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 27320

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 14096	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13667	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
3	2	adantl 487	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
4	3	nnrecred 12370	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
5	1, 4	fsumrecl 15880	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
6	5	recnd 11318	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
7		relogcl 26885	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
8	7	recnd 11318	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
9		emre 27315	. . . . . 6 ⊢ γ ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℝ)
11	10	recnd 11318	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℂ)
12	6, 8, 11	subsub4d 11681	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))
13	12	fveq2d 6881	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
14		rpreccl 13129	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
15	14	rpred 13145	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
16		resubcl 11603	. . . . 5 ⊢ ((γ ∈ ℝ ∧ (1 / 𝐴) ∈ ℝ) → (γ − (1 / 𝐴)) ∈ ℝ)
17	9, 15, 16	sylancr 599	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ∈ ℝ)
18		rprege0 13117	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
19		flge0nn0 13940	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
20	18, 19	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
21		nn0p1nn 12626	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
22	20, 21	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
23	22	nnrpd 13143	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
24		relogcl 26885	. . . . . 6 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
25	23, 24	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
26	5, 25	resubcld 11725	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
27	5, 7	resubcld 11725	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∈ ℝ)
28	22	nnrecred 12370	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℝ)
29		fzfid 14096	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (1...((⌊‘𝐴) + 1)) ∈ Fin)
30		elfznn 13667	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...((⌊‘𝐴) + 1)) → 𝑚 ∈ ℕ)
31	30	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → 𝑚 ∈ ℕ)
32	31	nnrecred 12370	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℝ)
33	29, 32	fsumrecl 15880	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℝ)
34	33, 25	resubcld 11725	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
35		harmonicbnd 27313	. . . . . . . 8 ⊢ (((⌊‘𝐴) + 1) ∈ ℕ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1))
36	22, 35	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (γ[,]1))
37		1re 11289	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38	9, 37	elicc2i 13524	. . . . . . . 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 1164	. . . . . . 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 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → γ ≤ (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
41		rpre 13110	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42		fllep1 13921	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
43	41, 42	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
44		rpregt0 13116	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45	22	nnred 12331	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ)
46	22	nngt0d 12368	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < ((⌊‘𝐴) + 1))
47		lerec 12181	. . . . . . . 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 850	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (1 / ((⌊‘𝐴) + 1)) ≤ (1 / 𝐴)))
49	43, 48	mpbid 235	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ≤ (1 / 𝐴))
50	10, 28, 34, 15, 40, 49	le2subd 11917	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))))
51	33	recnd 11318	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℂ)
52	25	recnd 11318	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℂ)
53	28	recnd 11318	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℂ)
54	51, 52, 53	sub32d 11682	. . . . . 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 12985	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
56	22, 55	eleqtrdi 2871	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ (ℤ_≥‘1))
57	32	recnd 11318	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℂ)
58		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝐴) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝐴) + 1)))
59	56, 57, 58	fsumm1 15897	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
60	20	nn0cnd 12650	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
61		ax-1cn 11239	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
62		pncan 11544	. . . . . . . . . . . . 13 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ) → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
63	60, 61, 62	sylancl 598	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) − 1) = (⌊‘𝐴))
64	63	oveq2d 7428	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1...(((⌊‘𝐴) + 1) − 1)) = (1...(⌊‘𝐴)))
65	64	sumeq1d 15847	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
66	65	oveq1d 7427	. . . . . . . . 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 2796	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
68	6, 53, 67	mvrraddd 11708	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
69	68	oveq1d 7427	. . . . . 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 2796	. . . . 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 5131	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
72		logleb 26913	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ ((⌊‘𝐴) + 1) ∈ ℝ⁺) → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
73	23, 72	mpdan 700	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ ((⌊‘𝐴) + 1) ↔ (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1))))
74	43, 73	mpbid 235	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ≤ (log‘((⌊‘𝐴) + 1)))
