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Theorem selberglem2 27038

Description: Lemma for selberg 27040. (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**
Step	Hyp	Ref	Expression
1		reex 11197	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12977	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5321	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		fzfid 13934	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
6		elfznn 13526	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
7	6	adantl 482	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8		mucl 26634	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
10	9	zred 12662	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
11	10	recnd 11238	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
12		fzfid 13934	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
13		elfznn 13526	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
14	13	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
15	14	nnrpd 13010	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
16	15	relogcld 26122	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
17	16	resqcld 14086	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℝ)
18	12, 17	fsumrecl 15676	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℝ)
19		simplr 767	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
20	18, 19	rerpdivcld 13043	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) ∈ ℝ)
21	20	recnd 11238	. . . . . . . 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 485	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
24	6	nnrpd 13010	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
25		rpdivcl 12995	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
26	23, 24, 25	syl2an 596	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
27	26	relogcld 26122	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
28	27	resqcld 14086	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
29		2re 12282	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
30		remulcl 11191	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (log‘(𝑥 / 𝑛)) ∈ ℝ) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
31	29, 27, 30	sylancr 587	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
32		resubcl 11520	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
33	29, 31, 32	sylancr 587	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
34	28, 33	readdcld 11239	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
35	34, 7	nndivred 12262	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛) ∈ ℝ)
36	22, 35	eqeltrid 2837	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℝ)
37	36	recnd 11238	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
38	21, 37	subcld 11567	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) ∈ ℂ)
39	11, 38	mulcld 11230	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
40	5, 39	fsumcl 15675	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
41	11, 37	mulcld 11230	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · 𝑇) ∈ ℂ)
42	5, 41	fsumcl 15675	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) ∈ ℂ)
43		2cn 12283	. . . . . . 7 ⊢ 2 ∈ ℂ
44		relogcl 26075	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
45	44	adantl 482	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
46	45	recnd 11238	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
47		mulcl 11190	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
48	43, 46, 47	sylancr 587	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 · (log‘𝑥)) ∈ ℂ)
49	42, 48	subcld 11567	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) ∈ ℂ)
50		eqidd 2733	. . . . 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 2733	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))))
52	4, 40, 49, 50, 51	offval2 7686	. . . 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))))))
53	40, 42, 48	addsubassd 11587	. . . . . 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 12988	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
55	54	adantl 482	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
56	55	simpld 495	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
57	10	adantr 481	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (μ‘𝑛) ∈ ℝ)
58	57, 17	remulcld 11240	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
59	12, 58	fsumrecl 15676	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
60	59	recnd 11238	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℂ)
61	55	simprd 496	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
62	5, 56, 60, 61	fsumdivc 15728	. . . . . . . 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))
63	17	recnd 11238	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℂ)
64	12, 63	fsumcl 15675	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℂ)
65	55	adantr 481	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66		divass 11886	. . . . . . . . . . 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)))
67	11, 64, 65, 66	syl3anc 1371	. . . . . . . . . 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)))
68	12, 11, 63	fsummulc2 15726	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)))
69	68	oveq1d 7420	. . . . . . . . . 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))
70	21, 37	npcand 11571	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) + 𝑇) = (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥))
71	70	oveq2d 7421	. . . . . . . . . . 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)))
72	11, 38, 37	adddid 11234	. . . . . . . . . . 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)))
73	71, 72	eqtr3d 2774	. . . . . . . . . 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)))
74	67, 69, 73	3eqtr3d 2780	. . . . . . . . 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)))
75	74	sumeq2dv 15645	. . . . . . . 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)))
76	5, 39, 41	fsumadd 15682	. . . . . . . 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)))
77	62, 75, 76	3eqtrrd 2777	. . . . . . 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))
78	77	oveq1d 7420	. . . . . 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))))
79	53, 78	eqtr3d 2774	. . . . 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))))
80	79	mpteq2dva 5247	. . . 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)))))
