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Theorem fsumharmonic 27056

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 14015	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fsumharmonic.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3		elfznn 13594	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5	4	nncnd 12283	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
6	4	nnne0d 12317	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
7	2, 5, 6	divcld 12044	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
8	1, 7	fsumcl 15770	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
9	8	abscld 15476	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
10	2	abscld 15476	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
11	10, 4	nndivred 12321	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
12	1, 11	fsumrecl 15771	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13		fsumharmonic.c	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
14	1, 13	fsumrecl 15771	. . 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 11265	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
20		1red 11263	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
21		0lt1 11786	. . . . . . . . 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 11422	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑇)
25	18, 24	elrpd 13075	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
26	25	relogcld 26666	. . . . 5 ⊢ (𝜑 → (log‘𝑇) ∈ ℝ)
27	26, 20	readdcld 11291	. . . 4 ⊢ (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
28	16, 27	remulcld 11292	. . 3 ⊢ (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
29	14, 28	readdcld 11291	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
30	1, 7	fsumabs 15838	. . 3 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
31	2, 5, 6	absdivd 15495	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
32	4	nnrpd 13076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
33	32	rprege0d 13085	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34		absid 15336	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
35	33, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
36	35	oveq2d 7448	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
37	31, 36	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
38	37	sumeq2dv 15739	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
39	30, 38	breqtrd 5168	. 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 13095	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ⁺)
42	41	rprege0d 13085	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43		flge0nn0 13861	. . . . . . . 8 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
45	44	nn0red 12590	. . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
46	45	ltp1d 12199	. . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47		fzdisj 13592	. . . . 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 12567	. . . . . . 7 ⊢ ((⌊‘(𝐴 / 𝑇)) ∈ ℕ₀ → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
50	44, 49	syl 17	. . . . . 6 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51		nnuz 12922	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
52	50, 51	eleqtrdi 2850	. . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1))
53	41	rpred 13078	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
54	40	rpred 13078	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	18, 24	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
56	40	rpregt0d 13084	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57		lediv2 12159	. . . . . . . . 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 1377	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
59	23, 58	mpbid 232	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
60	54	recnd 11290	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
61	60	div1d 12036	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
62	59, 61	breqtrd 5168	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63		flword2 13854	. . . . . 6 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
64	53, 54, 62, 63	syl3anc 1372	. . . . 5 ⊢ (𝜑 → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
65		fzsplit2 13590	. . . . 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 11290	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
68	48, 66, 1, 67	fsumsplit 15778	. . 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 14015	. . . . 5 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70		ssun1 4177	. . . . . . . 8 ⊢ (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
71	70, 66	sseqtrrid 4026	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
72	71	sselda 3982	. . . . . 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 15771	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75		fzfid 14015	. . . . 5 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76		ssun2 4178	. . . . . . . 8 ⊢ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
77	76, 66	sseqtrrid 4026	. . . . . . 7 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
78	77	sselda 3982	. . . . . 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 15771	. . . 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 15771	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83		fznnfl 13903	. . . . . . . . . . 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 13078	. . . . . . . . . . . 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 12150	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
90	86, 87, 88, 89	syl3anc 1372	. . . . . . . . . . 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 13127	. . . . . . . . . . 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 15476	. . . . . . . 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 13076	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
104	101, 81, 103	ledivmul2d 13132	. . . . . . 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 15836	. . . . 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 15833	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
109	74, 82, 14, 106, 108	letrd 11419	. . . 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 12318	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
112	75, 111	fsumrecl 15771	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
113	16, 112	remulcld 11292	. . . . 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 11290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116	110	nncnd 12283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117	110	nnne0d 12317	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118	115, 116, 117	divrecd 12047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119	114, 110	nndivred 12321	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120	118, 119	eqeltrrd 2841	. . . . . . 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 4337	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑛 ∈ ∅
124		elin 3966	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
125	48	eleq2d 2826	. . . . . . . . . . . . . . . . 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 12321	. . . . . . . . . . . 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 11409	. . . . . . . . . . 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 13136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147	146, 118	breqtrd 5168	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
148	75, 79, 120, 147	fsumle 15836	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
149	16	recnd 11290	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
150	111	recnd 11290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
151	75, 149, 150	fsummulc2 15821	. . . . . 6 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152	148, 151	breqtrrd 5170	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153	102	nnrecred 12318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
154	69, 153	fsumrecl 15771	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155	154	recnd 11290	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156	112	recnd 11290	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
157	4	nnrecred 12318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158	157	recnd 11290	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
159	48, 66, 1, 158	fsumsplit 15778	. . . . . . . 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 11677	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
161	1, 157	fsumrecl 15771	. . . . . . . . . . 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 11692	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165		0red 11265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ∈ ℝ)
166	27	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167		fzfid 14015	. . . . . . . . . . 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 13088	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ⁺)
170	169	rpred 13078	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171	169	rpge0d 13082	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
172	40	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
173	172	rpge0d 13082	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ 𝐴)
174		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1)
175		0p1e1 12389	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
176	174, 175	breqtrrdi 5184	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < (0 + 1))
177	54	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ)
178		0z 12626	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
179		flbi 13857	. . . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183		fz10 13586	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
184	182, 183	eqtrdi 2792	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185		0ss 4399	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186	184, 185	eqsstrdi 4027	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187	167, 170, 171, 186	fsumless 15833	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188	162, 163	suble0d 11855	. . . . . . . . . 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 26673	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (log‘𝑇))
191		0le1 11787	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
192	191	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 1)
193	26, 20, 190, 192	addge0d 11840	. . . . . . . . . 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 11419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196		harmonicubnd 27054	. . . . . . . . . . 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 27053	. . . . . . . . . . . 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 26666	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
202		peano2re 11435	. . . . . . . . . . . . 13 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203	201, 202	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
204	41	relogcld 26666	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205		le2sub 11763	. . . . . . . . . . . 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 11290	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
210	20	recnd 11290	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
211	26	recnd 11290	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝑇) ∈ ℂ)
212	209, 210, 211	pnncand 11660	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
213	40, 25	relogdivd 26669	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214	213	oveq2d 7448	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215		ax-1cn 11214	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
216		addcom 11448	. . . . . . . . . . . 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 2786	. . . . . . . . . 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 5168	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221	195, 220, 54, 20	ltlecasei 11370	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222	160, 221	eqbrtrrd 5166	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223		lemul2a 12123	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
225	80, 113, 28, 152, 224	letrd 11419	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
226	74, 80, 14, 28, 109, 225	le2addd 11883	. . 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 5164	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
228	9, 12, 29, 39, 227	letrd 11419	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 1539 ∈ wcel 2107 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 · cmul 11161 < clt 11296 ≤ cle 11297 − cmin 11493 / cdiv 11921 ℕcn 12267 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12879 ℝ⁺crp 13035 ...cfz 13548 ⌊cfl 13831 abscabs 15274 Σcsu 15723 logclog 26597

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-e 16105 df-sin 16106 df-cos 16107 df-tan 16108 df-pi 16109 df-dvds 16292 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-cmp 23396 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cncf 24905 df-limc 25902 df-dv 25903 df-ulm 26421 df-log 26599 df-atan 26911 df-em 27037

This theorem is referenced by: dchrvmasumlem2 27543 mulog2sumlem2 27580

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