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Theorem selberglem1 27048

Description: Lemma for selberg 27051. Estimation of the asymptotic part of selberglem3 27050. (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 13938	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
2		elfznn 13530	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
3	2	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
4		mucl 26645	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
6	5	zred 12666	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
7	6, 3	nndivred 12266	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
8	7	recnd 11242	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
9	2	nnrpd 13014	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
10		rpdivcl 12999	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
11	9, 10	sylan2 594	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
12		relogcl 26084	. . . . . . . . . 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 11242	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘(𝑥 / 𝑛)) ∈ ℂ)
15	14	sqcld 14109	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘(𝑥 / 𝑛))↑2) ∈ ℂ)
16	8, 15	mulcld 11234	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) ∈ ℂ)
17	1, 16	fsumcl 15679	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) ∈ ℂ)
18		2cn 12287	. . . . . . . . 9 ⊢ 2 ∈ ℂ
19	18	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 2 ∈ ℂ)
20	19, 14	mulcld 11234	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (log‘(𝑥 / 𝑛))) ∈ ℂ)
21	19, 20	subcld 11571	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 − (2 · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
22	8, 21	mulcld 11234	. . . . . 6 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
23	1, 22	fsumcl 15679	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
24		relogcl 26084	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
25	24	recnd 11242	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
26		mulcl 11194	. . . . . 6 ⊢ ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
27	18, 25, 26	sylancr 588	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) ∈ ℂ)
28	17, 23, 27	addsubd 11592	. . . 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 7420	. . . . . . . 8 ⊢ ((μ‘𝑛) · 𝑇) = ((μ‘𝑛) · ((((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) / 𝑛))
31	5	zcnd 12667	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℂ)
32	15, 21	addcld 11233	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((log‘(𝑥 / 𝑛))↑2) + (2 − (2 · (log‘(𝑥 / 𝑛))))) ∈ ℂ)
33	3	nnrpd 13014	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
34	33	rpcnne0d 13025	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
35		divass 11890	. . . . . . . . . . 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 11891	. . . . . . . . . . 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 2775	. . . . . . . . . 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 1372	. . . . . . . . 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 11238	. . . . . . . . 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 2773	. . . . . . . 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 2785	. . . . . . 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 15649	. . . . . 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 15686	. . . . . 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 2773	. . . . 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 7424	. . . 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 11234	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))) ∈ ℂ)
48	8, 47	subcld 11571	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((μ‘𝑛) / 𝑛) − (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ ℂ)
49	1, 46, 48	fsummulc2 15730	. . . . . 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 15734	. . . . . . 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 7425	. . . . . 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 11235	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · ((μ‘𝑛) / 𝑛)) = (((μ‘𝑛) / 𝑛) · 2))
53	19, 8, 14	mul12d 11423	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (2 · (((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) = (((μ‘𝑛) / 𝑛) · (2 · (log‘(𝑥 / 𝑛)))))
54	52, 53	oveq12d 7427	. . . . . . . 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 11670	. . . . . . . 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 11670	. . . . . . . 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 2783	. . . . . . 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 15649	. . . . . 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 2781	. . . . 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 7425	. . . 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 2783	. . 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 5252	. 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 7444	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · ((log‘(𝑥 / 𝑛))↑2)) − (2 · (log‘𝑥))) ∈ V)
64		ovexd 7444	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (2 · (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛))))) ∈ V)
65		mulog2sum 27040	. . . . 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 12289	. . . . . 6 ⊢ 2 ∈ V
68	67	a1i 11	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → 2 ∈ V)
69		ovexd 7444	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) − Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ V)
70		rpssre 12981	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
71		o1const 15564	. . . . . . 7 ⊢ ((ℝ⁺ ⊆ ℝ ∧ 2 ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
72	70, 18, 71	mp2an 691	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1)
73	72	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ 2) ∈ 𝑂(1))
74		reex 11201	. . . . . . . . 9 ⊢ ℝ ∈ V
75	74, 70	ssexi 5323	. . . . . . . 8 ⊢ ℝ⁺ ∈ V
76	75	a1i 11	. . . . . . 7 ⊢ (⊤ → ℝ⁺ ∈ V)
77		sumex 15634	. . . . . . . 8 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ V
78	77	a1i 11	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛) ∈ V)
79		sumex 15634	. . . . . . . 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 2734	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)))
82		eqidd 2734	. . . . . . 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 7690	. . . . . 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 27033	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) / 𝑛)) ∈ 𝑂(1)
85		mulogsum 27035	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))(((μ‘𝑛) / 𝑛) · (log‘(𝑥 / 𝑛)))) ∈ 𝑂(1)
86		o1sub 15560	. . . . . . 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 691	. . . . . 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 2843	. . . . 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 15569	. . . 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 15568	. . 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 1549	. 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 2830	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((μ‘𝑛) · 𝑇) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ⊆ wss 3949 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 ∘_f cof 7668 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 − cmin 11444 / cdiv 11871 ℕcn 12212 2c2 12267 ℤcz 12558 ℝ⁺crp 12974 ...cfz 13484 ⌊cfl 13755 ↑cexp 14027 𝑂(1)co1 15430 Σcsu 15632 logclog 26063 μcmu 26599

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-o1 15434 df-lo1 15435 df-sum 15633 df-ef 16011 df-e 16012 df-sin 16013 df-cos 16014 df-tan 16015 df-pi 16016 df-dvds 16198 df-gcd 16436 df-prm 16609 df-pc 16770 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-limc 25383 df-dv 25384 df-ulm 25889 df-log 26065 df-cxp 26066 df-atan 26372 df-em 26497 df-mu 26605

This theorem is referenced by: selberglem2 27049

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