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Theorem selbergr 27630

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 11275	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 13064	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5340	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6		ovexd 7483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)) ∈ V)
7		eqidd 2741	. . . . 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 2741	. . . . 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 7734	. . . 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 1544	. . 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 27625	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7117	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	13	recnd 11318	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℂ)
15		relogcl 26635	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
16	15	recnd 11318	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
17	14, 16	mulcld 11310	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ)
18		fzfid 14024	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
19		elfznn 13613	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
21		vmacl 27179	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
23	22	recnd 11318	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
24		rpre 13065	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
25		nndivre 12334	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ ∧ 𝑑 ∈ ℕ) → (𝑥 / 𝑑) ∈ ℝ)
26	24, 19, 25	syl2an 595	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ)
27		chpcl 27185	. . . . . . . . . . . 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 11318	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℂ)
30	23, 29	mulcld 11310	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
31	18, 30	fsumcl 15781	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
32	17, 31	addcld 11309	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
33		rpcn 13067	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
34		rpne0 13073	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
35	32, 33, 34	divcld 12070	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
36	22, 20	nndivred 12347	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℝ)
37	36	recnd 11318	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℂ)
38	18, 37	fsumcl 15781	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) ∈ ℂ)
39	35, 38, 16	nnncan2d 11682	. . . . 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 27185	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
41	24, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
42	41	recnd 11318	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
43	42, 16	mulcld 11310	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
44	43, 31	addcld 11309	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
45	44, 33, 34	divcld 12070	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
46	45, 16, 16	subsub4d 11678	. . . . . . 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 27624	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
48	47	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
49	42, 33, 16	subdird 11747	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
50	48, 49	eqtrd 2780	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
51	50	oveq1d 7463	. . . . . . . . . . 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 11310	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · (log‘𝑥)) ∈ ℂ)
53	43, 31, 52	addsubd 11668	. . . . . . . . . . 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 2783	. . . . . . . . . 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 7463	. . . . . . . . 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 13075	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
57		divsubdir 11988	. . . . . . . . . 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 1371	. . . . . . . . 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 12075	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
60	59	oveq2d 7464	. . . . . . . . 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 2784	. . . . . . . 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 7463	. . . . . . 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 12536	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥)))
64	63	oveq2d 7464	. . . . . . 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 2790	. . . . . 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 7463	. . . . 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 11310	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ)
68		divsubdir 11988	. . . . . . 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 1371	. . . . . 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 11667	. . . . . . . 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 11310	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) ∈ ℂ)
73	18, 30, 72	fsumsub 15836	. . . . . . . . . 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 11318	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℂ)
75	23, 29, 74	subdid 11746	. . . . . . . . . . . 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 13097	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
77		rpdivcl 13082	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ ℝ⁺) → (𝑥 / 𝑑) ∈ ℝ⁺)
78	76, 77	sylan2 592	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℝ⁺)
79	11	pntrval 27624	. . . . . . . . . . . . . 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 7464	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))))
82	20	nnrpd 13097	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
83		rpcnne0 13075	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℝ⁺ → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
85		div12 11971	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℂ ∧ (Λ‘𝑑) ∈ ℂ ∧ (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0)) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
86	71, 23, 84, 85	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) = ((Λ‘𝑑) · (𝑥 / 𝑑)))
87	86	oveq2d 7464	. . . . . . . . . . . 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 2790	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
89	88	sumeq2dv 15750	. . . . . . . . . 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 15832	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑)))
91	90	oveq2d 7464	. . . . . . . . . 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 2791	. . . . . . . . 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 7464	. . . . . . . 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 2780	. . . . . . 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 7463	. . . . . 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 12075	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))
97	96	oveq2d 7464	. . . . . 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 2789	. . . . 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 2788	. . . 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 5269	. . 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 2768	. 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 27613	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
103		vmadivsum 27544	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)
104		o1sub 15662	. . 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 691	. 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 2841	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 1537 ⊤wtru 1538 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ∘_f cof 7712 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 − cmin 11520 / cdiv 11947 ℕcn 12293 2c2 12348 ℝ⁺crp 13057 ...cfz 13567 ⌊cfl 13841 𝑂(1)co1 15532 Σcsu 15734 logclog 26614 Λcvma 27153 ψcchp 27154

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-fac 14323 df-bc 14352 df-hash 14380 df-shft 15116 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-o1 15536 df-lo1 15537 df-sum 15735 df-ef 16115 df-e 16116 df-sin 16117 df-cos 16118 df-tan 16119 df-pi 16120 df-dvds 16303 df-gcd 16541 df-prm 16719 df-pc 16884 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-ulm 26438 df-log 26616 df-cxp 26617 df-atan 26928 df-em 27054 df-cht 27158 df-vma 27159 df-chp 27160 df-ppi 27161 df-mu 27162

This theorem is referenced by: (None)

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