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Theorem vmadivsum 25171

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 10027	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 11843	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 4803	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 6680	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
6		ovexd 6680	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
7		eqidd 2623	. . . . 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 2623	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
9	4, 5, 6, 7, 8	offval2 6914	. . . 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	trud 1493	. . 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 12772	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
12		elfznn 12370	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
13	12	adantl 482	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14		vmacl 24844	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
16	15, 13	nndivred 11069	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
17	11, 16	fsumrecl 14465	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
18	17	recnd 10068	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
19		relogcl 24322	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
20	19	recnd 10068	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
21		rprege0 11847	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 12621	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23		faccl 13070	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (!‘(⌊‘𝑥)) ∈ ℕ)
24	21, 22, 23	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℕ)
25	24	nnrpd 11870	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℝ⁺)
26	25	relogcld 24369	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
27		rerpdivcl 11861	. . . . . . 7 ⊢ (((log‘(!‘(⌊‘𝑥))) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
28	26, 27	mpancom 703	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
29	28	recnd 10068	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
30	18, 20, 29	nnncan2d 10427	. . . 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 4740	. . 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 2644	. 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 10055	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
34		chpo1ub 25169	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
35	34	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		rpre 11839	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
37		chpcl 24850	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
38	36, 37	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
39		rerpdivcl 11861	. . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	38, 39	mpancom 703	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
41	40	recnd 10068	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
42	41	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
43	18, 29	subcld 10392	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
44	43	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
45	36	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
46	16, 45	remulcld 10070	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
47		nndivre 11056	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
48	36, 12, 47	syl2an 494	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
49		reflcl 12597	. . . . . . . . . . . . 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 10070	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
52	46, 51	resubcld 10458	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
53	48, 50	resubcld 10458	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
54		1red 10055	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
55		vmage0 24847	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
56	13, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
57		fracle1 12604	. . . . . . . . . . . . 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 10964	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) · 1))
60	15	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	48	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
62	50	recnd 10068	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 10486	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (𝑥 / 𝑛)) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
64		rpcn 11841	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
65	64	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
66	13	nnrpd 11870	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
67		rpcnne0 11850	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
69		div23 10704	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = (((Λ‘𝑛) / 𝑛) · 𝑥))
70		divass 10703	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
71	69, 70	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
72	60, 65, 68, 71	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
73	72	oveq1d 6665	. . . . . . . . . . . 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 2659	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
75	60	mulid1d 10057	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · 1) = (Λ‘𝑛))
76	59, 74, 75	3brtr3d 4684	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ (Λ‘𝑛))
77	11, 52, 15, 76	fsumle 14531	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
78	16	recnd 10068	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
79	11, 64, 78	fsummulc1 14517	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥))
80		logfac2 24942	. . . . . . . . . . . 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 6668	. . . . . . . . . 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 10068	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
84	51	recnd 10068	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
85	11, 83, 84	fsumsub 14520	. . . . . . . . . 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 2659	. . . . . . . . 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 24848	. . . . . . . . . 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 4685	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥))
90	17, 36	remulcld 10070	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
91	90, 26	resubcld 10458	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ)
92		rpregt0 11846	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
93		lediv1 10888	. . . . . . . . 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 1326	. . . . . . . 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 222	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ≤ ((ψ‘𝑥) / 𝑥))
96	90	recnd 10068	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
97	26	recnd 10068	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
98		rpcnne0 11850	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
99		divsubdir 10721	. . . . . . . . . . 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 1326	. . . . . . . . . 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 11848	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
102	18, 64, 101	divcan4d 10807	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
103	102	oveq1d 6665	. . . . . . . . . 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 2657	. . . . . . . . 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 6195	. . . . . . . 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 11861	. . . . . . . . . 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 703	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ∈ ℝ)
108		flle 12600	. . . . . . . . . . . . . . . 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 10617	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ↔ (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛)))
111	109, 110	mpbird 247	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
112	15, 53, 56, 111	mulge0d 10604	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
113	112, 74	breqtrd 4679	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
114	11, 52, 113	fsumge0 14527	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
115	114, 86	breqtrrd 4681	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))))
116		divge0 10892	. . . . . . . . . 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 1325	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
118	107, 117	absidd 14161	. . . . . . . 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 2656	. . . . . . 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 24852	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
121	36, 120	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑥))
122		divge0 10892	. . . . . . . . 9 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
123	38, 121, 92, 122	syl21anc 1325	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑥) / 𝑥))
124	40, 123	absidd 14161	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘((ψ‘𝑥) / 𝑥)) = ((ψ‘𝑥) / 𝑥))
125	95, 119, 124	3brtr4d 4685	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ≤ (abs‘((ψ‘𝑥) / 𝑥)))
126	125	ad2antrl 764	. . . . 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 14383	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1))
128	127	trud 1493	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)
129		logfacrlim 24949	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1
130		rlimo1 14347	. . . 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 14346	. . 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 708	. 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 2698	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ∘_𝑓 cof 6895 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℝ⁺crp 11832 ...cfz 12326 ⌊cfl 12591 !cfa 13060 abscabs 13974 ⇝_𝑟 crli 14216 𝑂(1)co1 14217 Σcsu 14416 logclog 24301 Λcvma 24818 ψcchp 24819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-o1 14221 df-lo1 14222 df-sum 14417 df-ef 14798 df-e 14799 df-sin 14800 df-cos 14801 df-pi 14803 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-cxp 24304 df-cht 24823 df-vma 24824 df-chp 24825 df-ppi 24826

This theorem is referenced by: vmadivsumb 25172 rpvmasumlem 25176 vmalogdivsum2 25227 vmalogdivsum 25228 2vmadivsumlem 25229 selberg3lem1 25246 selberg4lem1 25249 pntrsumo1 25254 selbergr 25257

W3C validator