harmonicbnd4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > harmonicbnd4		Structured version Visualization version GIF version

Theorem harmonicbnd4 26988

Description: The asymptotic behavior of Σ𝑚 ≤ 𝐴, 1 / 𝑚 = log𝐴 + γ + 𝑂(1 / 𝐴). (Contributed by Mario Carneiro, 14-May-2016.)

**Assertion**
Ref	Expression
harmonicbnd4	⊢ (𝐴 ∈ ℝ⁺ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))

Distinct variable group: 𝐴,𝑚

**Proof of Theorem harmonicbnd4**
Step	Hyp	Ref	Expression
1		fzfid 13926	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13498	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
4	3	nnrecred 12219	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
5	1, 4	fsumrecl 15687	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
6	5	recnd 11164	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
7		relogcl 26552	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
8	7	recnd 11164	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
9		emre 26983	. . . . . 6 ⊢ γ ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℝ)
11	10	recnd 11164	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℂ)
12	6, 8, 11	subsub4d 11527	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))
13	12	fveq2d 6838	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
14		rpreccl 12961	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
15	14	rpred 12977	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
16		resubcl 11449	. . . . 5 ⊢ ((γ ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (γ − (1 / 𝐴)) ∈ ℝ)
17	9, 15, 16	sylancr 588	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ∈ ℝ)
18		rprege0 12949	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
19		flge0nn0 13770	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
21		nn0p1nn 12467	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
23	22	nnrpd 12975	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
24		relogcl 26552	. . . . . 6 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
25	23, 24	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
26	5, 25	resubcld 11569	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
27	5, 7	resubcld 11569	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∈ ℝ)
28	22	nnrecred 12219	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℝ)
29		fzfid 13926	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (1...((⌊‘𝐴) + 1)) ∈ Fin)
30		elfznn 13498	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...((⌊‘𝐴) + 1)) → 𝑚 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → 𝑚 ∈ ℕ)
32	31	nnrecred 12219	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℝ)
33	29, 32	fsumrecl 15687	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℝ)
34	33, 25	resubcld 11569	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
35		harmonicbnd 26981	. . . . . . . 8 ⊢ (((⌊‘𝐴) + 1) ∈ ℕ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1))
36	22, 35	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1))
37		1re 11135	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38	9, 37	elicc2i 13356	. . . . . . . 8 ⊢ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1) ↔ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ ∧ γ ≤ (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∧ (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ 1))
39	38	simp2bi 1147	. . . . . . 7 ⊢ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1) → γ ≤ (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
40	36, 39	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → γ ≤ (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
41		rpre 12942	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42		fllep1 13751	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
43	41, 42	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
44		rpregt0 12948	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45	22	nnred 12180	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ)
46	22	nngt0d 12217	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < ((⌊‘𝐴) + 1))
47		lerec 12030	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (((⌊‘𝐴) + 1) ∈ ℝ ∧ 0 < ((⌊‘𝐴) + 1))) → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (1 / ((⌊‘𝐴) + 1)) ≤ (1 / 𝐴)))
48	44, 45, 46, 47	syl12anc 837	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (1 / ((⌊‘𝐴) + 1)) ≤ (1 / 𝐴)))
49	43, 48	mpbid 232	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ≤ (1 / 𝐴))
50	10, 28, 34, 15, 40, 49	le2subd 11761	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))))
51	33	recnd 11164	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℂ)
52	25	recnd 11164	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℂ)
53	28	recnd 11164	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℂ)
54	51, 52, 53	sub32d 11528	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))) = ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) − (log‘((⌊‘𝐴) + 1))))
55		nnuz 12818	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
56	22, 55	eleqtrdi 2847	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ (ℤ_≥‘1))
57	32	recnd 11164	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℂ)
58		oveq2 7368	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝐴) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝐴) + 1)))
59	56, 57, 58	fsumm1 15704	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
60	20	nn0cnd 12491	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
61		ax-1cn 11087	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
62		pncan 11390	. . . . . . . . . . . . 13 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
63	60, 61, 62	sylancl 587	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
64	63	oveq2d 7376	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1...(((⌊‘𝐴) + 1) − 1)) = (1...(⌊‘𝐴)))
65	64	sumeq1d 15653	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
66	65	oveq1d 7375	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
67	59, 66	eqtrd 2772	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
68	6, 53, 67	mvrraddd 11553	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
69	68	oveq1d 7375	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) − (log‘((⌊‘𝐴) + 1))) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
70	54, 69	eqtrd 2772	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
71	50, 70	breqtrd 5112	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
72		logleb 26580	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ ((⌊‘𝐴) + 1) ∈ ℝ⁺) → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
73	23, 72	mpdan 688	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
74	43, 73	mpbid 232	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1)))
