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Theorem selberglem2 26599

Description: Lemma for selberg 26601. (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 10893	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12666	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5241	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		fzfid 13621	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
6		elfznn 13214	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
7	6	adantl 481	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
8		mucl 26195	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
10	9	zred 12355	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
11	10	recnd 10934	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
12		fzfid 13621	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑛))) ∈ Fin)
13		elfznn 13214	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛))) → 𝑚 ∈ ℕ)
14	13	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℕ)
15	14	nnrpd 12699	. . . . . . . . . . . . 13 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → 𝑚 ∈ ℝ⁺)
16	15	relogcld 25683	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (log‘𝑚) ∈ ℝ)
17	16	resqcld 13893	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℝ)
18	12, 17	fsumrecl 15374	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℝ)
19		simplr 765	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ⁺)
20	18, 19	rerpdivcld 12732	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) ∈ ℝ)
21	20	recnd 10934	. . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ⁺)
24	6	nnrpd 12699	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
25		rpdivcl 12684	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
26	23, 24, 25	syl2an 595	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
27	26	relogcld 25683	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
28	27	resqcld 13893	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
29		2re 11977	. . . . . . . . . . . . 13 ⊢ 2 ∈ ℝ
30		remulcl 10887	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (log‘(𝑥 / 𝑛)) ∈ ℝ) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
31	29, 27, 30	sylancr 586	. . . . . . . . . . . . 13 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ)
32		resubcl 11215	. . . . . . . . . . . . 13 ⊢ ((2 ∈ ℝ ∧ (2 · (log‘(𝑥 / 𝑛))) ∈ ℝ) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
33	29, 31, 32	sylancr 586	. . . . . . . . . . . 12 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℝ)
34	28, 33	readdcld 10935	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℝ)
35	34, 7	nndivred 11957	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛) ∈ ℝ)
36	22, 35	eqeltrid 2843	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℝ)
37	36	recnd 10934	. . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑇 ∈ ℂ)
38	21, 37	subcld 11262	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇) ∈ ℂ)
39	11, 38	mulcld 10926	. . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
40	5, 39	fsumcl 15373	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
41	11, 37	mulcld 10926	. . . . . . 7 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · 𝑇) ∈ ℂ)
42	5, 41	fsumcl 15373	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) ∈ ℂ)
43		2cn 11978	. . . . . . 7 ⊢ 2 ∈ ℂ
44		relogcl 25636	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
45	44	adantl 481	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℝ)
46	45	recnd 10934	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (log‘𝑥) ∈ ℂ)
47		mulcl 10886	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
48	43, 46, 47	sylancr 586	. . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 · (log‘𝑥)) ∈ ℂ)
49	42, 48	subcld 11262	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) ∈ ℂ)
50		eqidd 2739	. . . . 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 2739	. . . . 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 7531	. . . 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 11282	. . . . . 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 12677	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
55	54	adantl 481	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
56	55	simpld 494	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℂ)
57	10	adantr 480	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → (μ‘𝑛) ∈ ℝ)
58	57, 17	remulcld 10936	. . . . . . . . . . 11 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
59	12, 58	fsumrecl 15374	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℝ)
60	59	recnd 10934	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) ∈ ℂ)
61	55	simprd 495	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ≠ 0)
62	5, 56, 60, 61	fsumdivc 15426	. . . . . . . 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 10934	. . . . . . . . . . . 12 ⊢ ((((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))) → ((log‘𝑚)↑2) ∈ ℂ)
64	12, 63	fsumcl 15373	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) ∈ ℂ)
65	55	adantr 480	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
