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Theorem harmonicbnd4 26977

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 13896	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13469	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
4	3	nnrecred 12196	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
5	1, 4	fsumrecl 15657	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
6	5	recnd 11160	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
7		relogcl 26540	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
8	7	recnd 11160	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
9		emre 26972	. . . . . 6 ⊢ γ ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℝ)
11	10	recnd 11160	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℂ)
12	6, 8, 11	subsub4d 11523	. . 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 12933	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
15	14	rpred 12949	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
16		resubcl 11445	. . . . 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 12921	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
19		flge0nn0 13740	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
21		nn0p1nn 12440	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
23	22	nnrpd 12947	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
24		relogcl 26540	. . . . . 6 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
25	23, 24	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
26	5, 25	resubcld 11565	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
27	5, 7	resubcld 11565	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∈ ℝ)
28	22	nnrecred 12196	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℝ)
29		fzfid 13896	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (1...((⌊‘𝐴) + 1)) ∈ Fin)
30		elfznn 13469	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...((⌊‘𝐴) + 1)) → 𝑚 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → 𝑚 ∈ ℕ)
32	31	nnrecred 12196	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℝ)
33	29, 32	fsumrecl 15657	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℝ)
34	33, 25	resubcld 11565	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
35		harmonicbnd 26970	. . . . . . . 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 11132	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38	9, 37	elicc2i 13328	. . . . . . . 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 12914	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42		fllep1 13721	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
43	41, 42	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
44		rpregt0 12920	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45	22	nnred 12160	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ)
46	22	nngt0d 12194	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < ((⌊‘𝐴) + 1))
47		lerec 12025	. . . . . . . 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 11757	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))))
51	33	recnd 11160	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℂ)
52	25	recnd 11160	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℂ)
53	28	recnd 11160	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℂ)
54	51, 52, 53	sub32d 11524	. . . . . 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 12790	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
56	22, 55	eleqtrdi 2846	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ (ℤ_≥‘1))
57	32	recnd 11160	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℂ)
58		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝐴) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝐴) + 1)))
59	56, 57, 58	fsumm1 15674	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
60	20	nn0cnd 12464	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
61		ax-1cn 11084	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
62		pncan 11386	. . . . . . . . . . . . 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 7374	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1...(((⌊‘𝐴) + 1) − 1)) = (1...(⌊‘𝐴)))
65	64	sumeq1d 15623	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
66	65	oveq1d 7373	. . . . . . . . 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 2771	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
68	6, 53, 67	mvrraddd 11549	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
69	68	oveq1d 7373	. . . . . 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 2771	. . . . 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 5124	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
72		logleb 26568	. . . . . . 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 11754	. . . 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 11290	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
77	27, 15	resubcld 11565	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ∈ ℝ)
78	15	recnd 11160	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℂ)
79	6, 8, 78	subsub4d 11523	. . . . . 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 11161	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) + (1 / 𝐴)) ∈ ℝ)
81		id 22	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
82	23, 81	relogdivd 26591	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) = ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)))
83		rerpdivcl 12937	. . . . . . . . . . . . 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 11161	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ∈ ℝ)
87	15	reefcld 16011	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ)
88	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
89		rpcnne0 12924	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
90		divdir 11821	. . . . . . . . . . . . . 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 13716	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
93	41, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
94		rerpdivcl 12937	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
95	93, 94	mpancom 688	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
96		flle 13719	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
97	41, 96	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
98		rpcn 12916	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
99	98	mulridd 11149	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 1) = 𝐴)
100	97, 99	breqtrrd 5126	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ (𝐴 · 1))
101		ledivmul 12018	. . . . . . . . . . . . . . . 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 11751	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)) ≤ (1 + (1 / 𝐴)))
105	91, 104	eqbrtrd 5120	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (1 + (1 / 𝐴)))
106		efgt1p 16040	. . . . . . . . . . . . . 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 11281	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ≤ (exp‘(1 / 𝐴)))
109	84, 86, 87, 105, 108	letrd 11290	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)))
110		rpdivcl 12932	. . . . . . . . . . . . 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 16030	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ⁺)
113	111, 112	logled 26592	. . . . . . . . . . 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 26593	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / 𝐴))) = (1 / 𝐴))
116	114, 115	breqtrd 5124	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (1 / 𝐴))
117	82, 116	eqbrtrrd 5122	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
118	25, 7, 15	lesubadd2d 11736	. . . . . . . 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 11754	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
121	79, 120	eqbrtrd 5120	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
122		harmonicbnd3 26974	. . . . . . 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 11134	. . . . . . . 8 ⊢ 0 ∈ ℝ
125	124, 9	elicc2i 13328	. . . . . . 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 11290	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ)
129	27, 15, 10	lesubaddd 11734	. . . 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 15360	. . 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 5122	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 1541 ∈ wcel 2113 ≠ wne 2932 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 · cmul 11031 < clt 11166 ≤ cle 11167 − cmin 11364 / cdiv 11794 ℕcn 12145 ℕ₀cn0 12401 ℤ_≥cuz 12751 ℝ⁺crp 12905 [,]cicc 13264 ...cfz 13423 ⌊cfl 13710 abscabs 15157 Σcsu 15609 expce 15984 logclog 26519 γcem 26958

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 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 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-e 15991 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-dvds 16180 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-limc 25823 df-dv 25824 df-ulm 26342 df-log 26521 df-atan 26833 df-em 26959

This theorem is referenced by: mulogsumlem 27498 mulog2sumlem1 27501

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