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Theorem fsumharmonic 25194

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 13095	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2		fsumharmonic.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3		elfznn 12691	. . . . . . 7 ⊢ (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
4	3	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
5	4	nncnd 11396	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
6	4	nnne0d 11429	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
7	2, 5, 6	divcld 11153	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
8	1, 7	fsumcl 14875	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
9	8	abscld 14587	. 2 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
10	2	abscld 14587	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
11	10, 4	nndivred 11433	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
12	1, 11	fsumrecl 14876	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13		fsumharmonic.c	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
14	1, 13	fsumrecl 14876	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15		fsumharmonic.r	. . . . 5 ⊢ (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
16	15	simpld 490	. . . 4 ⊢ (𝜑 → 𝑅 ∈ ℝ)
17		fsumharmonic.t	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
18	17	simpld 490	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℝ)
19		0red 10382	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℝ)
20		1red 10379	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ ℝ)
21		0lt1 10899	. . . . . . . . 9 ⊢ 0 < 1
22	21	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 < 1)
23	17	simprd 491	. . . . . . . 8 ⊢ (𝜑 → 1 ≤ 𝑇)
24	19, 20, 18, 22, 23	ltletrd 10538	. . . . . . 7 ⊢ (𝜑 → 0 < 𝑇)
25	18, 24	elrpd 12182	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ ℝ⁺)
26	25	relogcld 24810	. . . . 5 ⊢ (𝜑 → (log‘𝑇) ∈ ℝ)
27	26, 20	readdcld 10408	. . . 4 ⊢ (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
28	16, 27	remulcld 10409	. . 3 ⊢ (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
29	14, 28	readdcld 10408	. 2 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
30	1, 7	fsumabs 14941	. . 3 ⊢ (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
31	2, 5, 6	absdivd 14606	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
32	4	nnrpd 12183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
33	32	rprege0d 12192	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34		absid 14447	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
35	33, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
36	35	oveq2d 6940	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
37	31, 36	eqtrd 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
38	37	sumeq2dv 14845	. . 3 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
39	30, 38	breqtrd 4914	. 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 12202	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ⁺)
42	41	rprege0d 12192	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43		flge0nn0 12944	. . . . . . . 8 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
44	42, 43	syl 17	. . . . . . 7 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ₀)
45	44	nn0red 11707	. . . . . 6 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
46	45	ltp1d 11310	. . . . 5 ⊢ (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47		fzdisj 12689	. . . . 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 11687	. . . . . . 7 ⊢ ((⌊‘(𝐴 / 𝑇)) ∈ ℕ₀ → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
50	44, 49	syl 17	. . . . . 6 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51		nnuz 12033	. . . . . 6 ⊢ ℕ = (ℤ_≥‘1)
52	50, 51	syl6eleq 2869	. . . . 5 ⊢ (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1))
53	41	rpred 12185	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
54	40	rpred 12185	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ)
55	18, 24	jca 507	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
56	40	rpregt0d 12191	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57		lediv2 11269	. . . . . . . . 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 1444	. . . . . . . 8 ⊢ (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
59	23, 58	mpbid 224	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
60	54	recnd 10407	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℂ)
61	60	div1d 11145	. . . . . . 7 ⊢ (𝜑 → (𝐴 / 1) = 𝐴)
62	59, 61	breqtrd 4914	. . . . . 6 ⊢ (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63		flword2 12937	. . . . . 6 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
64	53, 54, 62, 63	syl3anc 1439	. . . . 5 ⊢ (𝜑 → (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇))))
65		fzsplit2 12687	. . . . 5 ⊢ ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ_≥‘1) ∧ (⌊‘𝐴) ∈ (ℤ_≥‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
66	52, 64, 65	syl2anc 579	. . . 4 ⊢ (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
67	11	recnd 10407	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
68	48, 66, 1, 67	fsumsplit 14882	. . 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 13095	. . . . 5 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70		ssun1 3999	. . . . . . . 8 ⊢ (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
71	70, 66	syl5sseqr 3873	. . . . . . 7 ⊢ (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
72	71	sselda 3821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
73	72, 11	syldan 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
74	69, 73	fsumrecl 14876	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75		fzfid 13095	. . . . 5 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76		ssun2 4000	. . . . . . . 8 ⊢ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
77	76, 66	syl5sseqr 3873	. . . . . . 7 ⊢ (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
78	77	sselda 3821	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
79	78, 11	syldan 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
80	75, 79	fsumrecl 14876	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
81	72, 13	syldan 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
