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Theorem selberglem1 26693

Description: Lemma for selberg 26696. Estimation of the asymptotic part of selberglem3 26695. (Contributed by Mario Carneiro, 20-May-2016.)

**Hypothesis**
Ref	Expression
selberglem1.t	⊢ 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)

**Assertion**
Ref	Expression
selberglem1	⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Distinct variable group: 𝑥,𝑛 Allowed substitution hints: 𝑇(𝑥,𝑛)

**Proof of Theorem selberglem1**
Step	Hyp	Ref	Expression
1		fzfid 13693	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 13285	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
3	2	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
4		mucl 26290	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
6	5	zred 12426	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
7	6, 3	nndivred 12027	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
8	7	recnd 11003	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
9	2	nnrpd 12770	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
10		rpdivcl 12755	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
11	9, 10	sylan2 593	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
12		relogcl 25731	. . . . . . . . . 10 ⊢ ((𝑥 / 𝑛) ∈ ℝ⁺ → (log‘(𝑥 / 𝑛)) ∈ ℝ)
13	11, 12	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℝ)
14	13	recnd 11003	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
15	14	sqcld 13862	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
16	8, 15	mulcld 10995	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) ∈ ℂ)
17	1, 16	fsumcl 15445	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) ∈ ℂ)
18		2cn 12048	. . . . . . . . 9 ⊢ 2 ∈ ℂ
19	18	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
20	19, 14	mulcld 10995	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℂ)
21	19, 20	subcld 11332	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
22	8, 21	mulcld 10995	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
23	1, 22	fsumcl 15445	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
24		relogcl 25731	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
25	24	recnd 11003	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
26		mulcl 10955	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
27	18, 25, 26	sylancr 587	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) ∈ ℂ)
28	17, 23, 27	addsubd 11353	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))) − (2 · (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
29		selberglem1.t	. . . . . . . . 9 ⊢ 𝑇 = ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)
30	29	oveq2i 7286	. . . . . . . 8 ⊢ ((μ‘𝑛) · 𝑇) = ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
31	5	zcnd 12427	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
32	15, 21	addcld 10994	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
33	3	nnrpd 12770	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
34	33	rpcnne0d 12781	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
35		divass 11651	. . . . . . . . . . 11 ⊢ (((μ‘𝑛) ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((μ‘𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) / 𝑛) = ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)))
36		div23 11652	. . . . . . . . . . 11 ⊢ (((μ‘𝑛) ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((μ‘𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) / 𝑛) = (((μ‘𝑛) / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
37	35, 36	eqtr3d 2780	. . . . . . . . . 10 ⊢ (((μ‘𝑛) ∈ ℂ ∧ (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)) = (((μ‘𝑛) / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
38	31, 32, 34, 37	syl3anc 1370	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)) = (((μ‘𝑛) / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))))
39	8, 15, 21	adddid 10999	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛)))))) = ((((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
40	38, 39	eqtrd 2778	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛)) = ((((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
41	30, 40	eqtrid 2790	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) · 𝑇) = ((((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
42	41	sumeq2dv 15415	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
43	1, 16, 22	fsumadd 15452	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
44	42, 43	eqtrd 2778	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
45	44	oveq1d 7290	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))) − (2 · (log‘𝑥))))
46	18	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 2 ∈ ℂ)
47	8, 14	mulcld 10995	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
48	8, 47	subcld 11332	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
49	1, 46, 48	fsummulc2 15496	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
50	1, 8, 47	fsumsub 15500	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
51	50	oveq2d 7291	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (2 · Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
52	19, 8	mulcomd 10996	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((μ‘𝑛) / 𝑛)) = (((μ‘𝑛) / 𝑛) · 2))
53	19, 8, 14	mul12d 11184	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (((μ‘𝑛) / 𝑛) · (2 · (log‘(𝑥 / 𝑛)))))
54	52, 53	oveq12d 7293	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((2 · ((μ‘𝑛) / 𝑛)) − (2 · (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = ((((μ‘𝑛) / 𝑛) · 2) − (((μ‘𝑛) / 𝑛) · (2 · (log‘(𝑥 / 𝑛))))))
55	19, 8, 47	subdid 11431	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = ((2 · ((μ‘𝑛) / 𝑛)) − (2 · (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
56	8, 19, 20	subdid 11431	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))) = ((((μ‘𝑛) / 𝑛) · 2) − (((μ‘𝑛) / 𝑛) · (2 · (log‘(𝑥 / 𝑛))))))
57	54, 55, 56	3eqtr4d 2788	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))))
58	57	sumeq2dv 15415	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(2 · (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))))
59	49, 51, 58	3eqtr3d 2786	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))))
60	59	oveq2d 7291	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛)))))))
61	28, 45, 60	3eqtr4d 2788	. . 3 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))))
62	61	mpteq2ia 5177	. 2 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))))
63		ovexd 7310	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) ∈ V)
64		ovexd 7310	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ V)
65		mulog2sum 26685	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
66	65	a1i 11	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥)))) ∈ 𝑂(1))
67		2ex 12050	. . . . . 6 ⊢ 2 ∈ V
68	67	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ V)
69		ovexd 7310	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ V)
70		rpssre 12737	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
71		o1const 15329	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
72	70, 18, 71	mp2an 689	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
73	72	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
74		reex 10962	. . . . . . . . 9 ⊢ ℝ ∈ V
75	74, 70	ssexi 5246	. . . . . . . 8 ⊢ ℝ⁺ ∈ V
76	75	a1i 11	. . . . . . 7 ⊢ (⊤ → ℝ⁺ ∈ V)
77		sumex 15399	. . . . . . . 8 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ V)
79		sumex 15399	. . . . . . . 8 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ V
80	79	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ V)
81		eqidd 2739	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)))
82		eqidd 2739	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))
83	76, 78, 80, 81, 82	offval2 7553	. . . . . 6 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∘_f − (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))
84		mudivsum 26678	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
85		mulogsum 26680	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)
86		o1sub 15325	. . . . . . 7 ⊢ (((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∘_f − (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
87	84, 85, 86	mp2an 689	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∘_f − (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1)
88	83, 87	eqeltrrdi 2848	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ 𝑂(1))
89	68, 69, 73, 88	o1mul2 15334	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))))) ∈ 𝑂(1))
90	63, 64, 66, 89	o1add2 15333	. . 3 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))) ∈ 𝑂(1))
91	90	mptru 1546	. 2 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) + (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))))) ∈ 𝑂(1)
92	62, 91	eqeltri 2835	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 ∧ w3a 1086 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ⊆ wss 3887 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ∘_f cof 7531 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 − cmin 11205 / cdiv 11632 ℕcn 11973 2c2 12028 ℤcz 12319 ℝ⁺crp 12730 ...cfz 13239 ⌊cfl 13510 ↑cexp 13782 𝑂(1)co1 15195 Σcsu 15397 logclog 25710 μcmu 26244

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-o1 15199 df-lo1 15200 df-sum 15398 df-ef 15777 df-e 15778 df-sin 15779 df-cos 15780 df-tan 15781 df-pi 15782 df-dvds 15964 df-gcd 16202 df-prm 16377 df-pc 16538 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-ulm 25536 df-log 25712 df-cxp 25713 df-atan 26017 df-em 26142 df-mu 26250

This theorem is referenced by: selberglem2 26694

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