75	7, 25, 5, 74	lesub2dd 11758	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
76	17, 26, 27, 71, 75	letrd 11294	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
77	27, 15	resubcld 11569	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ∈ ℝ)
78	15	recnd 11164	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℂ)
79	6, 8, 78	subsub4d 11527	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))))
80	7, 15	readdcld 11165	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) + (1 / 𝐴)) ∈ ℝ)
81		id 22	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
82	23, 81	relogdivd 26603	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) = ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)))
83		rerpdivcl 12965	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
84	45, 83	mpancom 689	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
85	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
86	85, 15	readdcld 11165	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ∈ ℝ)
87	15	reefcld 16044	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ)
88	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
89		rpcnne0 12952	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
90		divdir 11825	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
91	60, 88, 89, 90	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
92		reflcl 13746	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
93	41, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
94		rerpdivcl 12965	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
95	93, 94	mpancom 689	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
96		flle 13749	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
97	41, 96	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
98		rpcn 12944	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
99	98	mulridd 11153	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 1) = 𝐴)
100	97, 99	breqtrrd 5114	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ (𝐴 · 1))
101		ledivmul 12023	. . . . . . . . . . . . . . . 16 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (((⌊‘𝐴) / 𝐴) ≤ 1 ↔ (⌊‘𝐴) ≤ (𝐴 · 1)))
102	93, 85, 44, 101	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) ≤ 1 ↔ (⌊‘𝐴) ≤ (𝐴 · 1)))
103	100, 102	mpbird 257	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ≤ 1)
104	95, 85, 15, 103	leadd1dd 11755	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)) ≤ (1 + (1 / 𝐴)))
105	91, 104	eqbrtrd 5108	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (1 + (1 / 𝐴)))
106		efgt1p 16073	. . . . . . . . . . . . . 14 ⊢ ((1 / 𝐴) ∈ ℝ⁺ → (1 + (1 / 𝐴)) < (exp‘(1 / 𝐴)))
107	14, 106	syl 17	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) < (exp‘(1 / 𝐴)))
108	86, 87, 107	ltled 11285	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ≤ (exp‘(1 / 𝐴)))
109	84, 86, 87, 105, 108	letrd 11294	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)))
110		rpdivcl 12960	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
111	23, 110	mpancom 689	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
112	15	rpefcld 16063	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ⁺)
113	111, 112	logled 26604	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → ((((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)) ↔ (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (log‘(exp‘(1 / 𝐴)))))
114	109, 113	mpbid 232	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (log‘(exp‘(1 / 𝐴))))
115	15	relogefd 26605	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / 𝐴))) = (1 / 𝐴))
116	114, 115	breqtrd 5112	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (1 / 𝐴))
117	82, 116	eqbrtrrd 5110	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
118	25, 7, 15	lesubadd2d 11740	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴) ↔ (log‘((⌊‘𝐴) + 1)) ≤ ((log‘𝐴) + (1 / 𝐴))))
119	117, 118	mpbid 232	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ≤ ((log‘𝐴) + (1 / 𝐴)))
120	25, 80, 5, 119	lesub2dd 11758	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
121	79, 120	eqbrtrd 5108	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
122		harmonicbnd3 26985	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
123	20, 122	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
124		0re 11137	. . . . . . . 8 ⊢ 0 ∈ ℝ
125	124, 9	elicc2i 13356	. . . . . . 7 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) ↔ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ ∧ 0 ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∧ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ))
126	125	simp3bi 1148	. . . . . 6 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ)
127	123, 126	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ)
128	77, 26, 10, 121, 127	letrd 11294	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ)
129	27, 15, 10	lesubaddd 11738	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ ↔ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ≤ (γ + (1 / 𝐴))))
130	128, 129	mpbid 232	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ≤ (γ + (1 / 𝐴)))
131	27, 10, 15	absdifled 15390	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) ≤ (1 / 𝐴) ↔ ((γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∧ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ≤ (γ + (1 / 𝐴)))))
132	76, 130, 131	mpbir2and 714	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) ≤ (1 / 𝐴))
133	13, 132	eqbrtrrd 5110	1 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 ℕ₀cn0 12428 ℤ_≥cuz 12779 ℝ⁺crp 12933 [,]cicc 13292 ...cfz 13452 ⌊cfl 13740 abscabs 15187 Σcsu 15639 expce 16017 logclog 26531 γcem 26969

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 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-oadd 8402 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-fi 9317 df-sup 9348 df-inf 9349 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-fac 14227 df-bc 14256 df-hash 14284 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-ef 16023 df-e 16024 df-sin 16025 df-cos 16026 df-tan 16027 df-pi 16028 df-dvds 16213 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21336 df-xmet 21337 df-met 21338 df-bl 21339 df-mopn 21340 df-fbas 21341 df-fg 21342 df-cnfld 21345 df-top 22869 df-topon 22886 df-topsp 22908 df-bases 22921 df-cld 22994 df-ntr 22995 df-cls 22996 df-nei 23073 df-lp 23111 df-perf 23112 df-cn 23202 df-cnp 23203 df-haus 23290 df-cmp 23362 df-tx 23537 df-hmeo 23730 df-fil 23821 df-fm 23913 df-flim 23914 df-flf 23915 df-xms 24295 df-ms 24296 df-tms 24297 df-cncf 24855 df-limc 25843 df-dv 25844 df-ulm 26355 df-log 26533 df-atan 26844 df-em 26970

This theorem is referenced by: mulogsumlem 27508 mulog2sumlem1 27511

W3C validator