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Theorem selberglem2 25135
Description: Lemma for selberg 25137. (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 9971 . . . . . . 7 ℝ ∈ V
2 rpssre 11787 . . . . . . 7 + ⊆ ℝ
31, 2ssexi 4763 . . . . . 6 + ∈ V
43a1i 11 . . . . 5 (⊤ → ℝ+ ∈ V)
5 fzfid 12712 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin)
6 elfznn 12312 . . . . . . . . . . 11 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
76adantl 482 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8 mucl 24767 . . . . . . . . . 10 (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
97, 8syl 17 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
109zred 11426 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
1110recnd 10012 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
12 fzfid 12712 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
13 elfznn 12312 . . . . . . . . . . . . . . 15 (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
1413adantl 482 . . . . . . . . . . . . . 14 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
1514nnrpd 11814 . . . . . . . . . . . . 13 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ+)
1615relogcld 24273 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
1716resqcld 12975 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℝ)
1812, 17fsumrecl 14398 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℝ)
19 simplr 791 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ+)
2018, 19rerpdivcld 11847 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) ∈ ℝ)
2120recnd 10012 . . . . . . . 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 477 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
246nnrpd 11814 . . . . . . . . . . . . . . 15 (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ+)
25 rpdivcl 11800 . . . . . . . . . . . . . . 15 ((𝑥 ∈ ℝ+𝑛 ∈ ℝ+) → (𝑥 / 𝑛) ∈ ℝ+)
2623, 24, 25syl2an 494 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ+)
2726relogcld 24273 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
2827resqcld 12975 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
29 2re 11034 . . . . . . . . . . . . 13 2 ∈ ℝ
30 remulcl 9965 . . . . . . . . . . . . . 14 ((2 ∈ ℝ ∧ (log‘(𝑥 / 𝑛)) ∈ ℝ) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
3129, 27, 30sylancr 694 . . . . . . . . . . . . 13 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
32 resubcl 10289 . . . . . . . . . . . . 13 ((2 ∈ ℝ ∧ (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
3329, 31, 32sylancr 694 . . . . . . . . . . . 12 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
3428, 33readdcld 10013 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
3534, 7nndivred 11013 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛) ∈ ℝ)
3622, 35syl5eqel 2702 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℝ)
3736recnd 10012 . . . . . . . 8 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
3821, 37subcld 10336 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) ∈ ℂ)
3911, 38mulcld 10004 . . . . . 6 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
405, 39fsumcl 14397 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
4111, 37mulcld 10004 . . . . . . 7 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · 𝑇) ∈ ℂ)
425, 41fsumcl 14397 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) ∈ ℂ)
43 2cn 11035 . . . . . . 7 2 ∈ ℂ
44 relogcl 24226 . . . . . . . . 9 (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ)
4544adantl 482 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ)
4645recnd 10012 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ)
47 mulcl 9964 . . . . . . 7 ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
4843, 46, 47sylancr 694 . . . . . 6 ((⊤ ∧ 𝑥 ∈ ℝ+) → (2 · (log‘𝑥)) ∈ ℂ)
4942, 48subcld 10336 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) ∈ ℂ)
50 eqidd 2622 . . . . 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 2622 . . . . 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 6867 . . . 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 10356 . . . . . 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 11794 . . . . . . . . . . 11 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
5554adantl 482 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
5655simpld 475 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℂ)
5710adantr 481 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (μ‘𝑛) ∈ ℝ)
5857, 17remulcld 10014 . . . . . . . . . . 11 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
5912, 58fsumrecl 14398 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
6059recnd 10012 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℂ)
6155simprd 479 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0)
625, 56, 60, 61fsumdivc 14446 . . . . . . . 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 10012 . . . . . . . . . . . 12 ((((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℂ)
6412, 63fsumcl 14397 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℂ)
6555adantr 481 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66 divass 10647 . . . . . . . . . . 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 1323 . . . . . . . . . 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 14444 . . . . . . . . . . 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 6619 . . . . . . . . . 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 10340 . . . . . . . . . . . 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 6620 . . . . . . . . . . 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 10008 . . . . . . . . . . 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 2657 . . . . . . . . . 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 2663 . . . . . . . . 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 14367 . . . . . . . 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 14403 . . . . . . . 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 2660 . . . . . . 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 6619 . . . . . 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 2657 . . . . 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 4704 . . . 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 2655 . . 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 9999 . . . . 5 (⊤ → 1 ∈ ℝ)
