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Theorem vmadivsum 27369

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 11135	. . . . . . 7 ⊢ ℝ ∈ V
2		rpssre 12935	. . . . . . 7 ⊢ ℝ⁺ ⊆ ℝ
3	1, 2	ssexi 5272	. . . . . 6 ⊢ ℝ⁺ ∈ V
4	3	a1i 11	. . . . 5 ⊢ (⊤ → ℝ⁺ ∈ V)
5		ovexd 7404	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
6		ovexd 7404	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥)) ∈ V)
7		eqidd 2730	. . . . 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 2730	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) = (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))))
9	4, 5, 6, 7, 8	offval2 7653	. . . 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 1547	. . 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 13914	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (1...(⌊‘𝑥)) ∈ Fin)
12		elfznn 13490	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
13	12	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
14		vmacl 27004	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ)
15	13, 14	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℝ)
16	15, 13	nndivred 12216	. . . . . . 7 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ)
17	11, 16	fsumrecl 15676	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℝ)
18	17	recnd 11178	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) ∈ ℂ)
19		relogcl 26460	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℝ)
20	19	recnd 11178	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → (log‘𝑥) ∈ ℂ)
21		rprege0 12943	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥))
22		flge0nn0 13758	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ ∧ 0 ≤ 𝑥) → (⌊‘𝑥) ∈ ℕ₀)
23		faccl 14224	. . . . . . . . . 10 ⊢ ((⌊‘𝑥) ∈ ℕ₀ → (!‘(⌊‘𝑥)) ∈ ℕ)
24	21, 22, 23	3syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℕ)
25	24	nnrpd 12969	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (!‘(⌊‘𝑥)) ∈ ℝ⁺)
26	25	relogcld 26508	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℝ)
27		rerpdivcl 12959	. . . . . . 7 ⊢ (((log‘(!‘(⌊‘𝑥))) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
28	26, 27	mpancom 688	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℝ)
29	28	recnd 11178	. . . . 5 ⊢ (𝑥 ∈ ℝ⁺ → ((log‘(!‘(⌊‘𝑥))) / 𝑥) ∈ ℂ)
30	18, 20, 29	nnncan2d 11544	. . . 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 5197	. . 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 2752	. 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 11151	. . . . 5 ⊢ (⊤ → 1 ∈ ℝ)
34		chpo1ub 27367	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1)
35	34	a1i 11	. . . . 5 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ ((ψ‘𝑥) / 𝑥)) ∈ 𝑂(1))
36		rpre 12936	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
37		chpcl 27010	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → (ψ‘𝑥) ∈ ℝ)
38	36, 37	syl 17	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → (ψ‘𝑥) ∈ ℝ)
39		rerpdivcl 12959	. . . . . . . 8 ⊢ (((ψ‘𝑥) ∈ ℝ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
40	38, 39	mpancom 688	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℝ)
41	40	recnd 11178	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
42	41	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ⁺) → ((ψ‘𝑥) / 𝑥) ∈ ℂ)
43	18, 29	subcld 11509	. . . . . 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 11180	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
47		nndivre 12203	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑥 / 𝑛) ∈ ℝ)
48	36, 12, 47	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℝ)
49		reflcl 13734	. . . . . . . . . . . . 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 11180	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
52	46, 51	resubcld 11582	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ∈ ℝ)
53	48, 50	resubcld 11582	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛))) ∈ ℝ)
54		1red 11151	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 1 ∈ ℝ)
55		vmage0 27007	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → 0 ≤ (Λ‘𝑛))
56	13, 55	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ (Λ‘𝑛))
57		fracle1 13741	. . . . . . . . . . . . 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 12099	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) ≤ ((Λ‘𝑛) · 1))
60	15	recnd 11178	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (Λ‘𝑛) ∈ ℂ)
61	48	recnd 11178	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑥 / 𝑛) ∈ ℂ)
62	50	recnd 11178	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (⌊‘(𝑥 / 𝑛)) ∈ ℂ)
63	60, 61, 62	subdid 11610	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = (((Λ‘𝑛) · (𝑥 / 𝑛)) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
64		rpcn 12938	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℂ)
65	64	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑥 ∈ ℂ)
66	13	nnrpd 12969	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℝ⁺)
67		rpcnne0 12946	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ℝ⁺ → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0))
69		div23 11832	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = (((Λ‘𝑛) / 𝑛) · 𝑥))
70		divass 11831	. . . . . . . . . . . . . . 15 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) · 𝑥) / 𝑛) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
71	69, 70	eqtr3d 2766	. . . . . . . . . . . . . 14 ⊢ (((Λ‘𝑛) ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ (𝑛 ∈ ℂ ∧ 𝑛 ≠ 0)) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
72	60, 65, 68, 71	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) = ((Λ‘𝑛) · (𝑥 / 𝑛)))
73	72	oveq1d 7384	. . . . . . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))) = ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
75	60	mulridd 11167	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · 1) = (Λ‘𝑛))
76	59, 74, 75	3brtr3d 5133	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ (Λ‘𝑛))