66		divass 11581	. . . . . . . . . . 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 1369	. . . . . . . . . 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 15424	. . . . . . . . . . 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 7270	. . . . . . . . . 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 11266	. . . . . . . . . . . 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 7271	. . . . . . . . . . 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 10930	. . . . . . . . . . 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 2780	. . . . . . . . . 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 2786	. . . . . . . . 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 15343	. . . . . . . 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 15380	. . . . . . . 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 2783	. . . . . . 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 7270	. . . . . 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 2780	. . . . 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 5170	. . . 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 2778	. . 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 10907	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
83	5, 28	fsumrecl 15374	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℝ)
84	83, 23	rerpdivcld 12732	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℝ)
85	84	recnd 10934	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) ∈ ℂ)
86		2cnd 11981	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ ℂ)
87		2nn0 12180	. . . . . . . 8 ⊢ 2 ∈ ℕ₀
88		logexprlim 26278	. . . . . . . 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 11981	. . . . . . . 8 ⊢ (⊤ → 2 ∈ ℂ)
91		rlimconst 15181	. . . . . . . 8 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ⇝_𝑟 2)
92	2, 90, 91	sylancr 586	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 2) ⇝_𝑟 2)
93	85, 86, 89, 92	rlimadd 15280	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ⇝_𝑟 ((!‘2) + 2))
94		rlimo1 15254	. . . . . 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 10885	. . . . . 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 585	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℝ)
98	40	abscld 15076	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
99	97	recnd 10934	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2) ∈ ℂ)
100	99	abscld 15076	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (abs‘((Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) / 𝑥) + 2)) ∈ ℝ)
101	39	abscld 15076	. . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
102	5, 101	fsumrecl 15374	. . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(abs‘((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ ℝ)
103	5, 39	fsumabs 15441	. . . . . . . 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 10885	. . . . . . . . . . . 12 ⊢ ((((log‘(𝑥 / 𝑛))↑2) ∈ ℝ ∧ 2 ∈ ℝ) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
105	28, 29, 104	sylancl 585	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
106	105, 19	rerpdivcld 12732	. . . . . . . . . 10 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
107	5, 106	fsumrecl 15374	. . . . . . . . 9 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) ∈ ℝ)
108	38	abscld 15076	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℝ)
109	11, 38	absmuld 15094	. . . . . . . . . . . 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 15076	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘(μ‘𝑛)) ∈ ℝ)
111		1red 10907	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
112	38	absge0d 15084	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)))
113		mule1 26202	. . . . . . . . . . . . . . 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 11844	. . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (abs‘((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇)) ∈ ℂ)
117	116	mulid2d 10924	. . . . . . . . . . . . 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 5096	. . . . . . . . . . . 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 5092	. . . . . . . . . . 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 494	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
121	120, 38	absmuld 15094	. . . . . . . . . . . . . 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 11361	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ≠ 0)
124	64, 120, 123	divcan2d 11683	. . . . . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
126	7	nnrpd 12699	. . . . . . . . . . . . . . . . . . 19 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
127		rpcnne0 12677	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
128	126, 127	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
129		divass 11581	. . . . . . . . . . . . . . . . . . . 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 7266	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 · 𝑇) = (𝑥 · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
131	129, 130	eqtr4di 2797	. . . . . . . . . . . . . . . . . . 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 11582	. . . . . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 · 𝑇) = ((𝑥 / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
135	124, 134	oveq12d 7273	. . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . 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 12674	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
139		absid 14936	. . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . 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 11919	. . . . . . . . . . . . . . . . 17 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℂ)
144	143	mulid2d 10924	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) = 𝑛)
145		rpre 12667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
146	145	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 𝑥 ∈ ℝ)
147		fznnfl 13510	. . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ≤ 𝑥)
150	144, 149	eqbrtrd 5092	. . . . . . . . . . . . . . 15 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1 · 𝑛) ≤ 𝑥)
151	19	rpred 12701	. . . . . . . . . . . . . . . 16 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