835, 28fsumrecl 14398 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
8483, 23rerpdivcld 11847 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ)
8584recnd 10012 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℂ)
86 2cnd 11037 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℂ)
87 2nn0 11253 . . . . . . . 8 2 ∈ ℕ0
88 logexprlim 24850 . . . . . . . 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 11037 . . . . . . . 8 (⊤ → 2 ∈ ℂ)
91 rlimconst 14209 . . . . . . . 8 ((ℝ+ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ 2) ⇝𝑟 2)
922, 90, 91sylancr 694 . . . . . . 7 (⊤ → (𝑥 ∈ ℝ+ ↦ 2) ⇝𝑟 2)
9385, 86, 89, 92rlimadd 14307 . . . . . 6 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ⇝𝑟 ((!‘2) + 2))
94 rlimo1 14281 . . . . . 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 9963 . . . . . 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 693 . . . . 5 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
9840abscld 14109 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
9997recnd 10012 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℂ)
10099abscld 14109 . . . . . . 7 ((⊤ ∧ 𝑥 ∈ ℝ+) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ ℝ)
10139abscld 14109 . . . . . . . . 9 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
1025, 101fsumrecl 14398 . . . . . . . 8 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
1035, 39fsumabs 14460 . . . . . . . 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 9963 . . . . . . . . . . . 12 ((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 2 ∈ ℝ) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
10528, 29, 104sylancl 693 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
106105, 19rerpdivcld 11847 . . . . . . . . . 10 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
1075, 106fsumrecl 14398 . . . . . . . . 9 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
10838abscld 14109 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℝ)
10911, 38absmuld 14127 . . . . . . . . . . . 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 14109 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
111 1red 9999 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
11238absge0d 14117 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
113 mule1 24774 . . . . . . . . . . . . . . 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 10907 . . . . . . . . . . . . 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 10012 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
117116mulid2d 10002 . . . . . . . . . . . . 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 4639 . . . . . . . . . . . 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 4635 . . . . . . . . . . 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 475 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
121120, 38absmuld 14127 . . . . . . . . . . . . . 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 10430 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
12464, 120, 123divcan2d 10747 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2))
12534recnd 10012 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
1267nnrpd 11814 . . . . . . . . . . . . . . . . . . 19 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ+)
127 rpcnne0 11794 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ+ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
128126, 127syl 17 . . . . . . . . . . . . . . . . . 18 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
129 divass 10647 . . . . . . . . . . . . . . . . . . . 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 6615 . . . . . . . . . . . . . . . . . . . 20 (𝑥 · 𝑇) = (𝑥 · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
131129, 130syl6eqr 2673 . . . . . . . . . . . . . . . . . . 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 10648 . . . . . . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · 𝑇) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
135124, 134oveq12d 6622 . . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . . . 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 6152 . . . . . . . . . . . . . 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 11791 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℝ+ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
139 absid 13970 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
14019, 138, 1393syl 18 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑥) = 𝑥)
141140oveq1d 6619 . . . . . . . . . . . . . 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 2663 . . . . . . . . . . . . 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 10980 . . . . . . . . . . . . . . . . 17 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
144143mulid2d 10002 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
145 rpre 11783 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
146145adantl 482 . . . . . . . . . . . . . . . . . 18 ((⊤ ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ)
147 fznnfl 12601 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
148146, 147syl 17 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑥)))
149148simplbda 653 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛𝑥)
150144, 149eqbrtrd 4635 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
15119rpred 11816 . . . . . . . . . . . . . . . 16 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
152111, 151, 126lemuldivd 11865 . . . . . . . . . . . . . . 15 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
153150, 152mpbid 222 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
154 log2sumbnd 25133 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 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 4637 . . . . . . . . . . . 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 11866 . . . . . . . . . . . 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 222 . . . . . . . . . . 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 10138 . . . . . . . . . 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 14458 . . . . . . . . 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 14398 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
