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Theorem harmonicbnd4 26973

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 13991	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13570	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
4	3	nnrecred 12291	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
5	1, 4	fsumrecl 15750	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
6	5	recnd 11263	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
7		relogcl 26536	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
8	7	recnd 11263	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
9		emre 26968	. . . . . 6 ⊢ γ ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℝ)
11	10	recnd 11263	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℂ)
12	6, 8, 11	subsub4d 11625	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))
13	12	fveq2d 6880	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
14		rpreccl 13035	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
15	14	rpred 13051	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
16		resubcl 11547	. . . . 5 ⊢ ((γ ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (γ − (1 / 𝐴)) ∈ ℝ)
17	9, 15, 16	sylancr 587	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ∈ ℝ)
18		rprege0 13024	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
19		flge0nn0 13837	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
21		nn0p1nn 12540	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
23	22	nnrpd 13049	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
24		relogcl 26536	. . . . . 6 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
25	23, 24	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
26	5, 25	resubcld 11665	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
27	5, 7	resubcld 11665	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∈ ℝ)
28	22	nnrecred 12291	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℝ)
29		fzfid 13991	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (1...((⌊‘𝐴) + 1)) ∈ Fin)
30		elfznn 13570	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...((⌊‘𝐴) + 1)) → 𝑚 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → 𝑚 ∈ ℕ)
32	31	nnrecred 12291	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℝ)
33	29, 32	fsumrecl 15750	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℝ)
34	33, 25	resubcld 11665	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
35		harmonicbnd 26966	. . . . . . . 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 11235	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38	9, 37	elicc2i 13429	. . . . . . . 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 1146	. . . . . . 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 13017	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42		fllep1 13818	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
43	41, 42	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
44		rpregt0 13023	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45	22	nnred 12255	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ)
46	22	nngt0d 12289	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < ((⌊‘𝐴) + 1))
47		lerec 12125	. . . . . . . 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 836	. . . . . . 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 11857	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))))
51	33	recnd 11263	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℂ)
52	25	recnd 11263	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℂ)
53	28	recnd 11263	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℂ)
54	51, 52, 53	sub32d 11626	. . . . . 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 12895	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
56	22, 55	eleqtrdi 2844	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ (ℤ_≥‘1))
57	32	recnd 11263	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℂ)
58		oveq2 7413	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝐴) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝐴) + 1)))
59	56, 57, 58	fsumm1 15767	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
60	20	nn0cnd 12564	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
61		ax-1cn 11187	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
62		pncan 11488	. . . . . . . . . . . . 13 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
63	60, 61, 62	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
64	63	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1...(((⌊‘𝐴) + 1) − 1)) = (1...(⌊‘𝐴)))
65	64	sumeq1d 15716	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
66	65	oveq1d 7420	. . . . . . . . 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 2770	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
68	6, 53, 67	mvrraddd 11649	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
69	68	oveq1d 7420	. . . . . 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 2770	. . . . 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 5145	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
72		logleb 26564	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ ((⌊‘𝐴) + 1) ∈ ℝ⁺) → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
73	23, 72	mpdan 687	. . . . . 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 11854	. . . 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 11392	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
77	27, 15	resubcld 11665	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ∈ ℝ)
78	15	recnd 11263	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℂ)
79	6, 8, 78	subsub4d 11625	. . . . . 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 11264	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) + (1 / 𝐴)) ∈ ℝ)
81		id 22	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
82	23, 81	relogdivd 26587	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) = ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)))
83		rerpdivcl 13039	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
84	45, 83	mpancom 688	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
85	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
86	85, 15	readdcld 11264	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ∈ ℝ)
87	15	reefcld 16104	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ)
88	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
89		rpcnne0 13027	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
90		divdir 11921	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
91	60, 88, 89, 90	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
92		reflcl 13813	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
93	41, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
94		rerpdivcl 13039	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
95	93, 94	mpancom 688	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
96		flle 13816	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
97	41, 96	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
98		rpcn 13019	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
99	98	mulridd 11252	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 1) = 𝐴)
100	97, 99	breqtrrd 5147	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ (𝐴 · 1))
101		ledivmul 12118	. . . . . . . . . . . . . . . 16 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (((⌊‘𝐴) / 𝐴) ≤ 1 ↔ (⌊‘𝐴) ≤ (𝐴 · 1)))
102	93, 85, 44, 101	syl3anc 1373	. . . . . . . . . . . . . . 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 11851	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)) ≤ (1 + (1 / 𝐴)))
105	91, 104	eqbrtrd 5141	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (1 + (1 / 𝐴)))
106		efgt1p 16133	. . . . . . . . . . . . . 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 11383	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ≤ (exp‘(1 / 𝐴)))
109	84, 86, 87, 105, 108	letrd 11392	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)))
110		rpdivcl 13034	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
111	23, 110	mpancom 688	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
112	15	rpefcld 16123	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ⁺)
113	111, 112	logled 26588	. . . . . . . . . . 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 26589	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / 𝐴))) = (1 / 𝐴))
116	114, 115	breqtrd 5145	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (1 / 𝐴))
117	82, 116	eqbrtrrd 5143	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
118	25, 7, 15	lesubadd2d 11836	. . . . . . . 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 11854	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
121	79, 120	eqbrtrd 5141	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
122		harmonicbnd3 26970	. . . . . . 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 11237	. . . . . . . 8 ⊢ 0 ∈ ℝ
125	124, 9	elicc2i 13429	. . . . . . 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 1147	. . . . . 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 11392	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ)
129	27, 15, 10	lesubaddd 11834	. . . 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 15453	. . 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 713	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) ≤ (1 / 𝐴))
133	13, 132	eqbrtrrd 5143	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 1540 ∈ wcel 2108 ≠ wne 2932 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 ℕcn 12240 ℕ₀cn0 12501 ℤ_≥cuz 12852 ℝ⁺crp 13008 [,]cicc 13365 ...cfz 13524 ⌊cfl 13807 abscabs 15253 Σcsu 15702 expce 16077 logclog 26515 γcem 26954

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-e 16084 df-sin 16085 df-cos 16086 df-tan 16087 df-pi 16088 df-dvds 16273 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-cmp 23325 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-ulm 26338 df-log 26517 df-atan 26829 df-em 26955

This theorem is referenced by: mulogsumlem 27494 mulog2sumlem1 27497

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