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Theorem fsumharmonic 25575
 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 (𝜑𝐴 ∈ ℝ+)
fsumharmonic.t (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
fsumharmonic.r (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
fsumharmonic.b ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
fsumharmonic.c ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
fsumharmonic.0 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)
fsumharmonic.1 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
fsumharmonic.2 (((𝜑𝑛 ∈ (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
StepHypRef Expression
1 fzfid 13324 . . . 4 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fsumharmonic.b . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3 elfznn 12919 . . . . . . 7 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 485 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
54nncnd 11631 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
64nnne0d 11665 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
72, 5, 6divcld 11393 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
81, 7fsumcl 15069 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
98abscld 14775 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
102abscld 14775 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
1110, 4nndivred 11669 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
121, 11fsumrecl 15070 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13 fsumharmonic.c . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
141, 13fsumrecl 15070 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15 fsumharmonic.r . . . . 5 (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
1615simpld 498 . . . 4 (𝜑𝑅 ∈ ℝ)
17 fsumharmonic.t . . . . . . . 8 (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
1817simpld 498 . . . . . . 7 (𝜑𝑇 ∈ ℝ)
19 0red 10621 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
20 1red 10619 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
21 0lt1 11139 . . . . . . . . 9 0 < 1
2221a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
2317simprd 499 . . . . . . . 8 (𝜑 → 1 ≤ 𝑇)
2419, 20, 18, 22, 23ltletrd 10777 . . . . . . 7 (𝜑 → 0 < 𝑇)
2518, 24elrpd 12406 . . . . . 6 (𝜑𝑇 ∈ ℝ+)
2625relogcld 25192 . . . . 5 (𝜑 → (log‘𝑇) ∈ ℝ)
2726, 20readdcld 10647 . . . 4 (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
2816, 27remulcld 10648 . . 3 (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
2914, 28readdcld 10647 . 2 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
301, 7fsumabs 15135 . . 3 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
312, 5, 6absdivd 14794 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
324nnrpd 12407 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
3332rprege0d 12416 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34 absid 14635 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
3533, 34syl 17 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
3635oveq2d 7146 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
3731, 36eqtrd 2856 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
3837sumeq2dv 15039 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
3930, 38breqtrd 5065 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40 fsumharmonic.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
4140, 25rpdivcld 12426 . . . . . . . . 9 (𝜑 → (𝐴 / 𝑇) ∈ ℝ+)
4241rprege0d 12416 . . . . . . . 8 (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43 flge0nn0 13173 . . . . . . . 8 (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4442, 43syl 17 . . . . . . 7 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4544nn0red 11934 . . . . . 6 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
4645ltp1d 11547 . . . . 5 (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47 fzdisj 12917 . . . . 5 ((⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1) → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
4846, 47syl 17 . . . 4 (𝜑 → ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) = ∅)
49 nn0p1nn 11914 . . . . . . 7 ((⌊‘(𝐴 / 𝑇)) ∈ ℕ0 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
5044, 49syl 17 . . . . . 6 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51 nnuz 12259 . . . . . 6 ℕ = (ℤ‘1)
5250, 51eleqtrdi 2922 . . . . 5 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1))
5341rpred 12409 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
5440rpred 12409 . . . . . 6 (𝜑𝐴 ∈ ℝ)
5518, 24jca 515 . . . . . . . . 9 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
5640rpregt0d 12415 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57 lediv2 11507 . . . . . . . . 9 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5820, 22, 55, 56, 57syl211anc 1373 . . . . . . . 8 (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5923, 58mpbid 235 . . . . . . 7 (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
6054recnd 10646 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
6160div1d 11385 . . . . . . 7 (𝜑 → (𝐴 / 1) = 𝐴)
6259, 61breqtrd 5065 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63 flword2 13166 . . . . . 6 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
6453, 54, 62, 63syl3anc 1368 . . . . 5 (𝜑 → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
65 fzsplit2 12915 . . . . 5 ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6652, 64, 65syl2anc 587 . . . 4 (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6711recnd 10646 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
6848, 66, 1, 67fsumsplit 15076 . . 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 13324 . . . . 5 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70 ssun1 4124 . . . . . . . 8 (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7170, 66sseqtrrid 3996 . . . . . . 7 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
7271sselda 3943 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7372, 11syldan 594 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
7469, 73fsumrecl 15070 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75 fzfid 13324 . . . . 5 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76 ssun2 4125 . . . . . . . 8 (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7776, 66sseqtrrid 3996 . . . . . . 7 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
7877sselda 3943 . . . . . 6 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7978, 11syldan 594 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
8075, 79fsumrecl 15070 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
8172, 13syldan 594 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
