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Theorem fsumharmonic 26978

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 13896	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fsumharmonic.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3		elfznn 13469	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5	4	nncnd 12161	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
6	4	nnne0d 12195	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
7	2, 5, 6	divcld 11917	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
8	1, 7	fsumcl 15656	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
9	8	abscld 15362	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
10	2	abscld 15362	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
11	10, 4	nndivred 12199	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
12	1, 11	fsumrecl 15657	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13		fsumharmonic.c	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
14	1, 13	fsumrecl 15657	. . 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 11135	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
20		1red 11133	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
21		0lt1 11659	. . . . . . . . 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 11293	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑇)
25	18, 24	elrpd 12946	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
26	25	relogcld 26588	. . . . 5 ⊢ (𝜑 → (log‘𝑇) ∈ ℝ)
27	26, 20	readdcld 11161	. . . 4 ⊢ (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
28	16, 27	remulcld 11162	. . 3 ⊢ (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
29	14, 28	readdcld 11161	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
30	1, 7	fsumabs 15724	. . 3 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
31	2, 5, 6	absdivd 15381	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
32	4	nnrpd 12947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
33	32	rprege0d 12956	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34		absid 15219	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
35	33, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
36	35	oveq2d 7374	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
37	31, 36	eqtrd 2771	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
38	37	sumeq2dv 15625	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
39	30, 38	breqtrd 5124	. 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 12966	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ⁺)
42	41	rprege0d 12956	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43		flge0nn0 13740	. . . . . . . 8 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
45	44	nn0red 12463	. . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
46	45	ltp1d 12072	. . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47		fzdisj 13467	. . . . 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 12440	. . . . . . 7 ⊢ ((⌊‘(𝐴 / 𝑇)) ∈ ℕ₀ → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
50	44, 49	syl 17	. . . . . 6 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51		nnuz 12790	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
52	50, 51	eleqtrdi 2846	. . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1))
53	41	rpred 12949	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
54	40	rpred 12949	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	18, 24	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
56	40	rpregt0d 12955	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57		lediv2 12032	. . . . . . . . 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 11160	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
61	60	div1d 11909	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
62	59, 61	breqtrd 5124	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63		flword2 13733	. . . . . 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 13465	. . . . 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 11160	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
68	48, 66, 1, 67	fsumsplit 15664	. . 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 13896	. . . . 5 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70		ssun1 4130	. . . . . . . 8 ⊢ (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
71	70, 66	sseqtrrid 3977	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
72	71	sselda 3933	. . . . . 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 15657	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75		fzfid 13896	. . . . 5 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76		ssun2 4131	. . . . . . . 8 ⊢ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
77	76, 66	sseqtrrid 3977	. . . . . . 7 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
78	77	sselda 3933	. . . . . 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 15657	. . . 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 15657	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83		fznnfl 13782	. . . . . . . . . . 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 12949	. . . . . . . . . . . 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 12023	. . . . . . . . . . . 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 12998	. . . . . . . . . . 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 15362	. . . . . . . 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 12947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
104	101, 81, 103	ledivmul2d 13003	. . . . . . 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 15722	. . . . 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 15719	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
109	74, 82, 14, 106, 108	letrd 11290	. . . 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 12196	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
112	75, 111	fsumrecl 15657	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
113	16, 112	remulcld 11162	. . . . 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 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116	110	nncnd 12161	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117	110	nnne0d 12195	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118	115, 116, 117	divrecd 11920	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119	114, 110	nndivred 12199	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120	118, 119	eqeltrrd 2837	. . . . . . 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 4290	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑛 ∈ ∅
124		elin 3917	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
125	48	eleq2d 2822	. . . . . . . . . . . . . . . . 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 12199	. . . . . . . . . . . 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 11280	. . . . . . . . . . 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 13007	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147	146, 118	breqtrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
148	75, 79, 120, 147	fsumle 15722	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
149	16	recnd 11160	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
150	111	recnd 11160	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
151	75, 149, 150	fsummulc2 15707	. . . . . 6 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152	148, 151	breqtrrd 5126	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153	102	nnrecred 12196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
154	69, 153	fsumrecl 15657	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155	154	recnd 11160	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156	112	recnd 11160	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
157	4	nnrecred 12196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158	157	recnd 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
159	48, 66, 1, 158	fsumsplit 15664	. . . . . . . 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 11550	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
161	1, 157	fsumrecl 15657	. . . . . . . . . . 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 11565	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165		0red 11135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ∈ ℝ)
166	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167		fzfid 13896	. . . . . . . . . . 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 12959	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ⁺)
170	169	rpred 12949	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171	169	rpge0d 12953	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
172	40	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
173	172	rpge0d 12953	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ 𝐴)
174		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1)
175		0p1e1 12262	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
176	174, 175	breqtrrdi 5140	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < (0 + 1))
177	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ)
178		0z 12499	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
179		flbi 13736	. . . . . . . . . . . . . . . 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 7374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183		fz10 13461	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
184	182, 183	eqtrdi 2787	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185		0ss 4352	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186	184, 185	eqsstrdi 3978	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187	167, 170, 171, 186	fsumless 15719	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188	162, 163	suble0d 11728	. . . . . . . . . 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 26595	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (log‘𝑇))
191		0le1 11660	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
192	191	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 1)
193	26, 20, 190, 192	addge0d 11713	. . . . . . . . . 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 11290	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196		harmonicubnd 26976	. . . . . . . . . . 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 26975	. . . . . . . . . . . 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 26588	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
202		peano2re 11306	. . . . . . . . . . . . 13 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203	201, 202	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
204	41	relogcld 26588	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205		le2sub 11636	. . . . . . . . . . . 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 11160	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
210	20	recnd 11160	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
211	26	recnd 11160	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝑇) ∈ ℂ)
212	209, 210, 211	pnncand 11531	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
213	40, 25	relogdivd 26591	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214	213	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215		ax-1cn 11084	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
216		addcom 11319	. . . . . . . . . . . 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 2781	. . . . . . . . . 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 5124	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221	195, 220, 54, 20	ltlecasei 11241	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222	160, 221	eqbrtrrd 5122	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223		lemul2a 11996	. . . . . 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 11290	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
226	74, 80, 14, 28, 109, 225	le2addd 11756	. . 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 5120	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
228	9, 12, 29, 39, 227	letrd 11290	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 1541 ∈ wcel 2113 ∪ cun 3899 ∩ cin 3900 ∅c0 4285 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 ℤcz 12488 ℤ_≥cuz 12751 ℝ⁺crp 12905 ...cfz 13423 ⌊cfl 13710 abscabs 15157 Σcsu 15609 logclog 26519

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: dchrvmasumlem2 27465 mulog2sumlem2 27502

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