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Theorem vmadivsum 26052

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 10622	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12390	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5218	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7185	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
6		ovexd 7185	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
7		eqidd 2822	. . . . 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 2822	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
9	4, 5, 6, 7, 8	offval2 7420	. . . 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 1540	. . 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 13335	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
12		elfznn 12930	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
13	12	adantl 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14		vmacl 25689	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
16	15, 13	nndivred 11685	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
17	11, 16	fsumrecl 15085	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
18	17	recnd 10663	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
19		relogcl 25153	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
20	19	recnd 10663	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
21		rprege0 12398	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13184	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23		faccl 13637	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (!‘(⌊‘𝑥)) ∈ ℕ)
24	21, 22, 23	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℕ)
25	24	nnrpd 12423	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℝ⁺)
26	25	relogcld 25200	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
27		rerpdivcl 12413	. . . . . . 7 ⊢ (((log‘(!‘(⌊‘𝑥))) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
28	26, 27	mpancom 686	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
29	28	recnd 10663	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
30	18, 20, 29	nnncan2d 11026	. . . 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 5149	. . 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 2844	. 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 10636	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
34		chpo1ub 26050	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
35	34	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		rpre 12391	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
37		chpcl 25695	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
38	36, 37	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
39		rerpdivcl 12413	. . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	38, 39	mpancom 686	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
41	40	recnd 10663	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
42	41	adantl 484	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
43	18, 29	subcld 10991	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
44	43	adantl 484	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ ℂ)
45	36	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℝ)
46	16, 45	remulcld 10665	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
47		nndivre 11672	. . . . . . . . . . . . . 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 13160	. . . . . . . . . . . . 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 10665	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
52	46, 51	resubcld 11062	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
53	48, 50	resubcld 11062	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
54		1red 10636	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
55		vmage0 25692	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
56	13, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
57		fracle1 13167	. . . . . . . . . . . . 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 11574	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) · 1))
60	15	recnd 10663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	48	recnd 10663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
62	50	recnd 10663	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11090	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (𝑥 / 𝑛)) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
64		rpcn 12393	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
65	64	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
66	13	nnrpd 12423	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
67		rpcnne0 12401	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
69		div23 11311	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = (((Λ‘𝑛) / 𝑛) · 𝑥))
70		divass 11310	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
71	69, 70	eqtr3d 2858	. . . . . . . . . . . . . 14 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
72	60, 65, 68, 71	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
73	72	oveq1d 7165	. . . . . . . . . . . 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 2859	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
75	60	mulid1d 10652	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · 1) = (Λ‘𝑛))
76	59, 74, 75	3brtr3d 5089	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ (Λ‘𝑛))
77	11, 52, 15, 76	fsumle 15148	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
78	16	recnd 10663	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
79	11, 64, 78	fsummulc1 15134	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥))
80		logfac2 25787	. . . . . . . . . . . 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 7168	. . . . . . . . . 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 10663	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
84	51	recnd 10663	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
85	11, 83, 84	fsumsub 15137	. . . . . . . . . 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 2859	. . . . . . . . 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 25693	. . . . . . . . . 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 5090	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥))
90	17, 36	remulcld 10665	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
91	90, 26	resubcld 11062	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ)
92		rpregt0 12397	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
93		lediv1 11499	. . . . . . . . 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 1367	. . . . . . . 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 234	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ≤ ((ψ‘𝑥) / 𝑥))
96	90	recnd 10663	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
97	26	recnd 10663	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
98		rpcnne0 12401	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
99		divsubdir 11328	. . . . . . . . . . 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 1367	. . . . . . . . . 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 12399	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
102	18, 64, 101	divcan4d 11416	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
103	102	oveq1d 7165	. . . . . . . . . 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 2857	. . . . . . . . 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 6668	. . . . . . . 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 12413	. . . . . . . . . 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 686	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ∈ ℝ)
108		flle 13163	. . . . . . . . . . . . . . . 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 11224	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ↔ (⌊‘(𝑥 / 𝑛)) ≤ (𝑥 / 𝑛)))
111	109, 110	mpbird 259	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))))
112	15, 53, 56, 111	mulge0d 11211	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
113	112, 74	breqtrd 5084	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
114	11, 52, 113	fsumge0 15144	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
115	114, 86	breqtrrd 5086	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))))
116		divge0 11503	. . . . . . . . . 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 835	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
118	107, 117	absidd 14776	. . . . . . . 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 2856	. . . . . . 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 25697	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
121	36, 120	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑥))
122		divge0 11503	. . . . . . . . 9 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
123	38, 121, 92, 122	syl21anc 835	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑥) / 𝑥))
124	40, 123	absidd 14776	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘((ψ‘𝑥) / 𝑥)) = ((ψ‘𝑥) / 𝑥))
125	95, 119, 124	3brtr4d 5090	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ≤ (abs‘((ψ‘𝑥) / 𝑥)))
126	125	ad2antrl 726	. . . . 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 15003	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1))
128	127	mptru 1540	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)
129		logfacrlim 25794	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1
130		rlimo1 14967	. . . 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 14966	. . 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 690	. 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 2910	1 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − (log‘𝑥))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ⊤wtru 1534 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ∘_f cof 7401 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℕcn 11632 ℕ₀cn0 11891 ℝ⁺crp 12383 ...cfz 12886 ⌊cfl 13154 !cfa 13627 abscabs 14587 ⇝_𝑟 crli 14836 𝑂(1)co1 14837 Σcsu 15036 logclog 25132 Λcvma 25663 ψcchp 25664

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-o1 14841 df-lo1 14842 df-sum 15037 df-ef 15415 df-e 15416 df-sin 15417 df-cos 15418 df-pi 15420 df-dvds 15602 df-gcd 15838 df-prm 16010 df-pc 16168 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-cxp 25135 df-cht 25668 df-vma 25669 df-chp 25670 df-ppi 25671

This theorem is referenced by: vmadivsumb 26053 rpvmasumlem 26057 vmalogdivsum2 26108 vmalogdivsum 26109 2vmadivsumlem 26110 selberg3lem1 26127 selberg4lem1 26130 pntrsumo1 26135 selbergr 26138

W3C validator