fsumharmonic - Metamath Proof Explorer

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

Theorem fsumharmonic 26922

Description: Bound a finite sum based on the harmonic series, where the "strong" bound 𝐶 only applies asymptotically, and there is a "weak" bound 𝑅 for the remaining values. (Contributed by Mario Carneiro, 18-May-2016.)

**Hypotheses**
Ref	Expression
fsumharmonic.a	⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
fsumharmonic.t	⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
fsumharmonic.r	⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
fsumharmonic.b	⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
fsumharmonic.c	⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
fsumharmonic.0	⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)
fsumharmonic.1	⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
fsumharmonic.2	⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)

**Assertion**
Ref	Expression
fsumharmonic	⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))

Distinct variable groups: 𝐴,𝑛 𝜑,𝑛 𝑅,𝑛 𝑇,𝑛 Allowed substitution hints: 𝐵(𝑛) 𝐶(𝑛)

**Proof of Theorem fsumharmonic**
Step	Hyp	Ref	Expression
1		fzfid 13938	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fsumharmonic.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3		elfznn 13514	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5	4	nncnd 12202	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
6	4	nnne0d 12236	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
7	2, 5, 6	divcld 11958	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
8	1, 7	fsumcl 15699	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
9	8	abscld 15405	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
10	2	abscld 15405	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
11	10, 4	nndivred 12240	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
12	1, 11	fsumrecl 15700	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13		fsumharmonic.c	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
14	1, 13	fsumrecl 15700	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15		fsumharmonic.r	. . . . 5 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	simpld 494	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17		fsumharmonic.t	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
18	17	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℝ)
19		0red 11177	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
20		1red 11175	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
21		0lt1 11700	. . . . . . . . 9 ⊢ 0 < 1
22	21	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
23	17	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑇)
24	19, 20, 18, 22, 23	ltletrd 11334	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑇)
25	18, 24	elrpd 12992	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
26	25	relogcld 26532	. . . . 5 ⊢ (𝜑 → (log‘𝑇) ∈ ℝ)
27	26, 20	readdcld 11203	. . . 4 ⊢ (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
28	16, 27	remulcld 11204	. . 3 ⊢ (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
29	14, 28	readdcld 11203	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
30	1, 7	fsumabs 15767	. . 3 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
31	2, 5, 6	absdivd 15424	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
32	4	nnrpd 12993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
33	32	rprege0d 13002	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34		absid 15262	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
35	33, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
36	35	oveq2d 7403	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
37	31, 36	eqtrd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
38	37	sumeq2dv 15668	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
39	30, 38	breqtrd 5133	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40		fsumharmonic.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ⁺)
41	40, 25	rpdivcld 13012	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ⁺)
42	41	rprege0d 13002	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43		flge0nn0 13782	. . . . . . . 8 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
45	44	nn0red 12504	. . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
46	45	ltp1d 12113	. . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47		fzdisj 13512	. . . . 5 ⊢ ((⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1) → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
48	46, 47	syl 17	. . . 4 ⊢ (𝜑 → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
49		nn0p1nn 12481	. . . . . . 7 ⊢ ((⌊‘(𝐴 / 𝑇)) ∈ ℕ₀ → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
50	44, 49	syl 17	. . . . . 6 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51		nnuz 12836	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
52	50, 51	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1))
53	41	rpred 12995	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
54	40	rpred 12995	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	18, 24	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
56	40	rpregt0d 13001	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57		lediv2 12073	. . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
58	20, 22, 55, 56, 57	syl211anc 1378	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
59	23, 58	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
60	54	recnd 11202	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
61	60	div1d 11950	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
62	59, 61	breqtrd 5133	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63		flword2 13775	. . . . . 6 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
64	53, 54, 62, 63	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
65		fzsplit2 13510	. . . . 5 ⊢ ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
66	52, 64, 65	syl2anc 584	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
67	11	recnd 11202	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
68	48, 66, 1, 67	fsumsplit 15707	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛)))
69		fzfid 13938	. . . . 5 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70		ssun1 4141	. . . . . . . 8 ⊢ (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
71	70, 66	sseqtrrid 3990	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
72	71	sselda 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
73	72, 11	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
74	69, 73	fsumrecl 15700	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75		fzfid 13938	. . . . 5 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76		ssun2 4142	. . . . . . . 8 ⊢ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
77	76, 66	sseqtrrid 3990	. . . . . . 7 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
78	77	sselda 3946	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
79	78, 11	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
80	75, 79	fsumrecl 15700	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
81	72, 13	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
82	69, 81	fsumrecl 15700	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83		fznnfl 13824	. . . . . . . . . . 11 ⊢ ((𝐴 / 𝑇) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
84	53, 83	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
85	84	simplbda 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
86	32	rpred 12995	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
87	54	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
88	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89		lemuldiv2 12064	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
90	86, 87, 88, 89	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
91	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
92	91, 87, 32	lemuldivd 13044	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑇 ≤ (𝐴 / 𝑛)))
93	90, 92	bitr3d 281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
94	72, 93	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
95	85, 94	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96		fsumharmonic.1	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
97	96	ex 412	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
98	72, 97	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
99	95, 98	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
100	72, 2	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101	100	abscld 15405	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
102	72, 3	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103	102	nnrpd 12993	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
104	101, 81, 103	ledivmul2d 13049	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
105	99, 104	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
106	69, 73, 81, 105	fsumle 15765	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶)
107		fsumharmonic.0	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)
108	1, 13, 107, 71	fsumless 15762	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
109	74, 82, 14, 106, 108	letrd 11331	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
110	78, 3	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
111	110	nnrecred 12237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
112	75, 111	fsumrecl 15700	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
113	16, 112	remulcld 11204	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
