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Theorem fsumharmonic 27385

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 14154	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fsumharmonic.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3		elfznn 13725	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5	4	nncnd 12389	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
6	4	nnne0d 12426	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
7	2, 5, 6	divcld 12131	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
8	1, 7	fsumcl 15940	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
9	8	abscld 15647	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
10	2	abscld 15647	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
11	10, 4	nndivred 12430	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
12	1, 11	fsumrecl 15941	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13		fsumharmonic.c	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
14	1, 13	fsumrecl 15941	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15		fsumharmonic.r	. . . . 5 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	simpld 500	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17		fsumharmonic.t	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
18	17	simpld 500	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℝ)
19		0red 11348	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
20		1red 11346	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
21		0lt1 11876	. . . . . . . . 9 ⊢ 0 < 1
22	21	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
23	17	simprd 501	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑇)
24	19, 20, 18, 22, 23	ltletrd 11507	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑇)
25	18, 24	elrpd 13199	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
26	25	relogcld 26997	. . . . 5 ⊢ (𝜑 → (log‘𝑇) ∈ ℝ)
27	26, 20	readdcld 11375	. . . 4 ⊢ (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
28	16, 27	remulcld 11376	. . 3 ⊢ (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
29	14, 28	readdcld 11375	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
30	1, 7	fsumabs 16009	. . 3 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
31	2, 5, 6	absdivd 15666	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
32	4	nnrpd 13200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
33	32	rprege0d 13209	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34		absid 15504	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
35	33, 34	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
36	35	oveq2d 7465	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
37	31, 36	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
38	37	sumeq2dv 15910	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
39	30, 38	breqtrd 5135	. 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 13219	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ⁺)
42	41	rprege0d 13209	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43		flge0nn0 13998	. . . . . . . 8 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
44	42, 43	syl 18	. . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
45	44	nn0red 12706	. . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
46	45	ltp1d 12285	. . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47		fzdisj 13723	. . . . 5 ⊢ ((⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1) → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
48	46, 47	syl 18	. . . 4 ⊢ (𝜑 → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
49		nn0p1nn 12683	. . . . . . 7 ⊢ ((⌊‘(𝐴 / 𝑇)) ∈ ℕ₀ → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
50	44, 49	syl 18	. . . . . 6 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51		nnuz 13042	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
52	50, 51	eleqtrdi 2874	. . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1))
53	41	rpred 13202	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
54	40	rpred 13202	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	18, 24	jca 521	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
56	40	rpregt0d 13208	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57		lediv2 12245	. . . . . . . . 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 1404	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
59	23, 58	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
60	54	recnd 11374	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
61	60	div1d 12123	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
62	59, 61	breqtrd 5135	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63		flword2 13991	. . . . . 6 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
64	53, 54, 62, 63	syl3anc 1399	. . . . 5 ⊢ (𝜑 → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
65		fzsplit2 13721	. . . . 5 ⊢ ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
66	52, 64, 65	syl2anc 596	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
67	11	recnd 11374	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
68	48, 66, 1, 67	fsumsplit 15948	. . 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 14154	. . . . 5 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70		ssun1 4179	. . . . . . . 8 ⊢ (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
71	70, 66	sseqtrrid 3956	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
72	71	sselda 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
73	72, 11	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
74	69, 73	fsumrecl 15941	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75		fzfid 14154	. . . . 5 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76		ssun2 4180	. . . . . . . 8 ⊢ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
77	76, 66	sseqtrrid 3956	. . . . . . 7 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
78	77	sselda 3913	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
79	78, 11	syldan 603	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
80	75, 79	fsumrecl 15941	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
81	72, 13	syldan 603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
82	69, 81	fsumrecl 15941	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83		fznnfl 14040	. . . . . . . . . . 11 ⊢ ((𝐴 / 𝑇) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
84	53, 83	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
85	84	simplbda 505	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
86	32	rpred 13202	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
87	54	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
88	55	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89		lemuldiv2 12236	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
90	86, 87, 88, 89	syl3anc 1399	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
91	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
92	91, 87, 32	lemuldivd 13251	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑇 ≤ (𝐴 / 𝑛)))
93	90, 92	bitr3d 284	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
94	72, 93	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
95	85, 94	mpbid 235	. . . . . . . 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 418	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
98	72, 97	syldan 603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
99	95, 98	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
100	72, 2	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101	100	abscld 15647	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
102	72, 3	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103	102	nnrpd 13200	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
104	101, 81, 103	ledivmul2d 13256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
105	99, 104	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
106	69, 73, 81, 105	fsumle 16007	. . . . 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 16004	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
109	74, 82, 14, 106, 108	letrd 11504	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
110	78, 3	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
111	110	nnrecred 12427	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
112	75, 111	fsumrecl 15941	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
113	16, 112	remulcld 11376	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
114	16	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115	114	recnd 11374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116	110	nncnd 12389	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117	110	nnne0d 12426	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118	115, 116, 117	divrecd 12134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119	114, 110	nndivred 12430	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120	118, 119	eqeltrrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
