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Theorem selberg 27880

Description: Selberg's symmetry formula. The statement has many forms, and this one is equivalent to the statement that Σ𝑛 ≤ 𝑥, Λ(𝑛)log𝑛 + Σ𝑚 · 𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛) = 2𝑥log𝑥 + 𝑂(𝑥). Equation 10.4.10 of [Shapiro], p. 419. (Contributed by Mario Carneiro, 23-May-2016.)

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

Distinct variable group: 𝑥,𝑛

Proof of Theorem selberg Dummy variables 𝑑 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (Λ‘𝑛) = (Λ‘𝑑))
2		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑑 → (𝑥 / 𝑛) = (𝑥 / 𝑑))
3	2	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑑 → (ψ‘(𝑥 / 𝑛)) = (ψ‘(𝑥 / 𝑑)))
4	1, 3	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑑 → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))))
5	4	cbvsumv 15868	. . . . . . . . . . 11 ⊢ Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))
6		fzfid 14119	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (1...(⌊‘(𝑥 / 𝑑))) ∈ Fin)
7		elfznn 13690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
8	7	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
9		vmacl 27450	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
10	8, 9	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
11	10	recnd 11340	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
12		elfznn 13690	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑))) → 𝑚 ∈ ℕ)
13	12	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℕ)
14		vmacl 27450	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ → (Λ‘𝑚) ∈ ℝ)
15	13, 14	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℝ)
16	15	recnd 11340	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘𝑚) ∈ ℂ)
17	6, 11, 16	fsummulc2 15955	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘𝑚)))
18	7	nnrpd 13165	. . . . . . . . . . . . . . . . 17 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
19		rpdivcl 13150	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
20	18, 19	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
21	20	rpred 13167	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
22		chpval 27454	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 / 𝑑) ∈ ℝ → (ψ‘(𝑥 / 𝑑)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚))
23	21, 22	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚))
24	23	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))(Λ‘𝑚)))
25	13	nncnd 12354	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑚 ∈ ℂ)
26	7	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℕ)
27	26	nncnd 12354	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ∈ ℂ)
28	26	nnne0d 12391	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → 𝑑 ≠ 0)
29	25, 27, 28	divcan3d 12101	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑑 · 𝑚) / 𝑑) = 𝑚)
30	29	fveq2d 6889	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (Λ‘((𝑑 · 𝑚) / 𝑑)) = (Λ‘𝑚))
31	30	oveq2d 7436	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = ((Λ‘𝑑) · (Λ‘𝑚)))
32	31	sumeq2dv 15874	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘𝑚)))
33	17, 24, 32	3eqtr4d 2805	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
34	33	sumeq2dv 15874	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
35	5, 34	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
36		fvoveq1 7443	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑑 · 𝑚) → (Λ‘(𝑛 / 𝑑)) = (Λ‘((𝑑 · 𝑚) / 𝑑)))
37	36	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑛 = (𝑑 · 𝑚) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = ((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
38		rpre 13132	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
39		ssrab2 4005	. . . . . . . . . . . . . . . . 17 ⊢ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ ℕ
40		simprr 785	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
41	39, 40	sselid 3906	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℕ)
42	41	anassrs 473	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → 𝑑 ∈ ℕ)
43	42, 9	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘𝑑) ∈ ℝ)
44		elfznn 13690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
45	44	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
46		dvdsdivcl 16491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
47	45, 46	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})
48	39, 47	sselid 3906	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (𝑛 / 𝑑) ∈ ℕ)
49		vmacl 27450	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 / 𝑑) ∈ ℕ → (Λ‘(𝑛 / 𝑑)) ∈ ℝ)
50	48, 49	syl 18	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → (Λ‘(𝑛 / 𝑑)) ∈ ℝ)
51	43, 50	remulcld 11342	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
52	51	recnd 11340	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛}) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
53	52	anasss 472	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
54	37, 38, 53	dvdsflsumcom 27520	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((Λ‘𝑑) · (Λ‘((𝑑 · 𝑚) / 𝑑))))
55	35, 54	eqtr4d 2798	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))))
56	55	oveq1d 7435	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
57		fzfid 14119	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
58		vmacl 27450	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
59	45, 58	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
60	59	recnd 11340	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	44	nnrpd 13165	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℝ⁺)
62		rpdivcl 13150	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ ℝ⁺) → (𝑥 / 𝑛) ∈ ℝ⁺)
63	61, 62	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ⁺)
64	63	rpred 13167	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
65		chpcl 27456	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
