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 26919

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 13880	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1...(⌊‘𝐴)) ∈ Fin)
2		elfznn 13456	. . . . . . . 8 ⊢ (𝑚 ∈ (1...(⌊‘𝐴)) → 𝑚 ∈ ℕ)
3	2	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → 𝑚 ∈ ℕ)
4	3	nnrecred 12179	. . . . . 6 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...(⌊‘𝐴))) → (1 / 𝑚) ∈ ℝ)
5	1, 4	fsumrecl 15641	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℝ)
6	5	recnd 11143	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) ∈ ℂ)
7		relogcl 26482	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℝ)
8	7	recnd 11143	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (log‘𝐴) ∈ ℂ)
9		emre 26914	. . . . . 6 ⊢ γ ∈ ℝ
10	9	a1i 11	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℝ)
11	10	recnd 11143	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → γ ∈ ℂ)
12	6, 8, 11	subsub4d 11506	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ)))
13	12	fveq2d 6826	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (abs‘((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − γ)) = (abs‘(Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + γ))))
14		rpreccl 12921	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ⁺)
15	14	rpred 12937	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℝ)
16		resubcl 11428	. . . . 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 12909	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴))
19		flge0nn0 13724	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (⌊‘𝐴) ∈ ℕ₀)
20	18, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℕ₀)
21		nn0p1nn 12423	. . . . . . . 8 ⊢ ((⌊‘𝐴) ∈ ℕ₀ → ((⌊‘𝐴) + 1) ∈ ℕ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℕ)
23	22	nnrpd 12935	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ⁺)
24		relogcl 26482	. . . . . 6 ⊢ (((⌊‘𝐴) + 1) ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
25	23, 24	syl 17	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℝ)
26	5, 25	resubcld 11548	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
27	5, 7	resubcld 11548	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) ∈ ℝ)
28	22	nnrecred 12179	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℝ)
29		fzfid 13880	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (1...((⌊‘𝐴) + 1)) ∈ Fin)
30		elfznn 13456	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...((⌊‘𝐴) + 1)) → 𝑚 ∈ ℕ)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → 𝑚 ∈ ℕ)
32	31	nnrecred 12179	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℝ)
33	29, 32	fsumrecl 15641	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℝ)
34	33, 25	resubcld 11548	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) ∈ ℝ)
35		harmonicbnd 26912	. . . . . . . 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 11115	. . . . . . . . 9 ⊢ 1 ∈ ℝ
38	9, 37	elicc2i 13315	. . . . . . . 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 12902	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
42		fllep1 13705	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ ((⌊‘𝐴) + 1))
43	41, 42	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ≤ ((⌊‘𝐴) + 1))
44		rpregt0 12908	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
45	22	nnred 12143	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ ℝ)
46	22	nngt0d 12177	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → 0 < ((⌊‘𝐴) + 1))
47		lerec 12008	. . . . . . . 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 11740	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ ((Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))) − (1 / ((⌊‘𝐴) + 1))))
51	33	recnd 11143	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) ∈ ℂ)
52	25	recnd 11143	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (log‘((⌊‘𝐴) + 1)) ∈ ℂ)
53	28	recnd 11143	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / ((⌊‘𝐴) + 1)) ∈ ℂ)
54	51, 52, 53	sub32d 11507	. . . . . 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 12778	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
56	22, 55	eleqtrdi 2838	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) + 1) ∈ (ℤ_≥‘1))
57	32	recnd 11143	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑚 ∈ (1...((⌊‘𝐴) + 1))) → (1 / 𝑚) ∈ ℂ)
58		oveq2 7357	. . . . . . . . . 10 ⊢ (𝑚 = ((⌊‘𝐴) + 1) → (1 / 𝑚) = (1 / ((⌊‘𝐴) + 1)))
59	56, 57, 58	fsumm1 15658	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
60	20	nn0cnd 12447	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℂ)
61		ax-1cn 11067	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
62		pncan 11369	. . . . . . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (1...(((⌊‘𝐴) + 1) − 1)) = (1...(⌊‘𝐴)))
65	64	sumeq1d 15607	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...(((⌊‘𝐴) + 1) − 1))(1 / 𝑚) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
66	65	oveq1d 7364	. . . . . . . . 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 2764	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) = (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) + (1 / ((⌊‘𝐴) + 1))))
68	6, 53, 67	mvrraddd 11532	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...((⌊‘𝐴) + 1))(1 / 𝑚) − (1 / ((⌊‘𝐴) + 1))) = Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚))