162 remulcl 9965 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑥 · 2) ∈ ℝ)
163146, 29, 162sylancl 693 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 · 2) ∈ ℝ)
16483, 163readdcld 10013 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) ∈ ℝ)
16528recnd 10012 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
166 2cnd 11037 . . . . . . . . . . . . . 14 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
1675, 165, 166fsumadd 14403 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2))
168 fsumconst 14450 . . . . . . . . . . . . . . . 16 (((1...(⌊‘𝑥)) ∈ Fin ∧ 2 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((#‘(1...(⌊‘𝑥))) · 2))
1695, 43, 168sylancl 693 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((#‘(1...(⌊‘𝑥))) · 2))
170138adantl 482 . . . . . . . . . . . . . . . . 17 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
171 flge0nn0 12561 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ0)
172 hashfz1 13074 . . . . . . . . . . . . . . . . 17 ((⌊‘𝑥) ∈ ℕ0 → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
173170, 171, 1723syl 18 . . . . . . . . . . . . . . . 16 ((⊤ ∧ 𝑥 ∈ ℝ+) → (#‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
174173oveq1d 6619 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((#‘(1...(⌊‘𝑥))) · 2) = ((⌊‘𝑥) · 2))
175169, 174eqtrd 2655 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((⌊‘𝑥) · 2))
176175oveq2d 6620 . . . . . . . . . . . . 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 2655 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
178 reflcl 12537 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
179146, 178syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → (⌊‘𝑥) ∈ ℝ)
18029a1i 11 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → 2 ∈ ℝ)
181179, 180remulcld 10014 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) · 2) ∈ ℝ)
182 flle 12540 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
183146, 182syl 17 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → (⌊‘𝑥) ≤ 𝑥)
184 2pos 11056 . . . . . . . . . . . . . . . . 17 0 < 2
18529, 184pm3.2i 471 . . . . . . . . . . . . . . . 16 (2 ∈ ℝ ∧ 0 < 2)
186185a1i 11 . . . . . . . . . . . . . . 15 ((⊤ ∧ 𝑥 ∈ ℝ+) → (2 ∈ ℝ ∧ 0 < 2))
187 lemul1 10819 . . . . . . . . . . . . . . 15 (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
188179, 146, 186, 187syl3anc 1323 . . . . . . . . . . . . . 14 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
189183, 188mpbid 222 . . . . . . . . . . . . 13 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((⌊‘𝑥) · 2) ≤ (𝑥 · 2))
190181, 163, 83, 189leadd2dd 10586 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
191177, 190eqbrtrd 4635 . . . . . . . . . . 11 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
192161, 164, 23, 191lediv1dd 11874 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥))
193105recnd 10012 . . . . . . . . . . 11 (((⊤ ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℂ)
1945, 56, 193, 61fsumdivc 14446 . . . . . . . . . 10 ((⊤ ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
19583recnd 10012 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
19656, 86mulcld 10004 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → (𝑥 · 2) ∈ ℂ)
197 divdir 10654 . . . . . . . . . . . 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 1323 . . . . . . . . . . 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 10750 . . . . . . . . . . . 12 ((⊤ ∧ 𝑥 ∈ ℝ+) → ((𝑥 · 2) / 𝑥) = 2)
200199oveq2d 6620 . . . . . . . . . . 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 2655 . . . . . . . . . 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 4644 . . . . . . . . 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 10138 . . . . . . . 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 10138 . . . . . . 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 14087 . . . . . . 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 10138 . . . . . 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 752 . . . . 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 14317 . . . 4 (⊤ → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ 𝑂(1))
20922selberglem1 25134 . . . 4 (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
210 o1add 14278 . . . 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 693 . . 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 2699 . 2 (⊤ → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
213212trud 1490 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 196  wa 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  wne 2790  Vcvv 3186  wss 3555   class class class wbr 4613  cmpt 4673  cfv 5847  (class class class)co 6604  𝑓 cof 6848  Fincfn 7899  cc 9878  cr 9879  0cc0 9880  1c1 9881   + caddc 9883   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  cn 10964  2c2 11014  0cn0 11236  cz 11321  +crp 11776  ...cfz 12268  cfl 12531  cexp 12800  !cfa 13000  #chash 13057  abscabs 13908  𝑟 crli 14150  𝑂(1)co1 14151  Σcsu 14350  logclog 24205  μcmu 24721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-xnn0 11308  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-ioo 12121  df-ioc 12122  df-ico 12123  df-icc 12124  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-seq 12742  df-exp 12801  df-fac 13001  df-bc 13030  df-hash 13058  df-shft 13741  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-o1 14155  df-lo1 14156  df-sum 14351  df-ef 14723  df-e 14724  df-sin 14725  df-cos 14726  df-pi 14728  df-dvds 14908  df-gcd 15141  df-prm 15310  df-pc 15466  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-cmp 21100  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537  df-log 24207  df-cxp 24208  df-em 24619  df-mu 24727
This theorem is referenced by:  selberglem3  25136  selberg  25137
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