121	78, 10	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
122	78, 32	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
123		noel 4286	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑛 ∈ ∅
124		elin 4096	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
125	48	eleq2d 2850	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126	124, 125	bitr3id 288	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127	123, 126	mtbiri 330	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128		imnan 405	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129	127, 128	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130	129	con2d 135	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131	130	imp 412	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
132	83	baibd 549	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
133	53, 3, 132	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134	133, 93	bitrd 282	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
135	78, 134	syldan 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136	131, 135	mtbid 327	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
137	54	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138	137, 110	nndivred 12430	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
139	18	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140	138, 139	ltnled 11494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141	136, 140	mpbird 260	. . . . . . . . . 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 418	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
144	78, 143	syldan 603	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
145	141, 144	mpd 16	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ≤ 𝑅)
146	121, 114, 122, 145	lediv1dd 13260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147	146, 118	breqtrd 5135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
148	75, 79, 120, 147	fsumle 16007	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
149	16	recnd 11374	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
150	111	recnd 11374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
151	75, 149, 150	fsummulc2 15991	. . . . . 6 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152	148, 151	breqtrrd 5137	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153	102	nnrecred 12427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
154	69, 153	fsumrecl 15941	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155	154	recnd 11374	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156	112	recnd 11374	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
157	4	nnrecred 12427	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158	157	recnd 11374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
159	48, 66, 1, 158	fsumsplit 15948	. . . . . . . 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 11765	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
161	1, 157	fsumrecl 15941	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162	161	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163	154	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164	162, 163	resubcld 11781	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165		0red 11348	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ∈ ℝ)
166	27	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167		fzfid 14154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168	103	adantlr 728	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
169	168	rpreccld 13212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ⁺)
170	169	rpred 13202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171	169	rpge0d 13206	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
172	40	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
173	172	rpge0d 13206	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ 𝐴)
174		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1)
175		0p1e1 12501	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
176	174, 175	breqtrrdi 5151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < (0 + 1))
177	54	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ)
178		0z 12742	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
179		flbi 13994	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
180	177, 178, 179	sylancl 598	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
181	173, 176, 180	mpbir2and 726	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < 1) → (⌊‘𝐴) = 0)
182	181	oveq2d 7465	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183		fz10 13716	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
184	182, 183	eqtrdi 2814	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185		0ss 4353	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186	184, 185	eqsstrdi 3957	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187	167, 170, 171, 186	fsumless 16004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188	162, 163	suble0d 11945	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
190	18, 23	logge0d 27004	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (log‘𝑇))
191		0le1 11877	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
192	191	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 1)
193	26, 20, 190, 192	addge0d 11930	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194	193	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195	164, 165, 166, 189, 194	letrd 11504	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196		harmonicubnd 27383	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
197	54, 196	sylan 592	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
198		harmoniclbnd 27382	. . . . . . . . . . . 12 ⊢ ((𝐴 / 𝑇) ∈ ℝ⁺ → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
199	41, 198	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
200	199	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
201	40	relogcld 26997	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
202		peano2re 11520	. . . . . . . . . . . . 13 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203	201, 202	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
204	41	relogcld 26997	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205		le2sub 11853	. . . . . . . . . . . 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 852	. . . . . . . . . . 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 486	. . . . . . . . . 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 712	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209	201	recnd 11374	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
210	20	recnd 11374	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
211	26	recnd 11374	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝑇) ∈ ℂ)
212	209, 210, 211	pnncand 11745	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
213	40, 25	relogdivd 27000	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214	213	oveq2d 7465	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215		ax-1cn 11295	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
216		addcom 11533	. . . . . . . . . . . 12 ⊢ (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
217	211, 215, 216	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218	212, 214, 217	3eqtr4d 2808	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219	218	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220	208, 219	breqtrd 5135	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221	195, 220, 54, 20	ltlecasei 11455	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222	160, 221	eqbrtrrd 5133	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223		lemul2a 12210	. . . . . 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 1401	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
225	80, 113, 28, 152, 224	letrd 11504	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
226	74, 80, 14, 28, 109, 225	le2addd 11973	. . 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 5131	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
228	9, 12, 29, 39, 227	letrd 11504	1 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∩ cin 4092 ∪ cun 4152 ∅c0 4281 class class class wbr 5106 ‘cfv 6564 (class class class)co 7449 ℂcc 11235 ℝcr 11236 0cc0 11237 1c1 11238 + caddc 11240 · cmul 11242 < clt 11380 ≤ cle 11381 − cmin 11578 / cdiv 12011 ℕcn 12373 ℕ₀cn0 12644 ℤcz 12731 ℤ_≥cuz 13003 ℝ⁺crp 13158 ...cfz 13677 ⌊cfl 13968 abscabs 15442 Σcsu 15894 logclog 26928

