selbergr - Metamath Proof Explorer

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

Theorem selbergr 27535

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 11117	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12913	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5267	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7393	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥))) ∈ V)
6		ovexd 7393	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥)) ∈ V)
7		eqidd 2737	. . . . 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 2737	. . . . 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 7642	. . . 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 1548	. . 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 27530	. . . . . . . . . . 11 ⊢ 𝑅:ℝ⁺⟶ℝ
13	12	ffvelcdmi 7028	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℝ)
14	13	recnd 11160	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) ∈ ℂ)
15		relogcl 26540	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
16	15	recnd 11160	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
17	14, 16	mulcld 11152	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) ∈ ℂ)
18		fzfid 13896	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
19		elfznn 13469	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℕ)
20	19	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℕ)
21		vmacl 27084	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ℕ → (Λ‘𝑑) ∈ ℝ)
22	20, 21	syl 17	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℝ)
23	22	recnd 11160	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑑) ∈ ℂ)
24		rpre 12914	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
25		nndivre 12186	. . . . . . . . . . . . 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 27090	. . . . . . . . . . . 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 11160	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (ψ‘(𝑥 / 𝑑)) ∈ ℂ)
30	23, 29	mulcld 11152	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
31	18, 30	fsumcl 15656	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) ∈ ℂ)
32	17, 31	addcld 11151	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
33		rpcn 12916	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
34		rpne0 12922	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
35	32, 33, 34	divcld 11917	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((((𝑅‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
36	22, 20	nndivred 12199	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℝ)
37	36	recnd 11160	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) / 𝑑) ∈ ℂ)
38	18, 37	fsumcl 15656	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) ∈ ℂ)
39	35, 38, 16	nnncan2d 11527	. . . . 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 27090	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
41	24, 40	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
42	41	recnd 11160	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℂ)
43	42, 16	mulcld 11152	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) · (log‘𝑥)) ∈ ℂ)
44	43, 31	addcld 11151	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) ∈ ℂ)
45	44, 33, 34	divcld 11917	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) ∈ ℂ)
46	45, 16, 16	subsub4d 11523	. . . . . . 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 27529	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ⁺ → (𝑅‘𝑥) = ((ψ‘𝑥) − 𝑥))
48	47	oveq1d 7373	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) − 𝑥) · (log‘𝑥)))
49	42, 33, 16	subdird 11594	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ⁺ → (((ψ‘𝑥) − 𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
50	48, 49	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑅‘𝑥) · (log‘𝑥)) = (((ψ‘𝑥) · (log‘𝑥)) − (𝑥 · (log‘𝑥))))
51	50	oveq1d 7373	. . . . . . . . . . 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 11152	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · (log‘𝑥)) ∈ ℂ)
53	43, 31, 52	addsubd 11513	. . . . . . . . . . 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 2774	. . . . . . . . . 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 7373	. . . . . . . . 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 12924	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
57		divsubdir 11835	. . . . . . . . . 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 11922	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · (log‘𝑥)) / 𝑥) = (log‘𝑥))
60	59	oveq2d 7374	. . . . . . . . 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 2775	. . . . . . . 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 7373	. . . . . . 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 12384	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (2 · (log‘𝑥)) = ((log‘𝑥) + (log‘𝑥)))
64	63	oveq2d 7374	. . . . . . 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 2781	. . . . . 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 7373	. . . . 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 11152	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) ∈ ℂ)
68		divsubdir 11835	. . . . . . 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 11512	. . . . . . . 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 11152	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 · ((Λ‘𝑑) / 𝑑)) ∈ ℂ)
73	18, 30, 72	fsumsub 15711	. . . . . . . . . 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 11160	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑑) ∈ ℂ)
75	23, 29, 74	subdid 11593	. . . . . . . . . . . 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 12947	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ (1...(⌊‘𝑥)) → 𝑑 ∈ ℝ⁺)
77		rpdivcl 12932	. . . . . . . . . . . . . . 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 27529	. . . . . . . . . . . . . 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 7374	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = ((Λ‘𝑑) · ((ψ‘(𝑥 / 𝑑)) − (𝑥 / 𝑑))))
82	20	nnrpd 12947	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → 𝑑 ∈ ℝ⁺)
83		rpcnne0 12924	. . . . . . . . . . . . . . 15 ⊢ (𝑑 ∈ ℝ⁺ → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
84	82, 83	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → (𝑑 ∈ ℂ ∧ 𝑑 ≠ 0))
85		div12 11818	. . . . . . . . . . . . . 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 7374	. . . . . . . . . . . 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 2781	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑑 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑑) · (𝑅‘(𝑥 / 𝑑))) = (((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑))) − (𝑥 · ((Λ‘𝑑) / 𝑑))))
89	88	sumeq2dv 15625	. . . . . . . . . 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 15707	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) = Σ𝑑 ∈ (1...(⌊‘𝑥))(𝑥 · ((Λ‘𝑑) / 𝑑)))
91	90	oveq2d 7374	. . . . . . . . . 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 2782	. . . . . . . . 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 7374	. . . . . . . 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 2771	. . . . . . 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 7373	. . . . . 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 11922	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((𝑥 · Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑)) / 𝑥) = Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑))
97	96	oveq2d 7374	. . . . . 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 2780	. . . . 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 2779	. . . 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 5193	. . 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 2759	. 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 27518	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (((((ψ‘𝑥) · (log‘𝑥)) + Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) · (ψ‘(𝑥 / 𝑑)))) / 𝑥) − (2 · (log‘𝑥)))) ∈ 𝑂(1)
103		vmadivsum 27449	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑑 ∈ (1...(⌊‘𝑥))((Λ‘𝑑) / 𝑑) − (log‘𝑥))) ∈ 𝑂(1)
104		o1sub 15539	. . 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 2833	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 1541 ⊤wtru 1542 ∈ wcel 2113 ≠ wne 2932 Vcvv 3440 ↦ cmpt 5179 ‘cfv 6492 (class class class)co 7358 ∘_f cof 7620 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 − cmin 11364 / cdiv 11794 ℕcn 12145 2c2 12200 ℝ⁺crp 12905 ...cfz 13423 ⌊cfl 13710 𝑂(1)co1 15409 Σcsu 15609 logclog 26519 Λcvma 27058 ψcchp 27059

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9813 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-o1 15413 df-lo1 15414 df-sum 15610 df-ef 15990 df-e 15991 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-dvds 16180 df-gcd 16422 df-prm 16599 df-pc 16765 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-ulm 26342 df-log 26521 df-cxp 26522 df-atan 26833 df-em 26959 df-cht 27063 df-vma 27064 df-chp 27065 df-ppi 27066 df-mu 27067

This theorem is referenced by: (None)

W3C validator