8269, 81fsumrecl 15070 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83 fznnfl 13213 . . . . . . . . . . 11 ((𝐴 / 𝑇) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
8453, 83syl 17 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
8584simplbda 503 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
8632rpred 12409 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
8754adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
8855adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89 lemuldiv2 11498 . . . . . . . . . . . 12 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9086, 87, 88, 89syl3anc 1368 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9118adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
9291, 87, 32lemuldivd 12458 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑇 ≤ (𝐴 / 𝑛)))
9390, 92bitr3d 284 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9472, 93syldan 594 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9585, 94mpbid 235 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96 fsumharmonic.1 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
9796ex 416 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9872, 97syldan 594 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9995, 98mpd 15 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
10072, 2syldan 594 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101100abscld 14775 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
10272, 3syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103102nnrpd 12407 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
104101, 81, 103ledivmul2d 12463 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
10599, 104mpbird 260 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
10669, 73, 81, 105fsumle 15133 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶)
107 fsumharmonic.0 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 0 ≤ 𝐶)
1081, 13, 107, 71fsumless 15130 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
10974, 82, 14, 106, 108letrd 10774 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
11078, 3syl 17 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
111110nnrecred 11666 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
11275, 111fsumrecl 15070 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
11316, 112remulcld 10648 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
11416adantr 484 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115114recnd 10646 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116110nncnd 11631 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117110nnne0d 11665 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118115, 116, 117divrecd 11396 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119114, 110nndivred 11669 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120118, 119eqeltrrd 2913 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
12178, 10syldan 594 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
12278, 32syldan 594 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
123 noel 4270 . . . . . . . . . . . . . . . 16 ¬ 𝑛 ∈ ∅
124 elin 3926 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
12548eleq2d 2897 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126124, 125syl5bbr 288 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127123, 126mtbiri 330 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128 imnan 403 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129127, 128sylibr 237 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130129con2d 136 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131130imp 410 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
13283baibd 543 . . . . . . . . . . . . . . 15 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
13353, 3, 132syl2an 598 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134133, 93bitrd 282 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
13578, 134syldan 594 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136131, 135mtbid 327 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
13754adantr 484 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138137, 110nndivred 11669 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
13918adantr 484 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140138, 139ltnled 10764 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141136, 140mpbird 260 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142 fsumharmonic.2 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143142ex 416 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
14478, 143syldan 594 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
145141, 144mpd 15 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ≤ 𝑅)
146121, 114, 122, 145lediv1dd 12467 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147146, 118breqtrd 5065 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
14875, 79, 120, 147fsumle 15133 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
14916recnd 10646 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
150111recnd 10646 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15175, 149, 150fsummulc2 15118 . . . . . 6 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152148, 151breqtrrd 5067 . . . . 5 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153102nnrecred 11666 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
15469, 153fsumrecl 15070 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155154recnd 10646 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156112recnd 10646 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
1574nnrecred 11666 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158157recnd 10646 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15948, 66, 1, 158fsumsplit 15076 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
160155, 156, 159mvrladdd 11030 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
1611, 157fsumrecl 15070 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162161adantr 484 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163154adantr 484 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164162, 163resubcld 11045 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165 0red 10621 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ∈ ℝ)
16627adantr 484 . . . . . . . . 9 ((𝜑𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167 fzfid 13324 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168103adantlr 714 . . . . . . . . . . . . 13 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
169168rpreccld 12419 . . . . . . . . . . . 12 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ+)
170169rpred 12409 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171169rpge0d 12413 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