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7781 ax-inf2 9677 ax-cnex 11293 ax-resscn 11294 ax-1cn 11295 ax-icn 11296 ax-addcl 11297 ax-addrcl 11298 ax-mulcl 11299 ax-mulrcl 11300 ax-mulcom 11301 ax-addass 11302 ax-mulass 11303 ax-distr 11304 ax-i2m1 11305 ax-1ne0 11306 ax-1rid 11307 ax-rnegex 11308 ax-rrecex 11309 ax-cnre 11310 ax-pre-lttri 11311 ax-pre-lttrn 11312 ax-pre-ltadd 11313 ax-pre-mulgt0 11314 ax-pre-sup 11315 ax-addf 11316

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6328 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6518 df-fun 6566 df-fn 6567 df-f 6568 df-f1 6569 df-fo 6570 df-f1o 6571 df-fv 6572 df-isom 6573 df-riota 7406 df-ov 7452 df-oprab 7453 df-mpo 7454 df-of 7723 df-om 7910 df-1st 8033 df-2nd 8034 df-supp 8213 df-frecs 8335 df-wrecs 8366 df-recs 8415 df-rdg 8454 df-1o 8512 df-2o 8513 df-oadd 8516 df-er 8753 df-map 8885 df-pm 8886 df-ixp 8962 df-en 9010 df-dom 9011 df-sdom 9012 df-fin 9013 df-fsupp 9390 df-fi 9438 df-sup 9469 df-inf 9470 df-oi 9539 df-card 10055 df-pnf 11382 df-mnf 11383 df-xr 11384 df-ltxr 11385 df-le 11386 df-sub 11580 df-neg 11581 df-div 12012 df-nn 12374 df-2 12443 df-3 12444 df-4 12445 df-5 12446 df-6 12447 df-7 12448 df-8 12449 df-9 12450 df-n0 12645 df-xnn0 12718 df-z 12732 df-dec 12853 df-uz 13004 df-q 13114 df-rp 13159 df-xneg 13279 df-xadd 13280 df-xmul 13281 df-ioo 13518 df-ioc 13519 df-ico 13520 df-icc 13521 df-fz 13678 df-fzo 13827 df-fl 13970 df-mod 14048 df-seq 14183 df-exp 14243 df-fac 14457 df-bc 14486 df-hash 14514 df-shft 15261 df-cj 15307 df-re 15308 df-im 15309 df-sqrt 15443 df-abs 15444 df-limsup 15679 df-clim 15696 df-rlim 15697 df-sum 15895 df-ef 16274 df-e 16275 df-sin 16276 df-cos 16277 df-tan 16278 df-pi 16279 df-dvds 16464 df-struct 17366 df-sets 17383 df-slot 17401 df-ndx 17413 df-base 17429 df-ress 17450 df-plusg 17482 df-mulr 17483 df-starv 17484 df-sca 17485 df-vsca 17486 df-ip 17487 df-tset 17488 df-ple 17489 df-ds 17491 df-unif 17492 df-hom 17493 df-cco 17494 df-rest 17634 df-topn 17635 df-0g 17653 df-gsum 17654 df-topgen 17655 df-pt 17656 df-prds 17659 df-xrs 17715 df-qtop 17720 df-imas 17721 df-xps 17723 df-mre 17797 df-mrc 17798 df-acs 17800 df-mgm 18857 df-sgrp 18949 df-mnd 18965 df-submnd 19020 df-mulg 19319 df-cntz 19573 df-cmn 20038 df-psmet 21712 df-xmet 21713 df-met 21714 df-bl 21715 df-mopn 21716 df-fbas 21717 df-fg 21718 df-cnfld 21721 df-top 23256 df-topon 23273 df-topsp 23295 df-bases 23308 df-cld 23381 df-ntr 23382 df-cls 23383 df-nei 23460 df-lp 23498 df-perf 23499 df-cn 23589 df-cnp 23590 df-haus 23677 df-cmp 23749 df-tx 23925 df-hmeo 24118 df-fil 24209 df-fm 24301 df-flim 24302 df-flf 24303 df-xms 24683 df-ms 24684 df-tms 24685 df-cncf 25243 df-limc 26230 df-dv 26231 df-ulm 26750 df-log 26930 df-atan 27241 df-em 27366

This theorem is used by: dchrvmasumlem2 27871 mulog2sumlem2 27908

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