82	69, 81	fsumrecl 14876	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83		fznnfl 12984	. . . . . . . . . . 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 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
86	32	rpred 12185	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
87	54	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
88	55	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89		lemuldiv2 11260	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
90	86, 87, 88, 89	syl3anc 1439	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑇)))
91	18	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
92	91, 87, 32	lemuldivd 12234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴 ↔ 𝑇 ≤ (𝐴 / 𝑛)))
93	90, 92	bitr3d 273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
94	72, 93	syldan 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
95	85, 94	mpbid 224	. . . . . . . 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 403	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
98	72, 97	syldan 585	. . . . . . . 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 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101	100	abscld 14587	. . . . . . . 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 12183	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
104	101, 81, 103	ledivmul2d 12239	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
105	99, 104	mpbird 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
106	69, 73, 81, 105	fsumle 14939	. . . . 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 14936	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
109	74, 82, 14, 106, 108	letrd 10535	. . . 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 11430	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
112	75, 111	fsumrecl 14876	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
113	16, 112	remulcld 10409	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
114	16	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115	114	recnd 10407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116	110	nncnd 11396	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117	110	nnne0d 11429	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118	115, 116, 117	divrecd 11156	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119	114, 110	nndivred 11433	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120	118, 119	eqeltrrd 2860	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
121	78, 10	syldan 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
122	78, 32	syldan 585	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ⁺)
123		noel 4146	. . . . . . . . . . . . . . . 16 ⊢ ¬ 𝑛 ∈ ∅
124		elin 4019	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
125	48	eleq2d 2845	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126	124, 125	syl5bbr 277	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127	123, 126	mtbiri 319	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128		imnan 390	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129	127, 128	sylibr 226	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130	129	con2d 132	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131	130	imp 397	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
132	83	baibd 535	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
133	53, 3, 132	syl2an 589	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134	133, 93	bitrd 271	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
135	78, 134	syldan 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136	131, 135	mtbid 316	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
137	54	adantr 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138	137, 110	nndivred 11433	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
139	18	adantr 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140	138, 139	ltnled 10525	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141	136, 140	mpbird 249	. . . . . . . . . 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 403	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
144	78, 143	syldan 585	. . . . . . . . . 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 12243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147	146, 118	breqtrd 4914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
148	75, 79, 120, 147	fsumle 14939	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
149	16	recnd 10407	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℂ)
150	111	recnd 10407	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
151	75, 149, 150	fsummulc2 14924	. . . . . 6 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152	148, 151	breqtrrd 4916	. . . . 5 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153	102	nnrecred 11430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
154	69, 153	fsumrecl 14876	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155	154	recnd 10407	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156	112	recnd 10407	. . . . . . . 8 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
157	4	nnrecred 11430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158	157	recnd 10407	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
159	48, 66, 1, 158	fsumsplit 14882	. . . . . . . 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 10790	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
161	1, 157	fsumrecl 14876	. . . . . . . . . . 11 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162	161	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163	154	adantr 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164	162, 163	resubcld 10805	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165		0red 10382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ∈ ℝ)
166	27	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167		fzfid 13095	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168	103	adantlr 705	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ⁺)
169	168	rpreccld 12195	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ⁺)
170	169	rpred 12185	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171	169	rpge0d 12189	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
172	40	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