75	7, 25, 5, 74	lesub2dd 11914	. . . 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 11448	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
77	27, 15	resubcld 11725	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ∈ ℝ)
78	15	recnd 11318	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℂ)
79	6, 8, 78	subsub4d 11681	. . . . . 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 11319	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) + (1 / 𝐴)) ∈ ℝ)
81		id 23	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
82	23, 81	relogdivd 26936	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) = ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)))
83		rerpdivcl 13133	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
84	45, 83	mpancom 701	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ)
85	37	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℝ)
86	85, 15	readdcld 11319	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ∈ ℝ)
87	15	reefcld 16234	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ)
88	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
89		rpcnne0 13120	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
90		divdir 11980	. . . . . . . . . . . . . 14 ⊢ (((⌊‘𝐴) ∈ ℂ ∧ 1 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
91	60, 88, 89, 90	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) = (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)))
92		reflcl 13916	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
93	41, 92	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
94		rerpdivcl 13133	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
95	93, 94	mpancom 701	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
96		flle 13919	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
97	41, 96	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
98		rpcn 13112	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
99	98	mulridd 11307	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 1) = 𝐴)
100	97, 99	breqtrrd 5133	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ (𝐴 · 1))
101		ledivmul 12174	. . . . . . . . . . . . . . . 16 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (((⌊‘𝐴) / 𝐴) ≤ 1 ↔ (⌊‘𝐴) ≤ (𝐴 · 1)))
102	93, 85, 44, 101	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) ≤ 1 ↔ (⌊‘𝐴) ≤ (𝐴 · 1)))
103	100, 102	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ≤ 1)
104	95, 85, 15, 103	leadd1dd 11911	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)) ≤ (1 + (1 / 𝐴)))
105	91, 104	eqbrtrd 5127	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (1 + (1 / 𝐴)))
106		efgt1p 16263	. . . . . . . . . . . . . 14 ⊢ ((1 / 𝐴) ∈ ℝ⁺ → (1 + (1 / 𝐴)) < (exp‘(1 / 𝐴)))
107	14, 106	syl 18	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) < (exp‘(1 / 𝐴)))
108	86, 87, 107	ltled 11439	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ≤ (exp‘(1 / 𝐴)))
109	84, 86, 87, 105, 108	letrd 11448	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)))
110		rpdivcl 13128	. . . . . . . . . . . . 13 ⊢ ((((⌊‘𝐴) + 1) ∈ ℝ⁺ ∧ 𝐴 ∈ ℝ⁺) → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
111	23, 110	mpancom 701	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ∈ ℝ⁺)
112	15	rpefcld 16253	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ⁺)
113	111, 112	logled 26937	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → ((((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)) ↔ (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (log‘(exp‘(1 / 𝐴)))))
114	109, 113	mpbid 235	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (log‘(exp‘(1 / 𝐴))))
115	15	relogefd 26938	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / 𝐴))) = (1 / 𝐴))
116	114, 115	breqtrd 5131	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (1 / 𝐴))
117	82, 116	eqbrtrrd 5129	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
118	25, 7, 15	lesubadd2d 11896	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴) ↔ (log‘((⌊‘𝐴) + 1)) ≤ ((log‘𝐴) + (1 / 𝐴))))
119	117, 118	mpbid 235	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ≤ ((log‘𝐴) + (1 / 𝐴)))
120	25, 80, 5, 119	lesub2dd 11914	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
121	79, 120	eqbrtrd 5127	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
122		harmonicbnd3 27317	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
123	20, 122	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ))
124		0re 11291	. . . . . . . 8 ⊢ 0 ∈ ℝ
125	124, 9	elicc2i 13524	. . . . . . 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 1165	. . . . . 6 ⊢ ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ (0[,]γ) → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ)
127	123, 126	syl 18	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ≤ γ)
128	77, 26, 10, 121, 127	letrd 11448	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ)
129	27, 15, 10	lesubaddd 11894	. . . 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 235	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ≤ (γ + (1 / 𝐴)))
131	27, 10, 15	absdifled 15584	. . 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 726	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) ≤ (1 / 𝐴))
133	13, 132	eqbrtrrd 5129	1 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))) ≤ (1 / 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 < clt 11324 ≤ cle 11325 − cmin 11522 / cdiv 11954 ℕcn 12316 ℕ₀cn0 12587 ℤ_≥cuz 12946 ℝ⁺crp 13101 [,]cicc 13460 ...cfz 13620 ⌊cfl 13910 abscabs 15381 Σcsu 15833 expce 16207 logclog 26864 γcem 27301

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 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 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-card 10001 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-xnn0 12661 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-e 16214 df-sin 16215 df-cos 16216 df-tan 16217 df-pi 16218 df-dvds 16403 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-cmp 23685 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-ulm 26686 df-log 26866 df-atan 27177 df-em 27302

This theorem is used by: mulogsumlem 27840 mulog2sumlem1 27843

W3C validator