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Theorem selbergr 27486

Description: Selberg's symmetry formula, using the residual of the second Chebyshev function. Equation 10.6.2 of [Shapiro], p. 428. (Contributed by Mario Carneiro, 16-Apr-2016.)

**Hypothesis**
Ref	Expression
pntrval.r	⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))

**Assertion**
Ref	Expression
selbergr	⊢ (𝑥 ∈ ℝ⁺ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)

Distinct variable groups: 𝑎,𝑑,𝑥 𝑅,𝑑,𝑥 Allowed substitution hint: 𝑅(𝑎)

**Proof of Theorem selbergr**
Step	Hyp	Ref	Expression
1		reex 11166	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12966	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5280	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7425	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6		ovexd 7425	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)) ∈ V)
7		eqidd 2731	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))))
8		eqidd 2731	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
9	4, 5, 6, 7, 8	offval2 7676	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))))
10	9	mptru 1547	. . 3 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
11		pntrval.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑎 ∈ ℝ⁺ ↦ ((ψ‘𝑎) − 𝑎))
12	11	pntrf 27481	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7058	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	13	recnd 11209	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℂ)
15		relogcl 26491	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
16	15	recnd 11209	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
17	14, 16	mulcld 11201	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ)
18		fzfid 13945	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
19		elfznn 13521	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
21		vmacl 27035	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
23	22	recnd 11209	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
24		rpre 12967	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
25		nndivre 12234	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
26	24, 19, 25	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
27		chpcl 27041	. . . . . . . . . . . 12 ⊢ ((𝑥 / 𝑑) ∈ ℝ → (ψ‘(𝑥 / 𝑑)) ∈ ℝ)
28	26, 27	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℝ)
29	28	recnd 11209	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℂ)
30	23, 29	mulcld 11201	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
31	18, 30	fsumcl 15706	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
32	17, 31	addcld 11200	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
33		rpcn 12969	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
34		rpne0 12975	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
35	32, 33, 34	divcld 11965	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
36	22, 20	nndivred 12247	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℝ)
37	36	recnd 11209	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℂ)
38	18, 37	fsumcl 15706	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) ∈ ℂ)
39	35, 38, 16	nnncan2d 11575	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))
40		chpcl 27041	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
41	24, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
42	41	recnd 11209	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
43	42, 16	mulcld 11201	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
44	43, 31	addcld 11200	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
45	44, 33, 34	divcld 11965	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
46	45, 16, 16	subsub4d 11571	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥))))
47	11	pntrval 27480	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
48	47	oveq1d 7405	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
49	42, 33, 16	subdird 11642	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
50	48, 49	eqtrd 2765	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
51	50	oveq1d 7405	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))))
52	33, 16	mulcld 11201	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · (log‘𝑥)) ∈ ℂ)
53	43, 31, 52	addsubd 11561	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) = ((((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))))
54	51, 53	eqtr4d 2768	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) = ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))))
55	54	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥))
56		rpcnne0 12977	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
57		divsubdir 11883	. . . . . . . . . 10 ⊢ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ ∧ (𝑥 · (log‘𝑥)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)))
58	44, 52, 56, 57	syl3anc 1373	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · (log‘𝑥))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)))
59	16, 33, 34	divcan3d 11970	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
60	59	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · (log‘𝑥)) / 𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)))
61	55, 58, 60	3eqtrd 2769	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)))
62	61	oveq1d 7405	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) = ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (log‘𝑥)))
63	16	2timesd 12432	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥)))
64	63	oveq2d 7406	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((log‘𝑥) + (log‘𝑥))))
65	46, 62, 64	3eqtr4d 2775	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) = (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))))
66	65	oveq1d 7405	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (log‘𝑥)) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))))
67	33, 38	mulcld 11201	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ)
68		divsubdir 11883	. . . . . . 7 ⊢ (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ ∧ (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)))
69	32, 67, 56, 68	syl3anc 1373	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)))
70	17, 31, 67	addsubassd 11560	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (((𝑅‘𝑥) · (log‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))))
71	33	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
72	71, 37	mulcld 11201	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) ∈ ℂ)
73	18, 30, 72	fsumsub 15761	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))(((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑))))
74	26	recnd 11209	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℂ)
75	23, 29, 74	subdid 11641	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − ((Λ‘𝑑) · (𝑥 / 𝑑))))
76	19	nnrpd 13000	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
77		rpdivcl 12985	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
78	76, 77	sylan2 593	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
79	11	pntrval 27480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 / 𝑑) ∈ ℝ⁺ → (𝑅‘(𝑥 / 𝑑)) = ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑)))
80	78, 79	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑅‘(𝑥 / 𝑑)) = ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑)))
81	80	oveq2d 7406	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))))
82	20	nnrpd 13000	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
83		rpcnne0 12977	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℝ⁺ → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
85		div12 11866	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (Λ‘𝑑) ∈ ℂ ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
86	71, 23, 84, 85	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
87	86	oveq2d 7406	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − ((Λ‘𝑑) · (𝑥 / 𝑑))))
88	75, 81, 87	3eqtr4d 2775	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
89	88	sumeq2dv 15675	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))(((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
90	18, 33, 37	fsummulc2 15757	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑)))
91	90	oveq2d 7406	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑))))
92	73, 89, 91	3eqtr4rd 2776	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))))
93	92	oveq2d 7406	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))) = (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))))
94	70, 93	eqtrd 2765	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) = (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))))
95	94	oveq1d 7405	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) − (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))) / 𝑥) = ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
96	38, 33, 34	divcan3d 11970	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))
97	96	oveq2d 7406	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥)) = (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)))
98	69, 95, 97	3eqtr3rd 2774	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
99	39, 66, 98	3eqtr3d 2773	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) = ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
100	99	mpteq2ia 5205	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) − (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
101	10, 100	eqtri 2753	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥))
102		selberg2 27469	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
103		vmadivsum 27400	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)
104		o1sub 15589	. . 3 ⊢ (((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) ∈ 𝑂(1))
105	102, 103, 104	mp2an 692	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)))) ∈ 𝑂(1)
106	101, 105	eqeltrri 2826	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑)))) / 𝑥)) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2926 Vcvv 3450 ↦ cmpt 5191 ‘cfv 6514 (class class class)co 7390 ∘_f cof 7654 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 − cmin 11412 / cdiv 11842 ℕcn 12193 2c2 12248 ℝ⁺crp 12958 ...cfz 13475 ⌊cfl 13759 𝑂(1)co1 15459 Σcsu 15659 logclog 26470 Λcvma 27009 ψcchp 27010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-o1 15463 df-lo1 15464 df-sum 15660 df-ef 16040 df-e 16041 df-sin 16042 df-cos 16043 df-tan 16044 df-pi 16045 df-dvds 16230 df-gcd 16472 df-prm 16649 df-pc 16815 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775 df-ulm 26293 df-log 26472 df-cxp 26473 df-atan 26784 df-em 26910 df-cht 27014 df-vma 27015 df-chp 27016 df-ppi 27017 df-mu 27018

This theorem is referenced by: (None)

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