77	11, 52, 15, 76	fsumle 15741	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))) ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))(Λ‘𝑛))
78	16	recnd 11178	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ)
79	11, 64, 78	fsummulc1 15727	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · 𝑥))
80		logfac2 27104	. . . . . . . . . . . 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 7387	. . . . . . . . . 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 11178	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
84	51	recnd 11178	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛))) ∈ ℂ)
85	11, 83, 84	fsumsub 15730	. . . . . . . . . 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 2767	. . . . . . . . 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 27008	. . . . . . . . . 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 5134	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ≤ (ψ‘𝑥))
90	17, 36	remulcld 11180	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℝ)
91	90, 26	resubcld 11582	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) ∈ ℝ)
92		rpregt0 12942	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℝ ∧ 0 < 𝑥))
93		lediv1 12024	. . . . . . . . 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 1373	. . . . . . . 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 11178	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) ∈ ℂ)
97	26	recnd 11178	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (log‘(!‘(⌊‘𝑥))) ∈ ℂ)
98		rpcnne0 12946	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0))
99		divsubdir 11852	. . . . . . . . . . 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 1373	. . . . . . . . . 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 12944	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ≠ 0)
102	18, 64, 101	divcan4d 11940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) / 𝑥) = Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛))
103	102	oveq1d 7384	. . . . . . . . . 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 2765	. . . . . . . . 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 12959	. . . . . . . . . 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 688	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥) ∈ ℝ)
108		flle 13737	. . . . . . . . . . . . . . . 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 11744	. . . . . . . . . . . . . . 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 11731	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((Λ‘𝑛) · ((𝑥 / 𝑛) − (⌊‘(𝑥 / 𝑛)))))
113	112, 74	breqtrd 5128	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℝ⁺ ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 0 ≤ ((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
114	11, 52, 113	fsumge0 15737	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ Σ𝑛 ∈ (1...(⌊‘𝑥))((((Λ‘𝑛) / 𝑛) · 𝑥) − ((Λ‘𝑛) · (⌊‘(𝑥 / 𝑛)))))
115	114, 86	breqtrrd 5130	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))))
116		divge0 12028	. . . . . . . . . 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 837	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (((Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) · 𝑥) − (log‘(!‘(⌊‘𝑥)))) / 𝑥))
118	107, 117	absidd 15365	. . . . . . . 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 2764	. . . . . . 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 27012	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℝ → 0 ≤ (ψ‘𝑥))
121	36, 120	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ (ψ‘𝑥))
122		divge0 12028	. . . . . . . . 9 ⊢ ((((ψ‘𝑥) ∈ ℝ ∧ 0 ≤ (ψ‘𝑥)) ∧ (𝑥 ∈ ℝ ∧ 0 < 𝑥)) → 0 ≤ ((ψ‘𝑥) / 𝑥))
123	38, 121, 92, 122	syl21anc 837	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ⁺ → 0 ≤ ((ψ‘𝑥) / 𝑥))
124	40, 123	absidd 15365	. . . . . . 7 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘((ψ‘𝑥) / 𝑥)) = ((ψ‘𝑥) / 𝑥))
125	95, 119, 124	3brtr4d 5134	. . . . . 6 ⊢ (𝑥 ∈ ℝ⁺ → (abs‘(Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ≤ (abs‘((ψ‘𝑥) / 𝑥)))
126	125	ad2antrl 728	. . . . 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 15595	. . . 4 ⊢ (⊤ → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1))
128	127	mptru 1547	. . 3 ⊢ (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((Λ‘𝑛) / 𝑛) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ∈ 𝑂(1)
129		logfacrlim 27111	. . . 4 ⊢ (𝑥 ∈ ℝ⁺ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝_𝑟 1
130		rlimo1 15559	. . . 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 15558	. . 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 692	. 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 2825	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 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 Vcvv 3444 class class class wbr 5102 ↦ cmpt 5183 ‘cfv 6499 (class class class)co 7369 ∘_f cof 7631 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 ℕcn 12162 ℕ₀cn0 12418 ℝ⁺crp 12927 ...cfz 13444 ⌊cfl 13728 !cfa 14214 abscabs 15176 ⇝_𝑟 crli 15427 𝑂(1)co1 15428 Σcsu 15628 logclog 26439 Λcvma 26978 ψcchp 26979

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 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

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 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-o1 15432 df-lo1 15433 df-sum 15629 df-ef 16009 df-e 16010 df-sin 16011 df-cos 16012 df-pi 16014 df-dvds 16199 df-gcd 16441 df-prm 16618 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-cmp 23250 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-log 26441 df-cxp 26442 df-cht 26983 df-vma 26984 df-chp 26985 df-ppi 26986

This theorem is referenced by: vmadivsumb 27370 rpvmasumlem 27374 vmalogdivsum2 27425 vmalogdivsum 27426 2vmadivsumlem 27427 selberg3lem1 27444 selberg4lem1 27447 pntrsumo1 27452 selbergr 27455

W3C validator