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Theorem vmadivsum 27445

Description: The sum of the von Mangoldt function over 𝑛 is asymptotic to log𝑥 + 𝑂(1). Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)

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

Distinct variable group: 𝑥,𝑛

**Proof of Theorem vmadivsum**
Step	Hyp	Ref	Expression
1		reex 11129	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12950	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5263	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7402	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
6		ovexd 7402	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
7		eqidd 2737	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
8		eqidd 2737	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
9	4, 5, 6, 7, 8	offval2 7651	. . . 4 ⊢ (⊤ → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))))
10	9	mptru 1549	. . 3 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
11		fzfid 13935	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
12		elfznn 13507	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
13	12	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14		vmacl 27081	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
16	15, 13	nndivred 12231	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
17	11, 16	fsumrecl 15696	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
18	17	recnd 11173	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
19		relogcl 26539	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
20	19	recnd 11173	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
21		rprege0 12958	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13779	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23		faccl 14245	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (!‘(⌊‘𝑥)) ∈ ℕ)
24	21, 22, 23	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℕ)
25	24	nnrpd 12984	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℝ⁺)
26	25	relogcld 26587	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
27		rerpdivcl 12974	. . . . . . 7 ⊢ (((log‘(!‘(⌊‘𝑥))) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
28	26, 27	mpancom 689	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
29	28	recnd 11173	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
30	18, 20, 29	nnncan2d 11540	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
31	30	mpteq2ia 5180	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) − ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
32	10, 31	eqtri 2759	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) = (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥)))
33		1red 11145	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
34		chpo1ub 27443	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
35	34	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		rpre 12951	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
37		chpcl 27087	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
38	36, 37	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
39		rerpdivcl 12974	. . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	38, 39	mpancom 689	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
41	40	recnd 11173	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
42	41	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
43	18, 29	subcld 11505	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
44	43	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
45	36	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
46	16, 45	remulcld 11175	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
47		nndivre 12218	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
48	36, 12, 47	syl2an 597	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
49		reflcl 13755	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℝ)
51	15, 50	remulcld 11175	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
52	46, 51	resubcld 11578	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
53	48, 50	resubcld 11578	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
54		1red 11145	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
55		vmage0 27084	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
56	13, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
57		fracle1 13762	. . . . . . . . . . . . 13 ⊢ ((𝑥 / 𝑛) ∈ ℝ → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
58	48, 57	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ≤ 1)
59	53, 54, 15, 56, 58	lemul2ad 12096	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) · 1))
60	15	recnd 11173	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	48	recnd 11173	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
62	50	recnd 11173	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11606	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (𝑥 / 𝑛)) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
64		rpcn 12953	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
65	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
66	13	nnrpd 12984	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
67		rpcnne0 12961	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
69		div23 11828	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = (((Λ‘𝑛) / 𝑛) · 𝑥))
70		divass 11827	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
71	69, 70	eqtr3d 2773	. . . . . . . . . . . . . 14 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
72	60, 65, 68, 71	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
73	72	oveq1d 7382	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (𝑥 / 𝑛)) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
74	63, 73	eqtr4d 2774	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
75	60	mulridd 11162	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · 1) = (Λ‘𝑛))
76	59, 74, 75	3brtr3d 5116	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ (Λ‘𝑛))
77	11, 52, 15, 76	fsumle 15762	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
78	16	recnd 11173	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
79	11, 64, 78	fsummulc1 15747	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥))
80		logfac2 27180	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (log‘(!‘(⌊‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))))
81	21, 80	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))))
82	79, 81	oveq12d 7385	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
83	46	recnd 11173	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
84	51	recnd 11173	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
85	11, 83, 84	fsumsub 15750	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) = (Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥) − Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