114	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115	114	recnd 11202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116	110	nncnd 12202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117	110	nnne0d 12236	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118	115, 116, 117	divrecd 11961	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119	114, 110	nndivred 12240	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120	118, 119	eqeltrrd 2829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
121	78, 10	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
122	78, 32	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
123		noel 4301	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑛 ∈ ∅
124		elin 3930	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
125	48	eleq2d 2814	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126	124, 125	bitr3id 285	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127	123, 126	mtbiri 327	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128		imnan 399	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129	127, 128	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130	129	con2d 134	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131	130	imp 406	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
132	83	baibd 539	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
133	53, 3, 132	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134	133, 93	bitrd 279	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
135	78, 134	syldan 591	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136	131, 135	mtbid 324	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
137	54	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138	137, 110	nndivred 12240	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
139	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140	138, 139	ltnled 11321	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141	136, 140	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142		fsumharmonic.2	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143	142	ex 412	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
144	78, 143	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
145	141, 144	mpd 15	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ≤ 𝑅)
146	121, 114, 122, 145	lediv1dd 13053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147	146, 118	breqtrd 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
148	75, 79, 120, 147	fsumle 15765	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
149	16	recnd 11202	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
150	111	recnd 11202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
151	75, 149, 150	fsummulc2 15750	. . . . . 6 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152	148, 151	breqtrrd 5135	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153	102	nnrecred 12237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
154	69, 153	fsumrecl 15700	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155	154	recnd 11202	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156	112	recnd 11202	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
157	4	nnrecred 12237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158	157	recnd 11202	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
159	48, 66, 1, 158	fsumsplit 15707	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
160	155, 156, 159	mvrladdd 11591	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
161	1, 157	fsumrecl 15700	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162	161	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163	154	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164	162, 163	resubcld 11606	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165		0red 11177	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ∈ ℝ)
166	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167		fzfid 13938	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168	103	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
169	168	rpreccld 13005	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ⁺)
170	169	rpred 12995	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171	169	rpge0d 12999	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
172	40	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
173	172	rpge0d 12999	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ 𝐴)
174		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1)
175		0p1e1 12303	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
176	174, 175	breqtrrdi 5149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < (0 + 1))
177	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ)
178		0z 12540	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
179		flbi 13778	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
180	177, 178, 179	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
181	173, 176, 180	mpbir2and 713	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < 1) → (⌊‘𝐴) = 0)
182	181	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183		fz10 13506	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
184	182, 183	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185		0ss 4363	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186	184, 185	eqsstrdi 3991	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187	167, 170, 171, 186	fsumless 15762	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188	162, 163	suble0d 11769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0 ↔ Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)))
189	187, 188	mpbird 257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
190	18, 23	logge0d 26539	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (log‘𝑇))
191		0le1 11701	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
192	191	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 1)
193	26, 20, 190, 192	addge0d 11754	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194	193	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195	164, 165, 166, 189, 194	letrd 11331	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196		harmonicubnd 26920	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
197	54, 196	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
198		harmoniclbnd 26919	. . . . . . . . . . . 12 ⊢ ((𝐴 / 𝑇) ∈ ℝ⁺ → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
199	41, 198	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
200	199	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
201	40	relogcld 26532	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
202		peano2re 11347	. . . . . . . . . . . . 13 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203	201, 202	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
204	41	relogcld 26532	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205		le2sub 11677	. . . . . . . . . . . 12 ⊢ (((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ ∧ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ) ∧ (((log‘𝐴) + 1) ∈ ℝ ∧ (log‘(𝐴 / 𝑇)) ∈ ℝ)) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1) ∧ (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇)))))
206	161, 154, 203, 204, 205	syl22anc 838	. . . . . . . . . . 11 ⊢ (𝜑 → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1) ∧ (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇)))))
207	206	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1) ∧ (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇)))))
208	197, 200, 207	mp2and 699	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209	201	recnd 11202	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
210	20	recnd 11202	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
211	26	recnd 11202	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝑇) ∈ ℂ)
212	209, 210, 211	pnncand 11572	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
213	40, 25	relogdivd 26535	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214	213	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215		ax-1cn 11126	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
216		addcom 11360	. . . . . . . . . . . 12 ⊢ (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
217	211, 215, 216	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218	212, 214, 217	3eqtr4d 2774	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219	218	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220	208, 219	breqtrd 5133	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221	195, 220, 54, 20	ltlecasei 11282	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222	160, 221	eqbrtrrd 5131	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223		lemul2a 12037	. . . . . 6 ⊢ (((Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ ∧ ((log‘𝑇) + 1) ∈ ℝ ∧ (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅)) ∧ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1)) → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
224	112, 27, 15, 222, 223	syl31anc 1375	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
225	80, 113, 28, 152, 224	letrd 11331	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
226	74, 80, 14, 28, 109, 225	le2addd 11797	. . 3 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
227	68, 226	eqbrtrd 5129	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
228	9, 12, 29, 39, 227	letrd 11331	1 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 ℕ₀cn0 12442 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 ...cfz 13468 ⌊cfl 13752 abscabs 15200 Σcsu 15652 logclog 26463

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-er 8671 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-sum 15653 df-ef 16033 df-e 16034 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-dvds 16223 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-ulm 26286 df-log 26465 df-atan 26777 df-em 26903

This theorem is referenced by: dchrvmasumlem2 27409 mulog2sumlem2 27446

W3C validator