . . . . . 16 (2 ∈ ℝ ∧ 0 < 2)
186185a1i 11 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → (2 ∈ ℝ ∧ 0 < 2))
187 lemul1 11582 . . . . . . . . . . . . . . 15 (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
188179, 146, 186, 187syl3anc 1372 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
189183, 188mpbid 235 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) · 2) ≤ (𝑥 · 2))
190181, 163, 83, 189leadd2dd 11345 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
191177, 190eqbrtrd 5062 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
192161, 164, 23, 191lediv1dd 12584 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥))
193105recnd 10759 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℂ)
1945, 56, 193, 61fsumdivc 15246 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
19583recnd 10759 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
19656, 86mulcld 10751 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 · 2) ∈ ℂ)
197 divdir 11413 . . . . . . . . . . . 12 ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℂ ∧ (𝑥 · 2) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + ((𝑥 · 2) / 𝑥)))
198195, 196, 55, 197syl3anc 1372 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + ((𝑥 · 2) / 𝑥)))
19986, 56, 61divcan3d 11511 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((𝑥 · 2) / 𝑥) = 2)
200199oveq2d 7198 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + ((𝑥 · 2) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
201198, 200eqtrd 2774 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
202192, 194, 2013brtr3d 5071 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
203102, 107, 97, 160, 202letrd 10887 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
20498, 102, 97, 103, 203letrd 10887 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
20597leabsd 14876 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)))
20698, 97, 100, 204, 205letrd 10887 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)))
207206adantrr 717 . . . . 5 ((⊤ ∧ (𝑥 ∈ ℝ+ ∧ 1 ≤ 𝑥)) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)))
20882, 95, 97, 40, 207o1le 15114 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ 𝑂(1))
20922selberglem1 26293 . . . 4 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
210 o1add 15073 . . . 4 (((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∘f + (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))) ∈ 𝑂(1))
211208, 209, 210sylancl 589 . . 3 (⊤ → ((𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∘f + (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))))) ∈ 𝑂(1))
21281, 211eqeltrrd 2835 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
213212mptru 1549 1 (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1088   = wceq 1542  wtru 1543  wcel 2114  wne 2935  Vcvv 3400  wss 3853   class class class wbr 5040  cmpt 5120  cfv 6349  (class class class)co 7182  f cof 7435  Fincfn 8567  cc 10625  cr 10626  0cc0 10627  1c1 10628   + caddc 10630   · cmul 10632   < clt 10765  cle 10766  cmin 10960   / cdiv 11387  cn 11728  2c2 11783  0cn0 11988  cz 12074  +crp 12484  ...cfz 12993  cfl 13263  cexp 13533  !cfa 13737  chash 13794  abscabs 14695  𝑟 crli 14944  𝑂(1)co1 14945  Σcsu 15147  logclog 25310  μcmu 25844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5242  ax-pr 5306  ax-un 7491  ax-inf2 9189  ax-cnex 10683  ax-resscn 10684  ax-1cn 10685  ax-icn 10686  ax-addcl 10687  ax-addrcl 10688  ax-mulcl 10689  ax-mulrcl 10690  ax-mulcom 10691  ax-addass 10692  ax-mulass 10693  ax-distr 10694  ax-i2m1 10695  ax-1ne0 10696  ax-1rid 10697  ax-rnegex 10698  ax-rrecex 10699  ax-cnre 10700  ax-pre-lttri 10701  ax-pre-lttrn 10702  ax-pre-ltadd 10703  ax-pre-mulgt0 10704  ax-pre-sup 10705  ax-addf 10706  ax-mulf 10707
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-int 4847  df-iun 4893  df-iin 4894  df-disj 5006  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-se 5494  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-pred 6139  df-ord 6185  df-on 6186  df-lim 6187  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7139  df-ov 7185  df-oprab 7186  df-mpo 7187  df-of 7437  df-om 7612  df-1st 7726  df-2nd 7727  df-supp 7869  df-wrecs 7988  df-recs 8049  df-rdg 8087  df-1o 8143  df-2o 8144  df-oadd 8147  df-er 8332  df-map 8451  df-pm 8452  df-ixp 8520  df-en 8568  df-dom 8569  df-sdom 8570  df-fin 8571  df-fsupp 8919  df-fi 8960  df-sup 8991  df-inf 8992  df-oi 9059  df-dju 9415  df-card 9453  df-pnf 10767  df-mnf 10768  df-xr 10769  df-ltxr 10770  df-le 10771  df-sub 10962  df-neg 10963  df-div 11388  df-nn 11729  df-2 11791  df-3 11792  df-4 11793  df-5 11794  df-6 11795  df-7 11796  df-8 11797  df-9 11798  df-n0 11989  df-xnn0 12061  df-z 12075  df-dec 12192  df-uz 12337  df-q 12443  df-rp 12485  df-xneg 12602  df-xadd 12603  df-xmul 12604  df-ioo 12837  df-ioc 12838  df-ico 12839  df-icc 12840  df-fz 12994  df-fzo 13137  df-fl 13265  df-mod 13341  df-seq 13473  df-exp 13534  df-fac 13738  df-bc 13767  df-hash 13795  df-shft 14528  df-cj 14560  df-re 14561  df-im 14562  df-sqrt 14696  df-abs 14697  df-limsup 14930  df-clim 14947  df-rlim 14948  df-o1 14949  df-lo1 14950  df-sum 15148  df-ef 15525  df-e 15526  df-sin 15527  df-cos 15528  df-tan 15529  df-pi 15530  df-dvds 15712  df-gcd 15950  df-prm 16125  df-pc 16286  df-struct 16600  df-ndx 16601  df-slot 16602  df-base 16604  df-sets 16605  df-ress 16606  df-plusg 16693  df-mulr 16694  df-starv 16695  df-sca 16696  df-vsca 16697  df-ip 16698  df-tset 16699  df-ple 16700  df-ds 16702  df-unif 16703  df-hom 16704  df-cco 16705  df-rest 16811  df-topn 16812  df-0g 16830  df-gsum 16831  df-topgen 16832  df-pt 16833  df-prds 16836  df-xrs 16890  df-qtop 16895  df-imas 16896  df-xps 16898  df-mre 16972  df-mrc 16973  df-acs 16975  df-mgm 17980  df-sgrp 18029  df-mnd 18040  df-submnd 18085  df-mulg 18355  df-cntz 18577  df-cmn 19038  df-psmet 20221  df-xmet 20222  df-met 20223  df-bl 20224  df-mopn 20225  df-fbas 20226  df-fg 20227  df-cnfld 20230  df-top 21657  df-topon 21674  df-topsp 21696  df-bases 21709  df-cld 21782  df-ntr 21783  df-cls 21784  df-nei 21861  df-lp 21899  df-perf 21900  df-cn 21990  df-cnp 21991  df-haus 22078  df-cmp 22150  df-tx 22325  df-hmeo 22518  df-fil 22609  df-fm 22701  df-flim 22702  df-flf 22703  df-xms 23085  df-ms 23086  df-tms 23087  df-cncf 23642  df-limc 24630  df-dv 24631  df-ulm 25136  df-log 25312  df-cxp 25313  df-atan 25617  df-em 25742  df-mu 25850
This theorem is referenced by:  selberglem3  26295  selberg  26296
  Copyright terms: Public domain W3C validator