selberg2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > selberg2		Structured version Visualization version GIF version

Theorem selberg2 26604

Description: Selberg's symmetry formula, using the second Chebyshev function. Equation 10.4.14 of [Shapiro], p. 420. (Contributed by Mario Carneiro, 23-May-2016.)

**Assertion**
Ref	Expression
selberg2	⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Distinct variable group: 𝑥,𝑛

**Proof of Theorem selberg2**
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		ovexd 7290	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6		ovexd 7290	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ V)
7		eqidd 2739	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))))
8		eqidd 2739	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
9	4, 5, 6, 7, 8	offval2 7531	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))))
10	9	mptru 1546	. . 3 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
11		fzfid 13621	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
12		elfznn 13214	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
13	12	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14		vmacl 26172	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
16	15	recnd 10934	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
17	13	nnrpd 12699	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
18		relogcl 25636	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
20	19	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
21		rpre 12667	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
22		nndivre 11944	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
23	21, 12, 22	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
24		chpcl 26178	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
26	25	recnd 10934	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
27	20, 26	addcld 10925	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
28	16, 27	mulcld 10926	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ)
29	11, 28	fsumcl 15373	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ)
30		rpcn 12669	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
31		rpne0 12675	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
32	29, 30, 31	divcld 11681	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) ∈ ℂ)
33		2cn 11978	. . . . . . 7 ⊢ 2 ∈ ℂ
34		relogcl 25636	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
35	34	recnd 10934	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
36		mulcl 10886	. . . . . . 7 ⊢ ((2 ∈ ℂ ∧ (log‘𝑥) ∈ ℂ) → (2 · (log‘𝑥)) ∈ ℂ)
37	33, 35, 36	sylancr 586	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) ∈ ℂ)
38	16, 20	mulcld 10926	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
39	11, 38	fsumcl 15373	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
40		chpcl 26178	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
41	21, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
42	41	recnd 10934	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
43	42, 35	mulcld 10926	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
44	39, 43	subcld 11262	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ)
45	44, 30, 31	divcld 11681	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥) ∈ ℂ)
46	32, 37, 45	sub32d 11294	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥))))
47		rpcnne0 12677	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
48		divsubdir 11599	. . . . . . . 8 ⊢ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) ∈ ℂ ∧ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
49	29, 44, 47, 48	syl3anc 1369	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)))
50	16, 20, 26	adddid 10930	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
51	50	sumeq2dv 15343	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
52	16, 26	mulcld 10926	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
53	11, 38, 52	fsumadd 15380	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (log‘𝑛)) + ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
54	51, 53	eqtrd 2778	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
55	54	oveq1d 7270	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))))
56	11, 52	fsumcl 15373	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
57	39, 56, 43	pnncand 11301	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥))))
58	56, 43	addcomd 11107	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((ψ‘𝑥) · (log‘𝑥))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
59	55, 57, 58	3eqtrd 2782	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) = (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))))
60	59	oveq1d 7270	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) − (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥)))) / 𝑥) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
61	49, 60	eqtr3d 2780	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥))
62	61	oveq1d 7270	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
63	46, 62	eqtrd 2778	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
64	63	mpteq2ia 5173	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) − ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
65	10, 64	eqtri 2766	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))))
66		selberg 26601	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
67		selberg2lem 26603	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)
68		o1sub 15253	. . 3 ⊢ (((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥)) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1))
69	66, 67, 68	mp2an 688	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛)) − ((ψ‘𝑥) · (log‘𝑥))) / 𝑥))) ∈ 𝑂(1)
70	65, 69	eqeltrri 2836	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2108 ≠ wne 2942 Vcvv 3422 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ∘_f cof 7509 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 ℝ⁺crp 12659 ...cfz 13168 ⌊cfl 13438 𝑂(1)co1 15123 Σcsu 15325 logclog 25615 Λcvma 26146 ψcchp 26147

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-cht 26151 df-vma 26152 df-chp 26153 df-ppi 26154 df-mu 26155

This theorem is referenced by: selberg2b 26605 selberg3 26612 selberg4 26614 selbergr 26621

W3C validator