66	64, 65	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℝ)
67	66	recnd 11340	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑛)) ∈ ℂ)
68	60, 67	mulcld 11332	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) ∈ ℂ)
69	45	nnrpd 13165	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
70		relogcl 26908	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℝ⁺ → (log‘𝑛) ∈ ℝ)
71	69, 70	syl 18	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℝ)
72	71	recnd 11340	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (log‘𝑛) ∈ ℂ)
73	60, 72	mulcld 11332	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (log‘𝑛)) ∈ ℂ)
74	57, 68, 73	fsumadd 15911	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
75		fzfid 14119	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (1...𝑛) ∈ Fin)
76		dvdsssfz1 16493	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
77	45, 76	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ⊆ (1...𝑛))
78	75, 77	ssfid 9263	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ∈ Fin)
79	78, 51	fsumrecl 15905	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℝ)
80	79	recnd 11340	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) ∈ ℂ)
81	57, 80, 73	fsumadd 15911	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (log‘𝑛))))
82	56, 74, 81	3eqtr4d 2805	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
83	72, 67	addcomd 11515	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((log‘𝑛) + (ψ‘(𝑥 / 𝑛))) = ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛)))
84	83	oveq2d 7436	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛))))
85	60, 67, 72	adddid 11336	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((ψ‘(𝑥 / 𝑛)) + (log‘𝑛))) = (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
86	84, 85	eqtrd 2795	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
87	86	sumeq2dv 15874	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) · (ψ‘(𝑥 / 𝑛))) + ((Λ‘𝑛) · (log‘𝑛))))
88		logsqvma2 27875	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
89	45, 88	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = (Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
90	89	sumeq2dv 15874	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((Λ‘𝑑) · (Λ‘(𝑛 / 𝑑))) + ((Λ‘𝑛) · (log‘𝑛))))
91	82, 87, 90	3eqtr4d 2805	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)))
92		fvoveq1 7443	. . . . . . . . 9 ⊢ (𝑛 = (𝑑 · 𝑚) → (log‘(𝑛 / 𝑑)) = (log‘((𝑑 · 𝑚) / 𝑑)))
93	92	oveq1d 7435	. . . . . . . 8 ⊢ (𝑛 = (𝑑 · 𝑚) → ((log‘(𝑛 / 𝑑))↑2) = ((log‘((𝑑 · 𝑚) / 𝑑))↑2))
94	93	oveq2d 7436	. . . . . . 7 ⊢ (𝑛 = (𝑑 · 𝑚) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
95		mucl 27473	. . . . . . . . . 10 ⊢ (𝑑 ∈ ℕ → (μ‘𝑑) ∈ ℤ)
96	41, 95	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℤ)
97	96	zcnd 12807	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (μ‘𝑑) ∈ ℂ)
98	61	ad2antrl 741	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑛 ∈ ℝ⁺)
99	41	nnrpd 13165	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → 𝑑 ∈ ℝ⁺)
100	98, 99	rpdivcld 13184	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (𝑛 / 𝑑) ∈ ℝ⁺)
101		relogcl 26908	. . . . . . . . . . 11 ⊢ ((𝑛 / 𝑑) ∈ ℝ⁺ → (log‘(𝑛 / 𝑑)) ∈ ℝ)
102	101	recnd 11340	. . . . . . . . . 10 ⊢ ((𝑛 / 𝑑) ∈ ℝ⁺ → (log‘(𝑛 / 𝑑)) ∈ ℂ)
103	100, 102	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → (log‘(𝑛 / 𝑑)) ∈ ℂ)
104	103	sqcld 14290	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((log‘(𝑛 / 𝑑))↑2) ∈ ℂ)
105	97, 104	mulcld 11332	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ (𝑛 ∈ (1...(⌊‘𝑥)) ∧ 𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛})) → ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) ∈ ℂ)
106	94, 38, 105	dvdsflsumcom 27520	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))Σ𝑑 ∈ {𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛} ((μ‘𝑑) · ((log‘(𝑛 / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)))
107	29	fveq2d 6889	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → (log‘((𝑑 · 𝑚) / 𝑑)) = (log‘𝑚))
108	107	oveq1d 7435	. . . . . . . . 9 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((log‘((𝑑 · 𝑚) / 𝑑))↑2) = ((log‘𝑚)↑2))
109	108	oveq2d 7436	. . . . . . . 8 ⊢ (((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) ∧ 𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = ((μ‘𝑑) · ((log‘𝑚)↑2)))
110	109	sumeq2dv 15874	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
111	110	sumeq2dv 15874	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘((𝑑 · 𝑚) / 𝑑))↑2)) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
112	91, 106, 111	3eqtrd 2799	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) = Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)))
113	112	oveq1d 7435	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) = (Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥))
114	113	oveq1d 7435	. . 3 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥))) = ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
115	114	mpteq2ia 5199	. 2 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥))))
116		eqid 2760	. . 3 ⊢ ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑) = ((((log‘(𝑥 / 𝑑))↑2) + (2 − (2 · (log‘(𝑥 / 𝑑))))) / 𝑑)
117	116	selberglem2 27878	. 2 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑑 ∈ (1...(⌊‘𝑥))Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑑)))((μ‘𝑑) · ((log‘𝑚)↑2)) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
118	115, 117	eqeltri 2856	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑥 / 𝑛)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ∈ wcel 2144 {crab 3412 ⊆ wss 3895 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6538 (class class class)co 7420 ℂcc 11201 ℝcr 11202 1c1 11204 + caddc 11206 · cmul 11208 − cmin 11544 / cdiv 11976 ℕcn 12338 2c2 12400 ℤcz 12696 ℝ⁺crp 13123 ...cfz 13642 ⌊cfl 13933 ↑cexp 14207 𝑂(1)co1 15658 Σcsu 15858 ∥ cdvds 16427 logclog 26887 Λcvma 27424 ψcchp 27425 μcmu 27427

