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Theorem selbergr 27479

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 11159	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12959	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5277	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7422	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6		ovexd 7422	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)) ∈ V)
7		eqidd 2730	. . . . 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 2730	. . . . 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 7673	. . . 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 27474	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7055	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	13	recnd 11202	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℂ)
15		relogcl 26484	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
16	15	recnd 11202	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
17	14, 16	mulcld 11194	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ)
18		fzfid 13938	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
19		elfznn 13514	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
21		vmacl 27028	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
23	22	recnd 11202	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
24		rpre 12960	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
25		nndivre 12227	. . . . . . . . . . . . 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 27034	. . . . . . . . . . . 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 11202	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℂ)
30	23, 29	mulcld 11194	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
31	18, 30	fsumcl 15699	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
32	17, 31	addcld 11193	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
33		rpcn 12962	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
34		rpne0 12968	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
35	32, 33, 34	divcld 11958	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
36	22, 20	nndivred 12240	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℝ)
37	36	recnd 11202	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℂ)
38	18, 37	fsumcl 15699	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) ∈ ℂ)
39	35, 38, 16	nnncan2d 11568	. . . . 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 27034	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
41	24, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
42	41	recnd 11202	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
43	42, 16	mulcld 11194	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
44	43, 31	addcld 11193	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
45	44, 33, 34	divcld 11958	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
46	45, 16, 16	subsub4d 11564	. . . . . . 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 27473	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
48	47	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
49	42, 33, 16	subdird 11635	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
50	48, 49	eqtrd 2764	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
51	50	oveq1d 7402	. . . . . . . . . . 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 11194	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · (log‘𝑥)) ∈ ℂ)
53	43, 31, 52	addsubd 11554	. . . . . . . . . . 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 2767	. . . . . . . . . 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 7402	. . . . . . . . 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 12970	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
57		divsubdir 11876	. . . . . . . . . 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 11963	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
60	59	oveq2d 7403	. . . . . . . . 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 2768	. . . . . . . 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 7402	. . . . . . 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 12425	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥)))
64	63	oveq2d 7403	. . . . . . 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 2774	. . . . . 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 7402	. . . . 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 11194	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ)
68		divsubdir 11876	. . . . . . 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 11553	. . . . . . . 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 11194	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) ∈ ℂ)
73	18, 30, 72	fsumsub 15754	. . . . . . . . . 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 11202	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℂ)
75	23, 29, 74	subdid 11634	. . . . . . . . . . . 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 12993	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
77		rpdivcl 12978	. . . . . . . . . . . . . . 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 27473	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))))
82	20	nnrpd 12993	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
83		rpcnne0 12970	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℝ⁺ → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
85		div12 11859	. . . . . . . . . . . . . 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 7403	. . . . . . . . . . . 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 2774	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
89	88	sumeq2dv 15668	. . . . . . . . . 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 15750	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑)))
91	90	oveq2d 7403	. . . . . . . . . 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 2775	. . . . . . . . 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 7403	. . . . . . . 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 2764	. . . . . . 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 7402	. . . . . 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 11963	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))
97	96	oveq2d 7403	. . . . . 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 2773	. . . . 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 2772	. . . 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 5202	. . 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 2752	. 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 27462	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
103		vmadivsum 27393	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)
104		o1sub 15582	. . 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 2825	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 2925 Vcvv 3447 ↦ cmpt 5188 ‘cfv 6511 (class class class)co 7387 ∘_f cof 7651 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 ℝ⁺crp 12951 ...cfz 13468 ⌊cfl 13752 𝑂(1)co1 15452 Σcsu 15652 logclog 26463 Λcvma 27002 ψcchp 27003

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 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-o1 15456 df-lo1 15457 df-sum 15653 df-ef 16033 df-e 16034 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-dvds 16223 df-gcd 16465 df-prm 16642 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-ulm 26286 df-log 26465 df-cxp 26466 df-atan 26777 df-em 26903 df-cht 27007 df-vma 27008 df-chp 27009 df-ppi 27010 df-mu 27011

This theorem is referenced by: (None)

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