81	52, 80	eqtrd 2772	. . 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 11211	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
83	5, 28	fsumrecl 15676	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
84	83, 23	rerpdivcld 13043	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ)
85	84	recnd 11238	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℂ)
86		2cnd 12286	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
87		2nn0 12485	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
88		logexprlim 26717	. . . . . . . 8 ⊢ (2 ∈ ℕ₀ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥)) ⇝_𝑟 (!‘2))
89	87, 88	mp1i 13	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥)) ⇝_𝑟 (!‘2))
90		2cnd 12286	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
91		rlimconst 15484	. . . . . . . 8 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ⇝_𝑟 2)
92	2, 90, 91	sylancr 587	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 2) ⇝_𝑟 2)
93	85, 86, 89, 92	rlimadd 15583	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ⇝_𝑟 ((!‘2) + 2))
94		rlimo1 15557	. . . . . 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))
95	93, 94	syl 17	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ 𝑂(1))
96		readdcl 11189	. . . . . 6 ⊢ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ ∧ 2 ∈ ℝ) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
97	84, 29, 96	sylancl 586	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
98	40	abscld 15379	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
99	97	recnd 11238	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℂ)
100	99	abscld 15379	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ ℝ)
101	39	abscld 15379	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
102	5, 101	fsumrecl 15676	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
103	5, 39	fsumabs 15743	. . . . . . . 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 11189	. . . . . . . . . . . 12 ⊢ ((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 2 ∈ ℝ) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
105	28, 29, 104	sylancl 586	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
106	105, 19	rerpdivcld 13043	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
107	5, 106	fsumrecl 15676	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
108	38	abscld 15379	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℝ)
109	11, 38	absmuld 15397	. . . . . . . . . . . 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))))
110	11	abscld 15379	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
111		1red 11211	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
112	38	absge0d 15387	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
113		mule1 26641	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℕ → (abs‘(μ‘𝑛)) ≤ 1)
114	7, 113	syl 17	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ≤ 1)
115	110, 111, 108, 112, 114	lemul1ad 12149	. . . . . . . . . . . . 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))))
116	108	recnd 11238	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
117	116	mullidd 11228	. . . . . . . . . . . . 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)))
118	115, 117	breqtrd 5173	. . . . . . . . . . . 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)))
119	109, 118	eqbrtrd 5169	. . . . . . . . . . 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)))
120	65	simpld 495	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
121	120, 38	absmuld 15397	. . . . . . . . . . . . . 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))))
122	120, 21, 37	subdid 11666	. . . . . . . . . . . . . . . 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)))
123	65	simprd 496	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
124	64, 120, 123	divcan2d 11988	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2))
125	34	recnd 11238	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
126	7	nnrpd 13010	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
127		rpcnne0 12988	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
129		divass 11886	. . . . . . . . . . . . . . . . . . . 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)))
130	22	oveq2i 7416	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 · 𝑇) = (𝑥 · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
131	129, 130	eqtr4di 2790	. . . . . . . . . . . . . . . . . . 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 11887	. . . . . . . . . . . . . . . . . . 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)))))))
133	131, 132	eqtr3d 2774	. . . . . . . . . . . . . . . . . 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)))))))
134	120, 125, 128, 133	syl3anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · 𝑇) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
135	124, 134	oveq12d 7423	. . . . . . . . . . . . . . . 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))))))))
136	122, 135	eqtrd 2772	. . . . . . . . . . . . . . 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))))))))
137	136	fveq2d 6892	. . . . . . . . . . . . . 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 12985	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
139		absid 15239	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (abs‘𝑥) = 𝑥)
140	19, 138, 139	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘𝑥) = 𝑥)
141	140	oveq1d 7420	. . . . . . . . . . . . . 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))))
142	121, 137, 141	3eqtr3d 2780	. . . . . . . . . . . . 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))))
143	7	nncnd 12224	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
144	143	mullidd 11228	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
145		rpre 12978	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
146	145	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
147		fznnfl 13823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
148	146, 147	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑛 ∈ (1...(⌊‘𝑥)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑥)))
149	148	simplbda 500	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
150	144, 149	eqbrtrd 5169	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
151	19	rpred 13012	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