173	172	rpge0d 12189	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ 𝐴)
174		simpr 479	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < 1)
175		0p1e1 11508	. . . . . . . . . . . . . . . 16 ⊢ (0 + 1) = 1
176	174, 175	syl6breqr 4930	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 < (0 + 1))
177	54	adantr 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝐴 < 1) → 𝐴 ∈ ℝ)
178		0z 11743	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
179		flbi 12940	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
180	177, 178, 179	sylancl 580	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴 ∧ 𝐴 < (0 + 1))))
181	173, 176, 180	mpbir2and 703	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐴 < 1) → (⌊‘𝐴) = 0)
182	181	oveq2d 6940	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183		fz10 12683	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
184	182, 183	syl6eq 2830	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185		0ss 4198	. . . . . . . . . . . 12 ⊢ ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186	184, 185	syl6eqss 3874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187	167, 170, 171, 186	fsumless 14936	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188	162, 163	suble0d 10968	. . . . . . . . . 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 249	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
190	18, 23	logge0d 24817	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (log‘𝑇))
191		0le1 10900	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
192	191	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 1)
193	26, 20, 190, 192	addge0d 10953	. . . . . . . . . 10 ⊢ (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194	193	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195	164, 165, 166, 189, 194	letrd 10535	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196		harmonicubnd 25192	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
197	54, 196	sylan 575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
198		harmoniclbnd 25191	. . . . . . . . . . . 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 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
201	40	relogcld 24810	. . . . . . . . . . . . 13 ⊢ (𝜑 → (log‘𝐴) ∈ ℝ)
202		peano2re 10551	. . . . . . . . . . . . 13 ⊢ ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203	201, 202	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
204	41	relogcld 24810	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205		le2sub 10876	. . . . . . . . . . . 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 829	. . . . . . . . . . 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 474	. . . . . . . . . 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 689	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209	201	recnd 10407	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝐴) ∈ ℂ)
210	20	recnd 10407	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℂ)
211	26	recnd 10407	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘𝑇) ∈ ℂ)
212	209, 210, 211	pnncand 10775	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
213	40, 25	relogdivd 24813	. . . . . . . . . . . 12 ⊢ (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214	213	oveq2d 6940	. . . . . . . . . . 11 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215		ax-1cn 10332	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
216		addcom 10564	. . . . . . . . . . . 12 ⊢ (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
217	211, 215, 216	sylancl 580	. . . . . . . . . . 11 ⊢ (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218	212, 214, 217	3eqtr4d 2824	. . . . . . . . . 10 ⊢ (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219	218	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220	208, 219	breqtrd 4914	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221	195, 220, 54, 20	ltlecasei 10486	. . . . . . 7 ⊢ (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222	160, 221	eqbrtrrd 4912	. . . . . 6 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223		lemul2a 11234	. . . . . 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 1441	. . . . 5 ⊢ (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
225	80, 113, 28, 152, 224	letrd 10535	. . . 4 ⊢ (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
226	74, 80, 14, 28, 109, 225	le2addd 10996	. . 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 4910	. 2 ⊢ (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
228	9, 12, 29, 39, 227	letrd 10535	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ∩ cin 3791 ∅c0 4141 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 1c1 10275 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 − cmin 10608 / cdiv 11034 ℕcn 11378 ℕ₀cn0 11646 ℤcz 11732 ℤ_≥cuz 11996 ℝ⁺crp 12141 ...cfz 12647 ⌊cfl 12914 abscabs 14385 Σcsu 14828 logclog 24742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-2o 7846 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-ixp 8197 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-cda 9327 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-ioo 12495 df-ioc 12496 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-fl 12916 df-mod 12992 df-seq 13124 df-exp 13183 df-fac 13383 df-bc 13412 df-hash 13440 df-shft 14218 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-limsup 14614 df-clim 14631 df-rlim 14632 df-sum 14829 df-ef 15204 df-e 15205 df-sin 15206 df-cos 15207 df-pi 15209 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-sca 16358 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-hom 16366 df-cco 16367 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-pt 16495 df-prds 16498 df-xrs 16552 df-qtop 16557 df-imas 16558 df-xps 16560 df-mre 16636 df-mrc 16637 df-acs 16639 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-mulg 17932 df-cntz 18137 df-cmn 18585 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lp 21352 df-perf 21353 df-cn 21443 df-cnp 21444 df-haus 21531 df-tx 21778 df-hmeo 21971 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-xms 22537 df-ms 22538 df-tms 22539 df-cncf 23093 df-limc 24071 df-dv 24072 df-log 24744 df-em 25175

This theorem is referenced by: dchrvmasumlem2 25643 mulog2sumlem2 25680

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