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9645 ax-cnex 11259 ax-resscn 11260 ax-1cn 11261 ax-icn 11262 ax-addcl 11263 ax-addrcl 11264 ax-mulcl 11265 ax-mulrcl 11266 ax-mulcom 11267 ax-addass 11268 ax-mulass 11269 ax-distr 11270 ax-i2m1 11271 ax-1ne0 11272 ax-1rid 11273 ax-rnegex 11274 ax-rrecex 11275 ax-cnre 11276 ax-pre-lttri 11277 ax-pre-lttrn 11278 ax-pre-ltadd 11279 ax-pre-mulgt0 11280 ax-pre-sup 11281 ax-addf 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-disj 5070 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8180 df-frecs 8302 df-wrecs 8333 df-recs 8382 df-rdg 8421 df-1o 8479 df-2o 8480 df-oadd 8483 df-er 8720 df-map 8852 df-pm 8853 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9357 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9507 df-dju 9985 df-card 10023 df-pnf 11348 df-mnf 11349 df-xr 11350 df-ltxr 11351 df-le 11352 df-sub 11546 df-neg 11547 df-div 11977 df-nn 12339 df-2 12408 df-3 12409 df-4 12410 df-5 12411 df-6 12412 df-7 12413 df-8 12414 df-9 12415 df-n0 12610 df-xnn0 12683 df-z 12697 df-dec 12818 df-uz 12969 df-q 13079 df-rp 13124 df-xneg 13244 df-xadd 13245 df-xmul 13246 df-ioo 13483 df-ioc 13484 df-ico 13485 df-icc 13486 df-fz 13643 df-fzo 13792 df-fl 13935 df-mod 14013 df-seq 14148 df-exp 14208 df-fac 14421 df-bc 14450 df-hash 14478 df-shft 15225 df-cj 15271 df-re 15272 df-im 15273 df-sqrt 15407 df-abs 15408 df-limsup 15643 df-clim 15660 df-rlim 15661 df-o1 15662 df-lo1 15663 df-sum 15859 df-ef 16238 df-e 16239 df-sin 16240 df-cos 16241 df-tan 16242 df-pi 16243 df-dvds 16428 df-gcd 16670 df-prm 16852 df-pc 17020 df-struct 17330 df-sets 17347 df-slot 17365 df-ndx 17377 df-base 17393 df-ress 17414 df-plusg 17446 df-mulr 17447 df-starv 17448 df-sca 17449 df-vsca 17450 df-ip 17451 df-tset 17452 df-ple 17453 df-ds 17455 df-unif 17456 df-hom 17457 df-cco 17458 df-rest 17598 df-topn 17599 df-0g 17617 df-gsum 17618 df-topgen 17619 df-pt 17620 df-prds 17623 df-xrs 17679 df-qtop 17684 df-imas 17685 df-xps 17687 df-mre 17761 df-mrc 17762 df-acs 17764 df-mgm 18821 df-sgrp 18913 df-mnd 18929 df-submnd 18984 df-mulg 19283 df-cntz 19536 df-cmn 20001 df-psmet 21675 df-xmet 21676 df-met 21677 df-bl 21678 df-mopn 21679 df-fbas 21680 df-fg 21681 df-cnfld 21684 df-top 23217 df-topon 23234 df-topsp 23256 df-bases 23269 df-cld 23342 df-ntr 23343 df-cls 23344 df-nei 23421 df-lp 23459 df-perf 23460 df-cn 23550 df-cnp 23551 df-haus 23638 df-cmp 23710 df-tx 23886 df-hmeo 24079 df-fil 24170 df-fm 24262 df-flim 24263 df-flf 24264 df-xms 24644 df-ms 24645 df-tms 24646 df-cncf 25204 df-limc 26191 df-dv 26192 df-ulm 26709 df-log 26889 df-cxp 26890 df-atan 27200 df-em 27325 df-vma 27430 df-chp 27431 df-mu 27433

This theorem is used by: selbergb 27881 selberg2 27883 selbergs 27906

W3C validator