152	111, 151, 126	lemuldivd 13061	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((1 · 𝑛) ≤ 𝑥 ↔ 1 ≤ (𝑥 / 𝑛)))
153	150, 152	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ≤ (𝑥 / 𝑛))
154		log2sumbnd 27036	. . . . . . . . . . . . . 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))
155	26, 153, 154	syl2anc 584	. . . . . . . . . . . . 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))
156	142, 155	eqbrtrrd 5171	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ (((log‘(𝑥 / 𝑛))↑2) + 2))
157	108, 105, 19	lemuldiv2d 13062	. . . . . . . . . . . 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)))
158	156, 157	mpbid 231	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ≤ ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
159	101, 108, 106, 119, 158	letrd 11367	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ≤ ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
160	5, 101, 106, 159	fsumle 15741	. . . . . . . . 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))
161	5, 105	fsumrecl 15676	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
162		remulcl 11191	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑥 · 2) ∈ ℝ)
163	146, 29, 162	sylancl 586	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 2) ∈ ℝ)
164	83, 163	readdcld 11239	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) ∈ ℝ)
165	28	recnd 11238	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
166		2cnd 12286	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
167	5, 165, 166	fsumadd 15682	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2))
168		fsumconst 15732	. . . . . . . . . . . . . . . 16 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ 2 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
169	5, 43, 168	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
170	138	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
171		flge0nn0 13781	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
172		hashfz1 14302	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
173	170, 171, 172	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
174	173	oveq1d 7420	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((♯‘(1...(⌊‘𝑥))) · 2) = ((⌊‘𝑥) · 2))
175	169, 174	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((⌊‘𝑥) · 2))
176	175	oveq2d 7421	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
177	167, 176	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
178		reflcl 13757	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ∈ ℝ)
179	146, 178	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ∈ ℝ)
180	29	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℝ)
181	179, 180	remulcld 11240	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) · 2) ∈ ℝ)
182		flle 13760	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
183	146, 182	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
184		2pos 12311	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
185	29, 184	pm3.2i 471	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℝ ∧ 0 < 2)
186	185	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 ∈ ℝ ∧ 0 < 2))
187		lemul1 12062	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
188	179, 146, 186, 187	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
189	183, 188	mpbid 231	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) · 2) ≤ (𝑥 · 2))
190	181, 163, 83, 189	leadd2dd 11825	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
191	177, 190	eqbrtrd 5169	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ≤ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)))
192	161, 164, 23, 191	lediv1dd 13070	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥))
193	105	recnd 11238	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℂ)
194	5, 56, 193, 61	fsumdivc 15728	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
195	83	recnd 11238	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
196	56, 86	mulcld 11230	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 2) ∈ ℂ)
197		divdir 11893	. . . . . . . . . . . 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)))
198	195, 196, 55, 197	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + ((𝑥 · 2) / 𝑥)))
199	86, 56, 61	divcan3d 11991	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 · 2) / 𝑥) = 2)
200	199	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + ((𝑥 · 2) / 𝑥)) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
201	198, 200	eqtrd 2772	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
202	192, 194, 201	3brtr3d 5178	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2))
203	102, 107, 97, 160, 202	letrd 11367	. . . . . . . 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))
204	98, 102, 97, 103, 203	letrd 11367	. . . . . . 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))
205	97	leabsd 15357	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ≤ (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)))
206	98, 97, 100, 204, 205	letrd 11367	. . . . . 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)))
207	206	adantrr 715	. . . . 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)))
208	82, 95, 97, 40, 207	o1le 15595	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ 𝑂(1))
209	22	selberglem1 27037	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
210		o1add 15554	. . . 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))
211	208, 209, 210	sylancl 586	. . 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))
212	81, 211	eqeltrrd 2834	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
213	212	mptru 1548	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 Fincfn 8935 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 2c2 12263 ℕ₀cn0 12468 ℤcz 12554 ℝ⁺crp 12970 ...cfz 13480 ⌊cfl 13751 ↑cexp 14023 !cfa 14229 ♯chash 14286 abscabs 15177 ⇝_𝑟 crli 15425 𝑂(1)co1 15426 Σcsu 15628 logclog 26054 μcmu 26588

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-xnn0 12541 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-o1 15430 df-lo1 15431 df-sum 15629 df-ef 16007 df-e 16008 df-sin 16009 df-cos 16010 df-tan 16011 df-pi 16012 df-dvds 16194 df-gcd 16432 df-prm 16605 df-pc 16766 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-limc 25374 df-dv 25375 df-ulm 25880 df-log 26056 df-cxp 26057 df-atan 26361 df-em 26486 df-mu 26594

This theorem is referenced by: selberglem3 27039 selberg 27040

W3C validator