69	68	oveq1d 7364	. . . . . 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 2764	. . . . 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 5118	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
72		logleb 26510	. . . . . . 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 11737	. . . 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 11273	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (γ − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)))
77	27, 15	resubcld 11548	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ∈ ℝ)
78	15	recnd 11143	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → (1 / 𝐴) ∈ ℂ)
79	6, 8, 78	subsub4d 11506	. . . . . 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 11144	. . . . . . 7 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘𝐴) + (1 / 𝐴)) ∈ ℝ)
81		id 22	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ⁺)
82	23, 81	relogdivd 26533	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) = ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)))
83		rerpdivcl 12925	. . . . . . . . . . . . 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 11144	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ∈ ℝ)
87	15	reefcld 15995	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ)
88	61	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → 1 ∈ ℂ)
89		rpcnne0 12912	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0))
90		divdir 11804	. . . . . . . . . . . . . 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 13700	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ∈ ℝ)
93	41, 92	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ∈ ℝ)
94		rerpdivcl 12925	. . . . . . . . . . . . . . 15 ⊢ (((⌊‘𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ⁺) → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
95	93, 94	mpancom 688	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ⁺ → ((⌊‘𝐴) / 𝐴) ∈ ℝ)
96		flle 13703	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) ≤ 𝐴)
97	41, 96	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ 𝐴)
98		rpcn 12904	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
99	98	mulridd 11132	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 · 1) = 𝐴)
100	97, 99	breqtrrd 5120	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ ℝ⁺ → (⌊‘𝐴) ≤ (𝐴 · 1))
101		ledivmul 12001	. . . . . . . . . . . . . . . 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 11734	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) / 𝐴) + (1 / 𝐴)) ≤ (1 + (1 / 𝐴)))
105	91, 104	eqbrtrd 5114	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (1 + (1 / 𝐴)))
106		efgt1p 16024	. . . . . . . . . . . . . 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 11264	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (1 + (1 / 𝐴)) ≤ (exp‘(1 / 𝐴)))
109	84, 86, 87, 105, 108	letrd 11273	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ⁺ → (((⌊‘𝐴) + 1) / 𝐴) ≤ (exp‘(1 / 𝐴)))
110		rpdivcl 12920	. . . . . . . . . . . . 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 16014	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ⁺ → (exp‘(1 / 𝐴)) ∈ ℝ⁺)
113	111, 112	logled 26534	. . . . . . . . . . 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 26535	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(exp‘(1 / 𝐴))) = (1 / 𝐴))
116	114, 115	breqtrd 5118	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ⁺ → (log‘(((⌊‘𝐴) + 1) / 𝐴)) ≤ (1 / 𝐴))
117	82, 116	eqbrtrrd 5116	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ⁺ → ((log‘((⌊‘𝐴) + 1)) − (log‘𝐴)) ≤ (1 / 𝐴))
118	25, 7, 15	lesubadd2d 11719	. . . . . . . 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 11737	. . . . . 6 ⊢ (𝐴 ∈ ℝ⁺ → (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − ((log‘𝐴) + (1 / 𝐴))) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
121	79, 120	eqbrtrd 5114	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ (Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘((⌊‘𝐴) + 1))))
122		harmonicbnd3 26916	. . . . . . 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 11117	. . . . . . . 8 ⊢ 0 ∈ ℝ
125	124, 9	elicc2i 13315	. . . . . . 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 11273	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → ((Σ𝑚 ∈ (1...(⌊‘𝐴))(1 / 𝑚) − (log‘𝐴)) − (1 / 𝐴)) ≤ γ)
129	27, 15, 10	lesubaddd 11717	. . . 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 15344	. . 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 5116	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 2109 ≠ wne 2925 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11149 ≤ cle 11150 − cmin 11347 / cdiv 11777 ℕcn 12128 ℕ₀cn0 12384 ℤ_≥cuz 12735 ℝ⁺crp 12893 [,]cicc 13251 ...cfz 13410 ⌊cfl 13694 abscabs 15141 Σcsu 15593 expce 15968 logclog 26461 γcem 26900

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

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 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-xnn0 12458 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-e 15975 df-sin 15976 df-cos 15977 df-tan 15978 df-pi 15979 df-dvds 16164 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-limc 25765 df-dv 25766 df-ulm 26284 df-log 26463 df-atan 26775 df-em 26901

This theorem is referenced by: mulogsumlem 27440 mulog2sumlem1 27443

W3C validator