86	82, 85	eqtr4d 2774	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) = Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
87		chpval 27085	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
88	36, 87	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
89	77, 86, 88	3brtr4d 5117	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥))
90	17, 36	remulcld 11175	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
91	90, 26	resubcld 11578	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ)
92		rpregt0 12957	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
93		lediv1 12021	. . . . . . . . 9 ⊢ ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ ∧ (ψ‘𝑥) ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥) ↔ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ≤ ((ψ‘𝑥) / 𝑥)))
94	91, 38, 92, 93	syl3anc 1374	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥) ↔ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ≤ ((ψ‘𝑥) / 𝑥)))
95	89, 94	mpbid 232	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ≤ ((ψ‘𝑥) / 𝑥))
96	90	recnd 11173	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
97	26	recnd 11173	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
98		rpcnne0 12961	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
99		divsubdir 11848	. . . . . . . . . . 11 ⊢ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ ∧ (log‘(!‘(⌊‘𝑥))) ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
100	96, 97, 98, 99	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
101		rpne0 12959	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
102	18, 64, 101	divcan4d 11937	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
103	102	oveq1d 7382	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))
104	100, 103	eqtr2d 2772	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
105	104	fveq2d 6844	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥)))
106		rerpdivcl 12974	. . . . . . . . . 10 ⊢ ((((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ∈ ℝ)
107	91, 106	mpancom 689	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ∈ ℝ)
108		flle 13758	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 / 𝑛) ∈ ℝ → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
109	48, 108	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛))
110	48, 50	subge0d 11740	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ↔ (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛)))
111	109, 110	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
112	15, 53, 56, 111	mulge0d 11727	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
113	112, 74	breqtrd 5111	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
114	11, 52, 113	fsumge0 15758	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
115	114, 86	breqtrrd 5113	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))))
116		divge0 12025	. . . . . . . . . 10 ⊢ (((((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ ∧ 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥))))) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
117	91, 115, 92, 116	syl21anc 838	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
118	107, 117	absidd 15385	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥)) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
119	105, 118	eqtrd 2771	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
120		chpge0 27089	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
121	36, 120	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑥))
122		divge0 12025	. . . . . . . . 9 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
123	38, 121, 92, 122	syl21anc 838	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑥) / 𝑥))
124	40, 123	absidd 15385	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘((ψ‘𝑥) / 𝑥)) = ((ψ‘𝑥) / 𝑥))
125	95, 119, 124	3brtr4d 5117	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ≤ (abs‘((ψ‘𝑥) / 𝑥)))
126	125	ad2antrl 729	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ ℝ⁺ ∧ 1 ≤ 𝑥)) → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ≤ (abs‘((ψ‘𝑥) / 𝑥)))
127	33, 35, 42, 44, 126	o1le 15615	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1))
128	127	mptru 1549	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)
129		logfacrlim 27187	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1
130		rlimo1 15579	. . . 4 ⊢ ((𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1 → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1))
131	129, 130	ax-mp 5	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)
132		o1sub 15578	. . 3 ⊢ (((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1) ∧ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)) → ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) ∈ 𝑂(1))
133	128, 131, 132	mp2an 693	. 2 ⊢ ((𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∘_f − (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)))) ∈ 𝑂(1)
134	32, 133	eqeltrri 2833	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2932 Vcvv 3429 class class class wbr 5085 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 ∘_f cof 7629 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 ℕ₀cn0 12437 ℝ⁺crp 12942 ...cfz 13461 ⌊cfl 13749 !cfa 14235 abscabs 15196 ⇝_𝑟 crli 15447 𝑂(1)co1 15448 Σcsu 15648 logclog 26518 Λcvma 27055 ψcchp 27056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 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 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-fi 9324 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-ioo 13302 df-ioc 13303 df-ico 13304 df-icc 13305 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15029 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15649 df-ef 16032 df-e 16033 df-sin 16034 df-cos 16035 df-pi 16037 df-dvds 16222 df-gcd 16464 df-prm 16641 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 17466 df-qtop 17471 df-imas 17472 df-xps 17474 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-mulg 19044 df-cntz 19292 df-cmn 19757 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-fbas 21349 df-fg 21350 df-cnfld 21353 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-lp 23101 df-perf 23102 df-cn 23192 df-cnp 23193 df-haus 23280 df-cmp 23352 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24285 df-ms 24286 df-tms 24287 df-cncf 24845 df-limc 25833 df-dv 25834 df-log 26520 df-cxp 26521 df-cht 27060 df-vma 27061 df-chp 27062 df-ppi 27063

This theorem is referenced by: vmadivsumb 27446 rpvmasumlem 27450 vmalogdivsum2 27501 vmalogdivsum 27502 2vmadivsumlem 27503 selberg3lem1 27520 selberg4lem1 27523 pntrsumo1 27528 selbergr 27531

W3C validator