152	111, 151, 126	lemuldivd 12750	. . . . . . . . . . . . . . 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 26597	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 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 5094	. . . . . . . . . . . 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 12751	. . . . . . . . . . . 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 11062	. . . . . . . . . 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 15439	. . . . . . . . 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 15374	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℝ)
162		remulcl 10887	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑥 · 2) ∈ ℝ)
163	146, 29, 162	sylancl 585	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 2) ∈ ℝ)
164	83, 163	readdcld 10935	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + (𝑥 · 2)) ∈ ℝ)
165	28	recnd 10934	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
166		2cnd 11981	. . . . . . . . . . . . . 14 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
167	5, 165, 166	fsumadd 15380	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + Σ𝑛 ∈ (1...(⌊‘𝑥))2))
168		fsumconst 15430	. . . . . . . . . . . . . . . 16 ⊢ (((1...(⌊‘𝑥)) ∈ Fin ∧ 2 ∈ ℂ) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
169	5, 43, 168	sylancl 585	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((♯‘(1...(⌊‘𝑥))) · 2))
170	138	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
171		flge0nn0 13468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
172		hashfz1 13988	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
173	170, 171, 172	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (♯‘(1...(⌊‘𝑥))) = (⌊‘𝑥))
174	173	oveq1d 7270	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((♯‘(1...(⌊‘𝑥))) · 2) = ((⌊‘𝑥) · 2))
175	169, 174	eqtrd 2778	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))2 = ((⌊‘𝑥) · 2))
176	175	oveq2d 7271	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) + ((⌊‘𝑥) · 2)))
178		reflcl 13444	. . . . . . . . . . . . . . 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 10936	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((⌊‘𝑥) · 2) ∈ ℝ)
182		flle 13447	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ → (⌊‘𝑥) ≤ 𝑥)
183	146, 182	syl 17	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (⌊‘𝑥) ≤ 𝑥)
184		2pos 12006	. . . . . . . . . . . . . . . . 17 ⊢ 0 < 2
185	29, 184	pm3.2i 470	. . . . . . . . . . . . . . . 16 ⊢ (2 ∈ ℝ ∧ 0 < 2)
186	185	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 ∈ ℝ ∧ 0 < 2))
187		lemul1 11757	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((⌊‘𝑥) ≤ 𝑥 ↔ ((⌊‘𝑥) · 2) ≤ (𝑥 · 2)))
188	179, 146, 186, 187	syl3anc 1369	. . . . . . . . . . . . . 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 11520	. . . . . . . . . . . 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 5092	. . . . . . . . . . 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 12759	. . . . . . . . . 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 10934	. . . . . . . . . . 11 ⊢ (((⊤ ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + 2) ∈ ℂ)
194	5, 56, 193, 61	fsumdivc 15426	. . . . . . . . . 10 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((log‘(𝑥 / 𝑛))↑2) + 2) / 𝑥))
195	83	recnd 10934	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
196	56, 86	mulcld 10926	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (𝑥 · 2) ∈ ℂ)
197		divdir 11588	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 11686	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((𝑥 · 2) / 𝑥) = 2)
200	199	oveq2d 7271	. . . . . . . . . . 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 2778	. . . . . . . . . 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 5101	. . . . . . . . 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 11062	. . . . . . . 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 11062	. . . . . . 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 15054	. . . . . . 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 11062	. . . . . 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 713	. . . . 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 15292	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · ((Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((log‘𝑚)↑2) / 𝑥) − 𝑇))) ∈ 𝑂(1))
209	22	selberglem1 26598	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
210		o1add 15251	. . . 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 585	. . 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 2840	. 2 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((μ‘𝑛) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
213	212	mptru 1546	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 395 ∧ w3a 1085 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ⊆ wss 3883 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 Fincfn 8691 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℕ₀cn0 12163 ℤcz 12249 ℝ⁺crp 12659 ...cfz 13168 ⌊cfl 13438 ↑cexp 13710 !cfa 13915 ♯chash 13972 abscabs 14873 ⇝_𝑟 crli 15122 𝑂(1)co1 15123 Σcsu 15325 logclog 25615 μcmu 26149

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-o1 15127 df-lo1 15128 df-sum 15326 df-ef 15705 df-e 15706 df-sin 15707 df-cos 15708 df-tan 15709 df-pi 15710 df-dvds 15892 df-gcd 16130 df-prm 16305 df-pc 16466 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-limc 24935 df-dv 24936 df-ulm 25441 df-log 25617 df-cxp 25618 df-atan 25922 df-em 26047 df-mu 26155

This theorem is referenced by: selberglem3 26600 selberg 26601

W3C validator