17240adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ+)
173172rpge0d 12413 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 0 ≤ 𝐴)
174 simpr 488 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 < 1)
175 0p1e1 11737 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
176174, 175breqtrrdi 5081 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 𝐴 < (0 + 1))
17754adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ)
178 0z 11970 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
179 flbi 13169 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
180177, 178, 179sylancl 589 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
181173, 176, 180mpbir2and 712 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 1) → (⌊‘𝐴) = 0)
182181oveq2d 7146 . . . . . . . . . . . . 13 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183 fz10 12911 . . . . . . . . . . . . 13 (1...0) = ∅
184182, 183syl6eq 2872 . . . . . . . . . . . 12 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185 0ss 4323 . . . . . . . . . . . 12 ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186184, 185eqsstrdi 3997 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187167, 170, 171, 186fsumless 15130 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188162, 163suble0d 11208 . . . . . . . . . 10 ((𝜑𝐴 < 1) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0 ↔ Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)))
189187, 188mpbird 260 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
19018, 23logge0d 25199 . . . . . . . . . . 11 (𝜑 → 0 ≤ (log‘𝑇))
191 0le1 11140 . . . . . . . . . . . 12 0 ≤ 1
192191a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 1)
19326, 20, 190, 192addge0d 11193 . . . . . . . . . 10 (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194193adantr 484 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195164, 165, 166, 189, 194letrd 10774 . . . . . . . 8 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196 harmonicubnd 25573 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
19754, 196sylan 583 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
198 harmoniclbnd 25572 . . . . . . . . . . . 12 ((𝐴 / 𝑇) ∈ ℝ+ → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
19941, 198syl 17 . . . . . . . . . . 11 (𝜑 → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
200199adantr 484 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20140relogcld 25192 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐴) ∈ ℝ)
202 peano2re 10790 . . . . . . . . . . . . 13 ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203201, 202syl 17 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
20441relogcld 25192 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205 le2sub 11116 . . . . . . . . . . . 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)))))
206161, 154, 203, 204, 205syl22anc 837 . . . . . . . . . . 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)))))
207206adantr 484 . . . . . . . . . 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)))))
208197, 200, 207mp2and 698 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209201recnd 10646 . . . . . . . . . . . 12 (𝜑 → (log‘𝐴) ∈ ℂ)
21020recnd 10646 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
21126recnd 10646 . . . . . . . . . . . 12 (𝜑 → (log‘𝑇) ∈ ℂ)
212209, 210, 211pnncand 11013 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
21340, 25relogdivd 25195 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214213oveq2d 7146 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215 ax-1cn 10572 . . . . . . . . . . . 12 1 ∈ ℂ
216 addcom 10803 . . . . . . . . . . . 12 (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
217211, 215, 216sylancl 589 . . . . . . . . . . 11 (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218212, 214, 2173eqtr4d 2866 . . . . . . . . . 10 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219218adantr 484 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220208, 219breqtrd 5065 . . . . . . . 8 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221195, 220, 54, 20ltlecasei 10725 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222160, 221eqbrtrrd 5063 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223 lemul2a 11472 . . . . . 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)))
224112, 27, 15, 222, 223syl31anc 1370 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
22580, 113, 28, 152, 224letrd 10774 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
22674, 80, 14, 28, 109, 225le2addd 11236 . . 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))))
22768, 226eqbrtrd 5061 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
2289, 12, 29, 39, 227letrd 10774 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ∪ cun 3908   ∩ cin 3909  ∅c0 4266   class class class wbr 5039  ‘cfv 6328  (class class class)co 7130  ℂcc 10512  ℝcr 10513  0cc0 10514  1c1 10515   + caddc 10517   · cmul 10519   < clt 10652   ≤ cle 10653   − cmin 10847   / cdiv 11274  ℕcn 11615  ℕ0cn0 11875  ℤcz 11959  ℤ≥cuz 12221  ℝ+crp 12367  ...cfz 12875  ⌊cfl 13143  abscabs 14572  Σcsu 15021  logclog 25124 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-inf2 9080  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591  ax-pre-sup 10592  ax-addf 10593  ax-mulf 10594 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-fi 8851  df-sup 8882  df-inf 8883  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-div 11275  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-xnn0 11946  df-z 11960  df-dec 12077  df-uz 12222  df-q 12327  df-rp 12368  df-xneg 12485  df-xadd 12486  df-xmul 12487  df-ioo 12720  df-ioc 12721  df-ico 12722  df-icc 12723  df-fz 12876  df-fzo 13017  df-fl 13145  df-mod 13221  df-seq 13353  df-exp 13414  df-fac 13618  df-bc 13647  df-hash 13675  df-shft 14405  df-cj 14437  df-re 14438  df-im 14439  df-sqrt 14573  df-abs 14574  df-limsup 14807  df-clim 14824  df-rlim 14825  df-sum 15022  df-ef 15400  df-e 15401  df-sin 15402  df-cos 15403  df-tan 15404  df-pi 15405  df-dvds 15587  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-starv 16558  df-sca 16559  df-vsca 16560  df-ip 16561  df-tset 16562  df-ple 16563  df-ds 16565  df-unif 16566  df-hom 16567  df-cco 16568  df-rest 16674  df-topn 16675  df-0g 16693  df-gsum 16694  df-topgen 16695  df-pt 16696  df-prds 16699  df-xrs 16753  df-qtop 16758  df-imas 16759  df-xps 16761  df-mre 16835  df-mrc 16836  df-acs 16838  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-submnd 17935  df-mulg 18203  df-cntz 18425  df-cmn 18886  df-psmet 20512  df-xmet 20513  df-met 20514  df-bl 20515  df-mopn 20516  df-fbas 20517  df-fg 20518  df-cnfld 20521  df-top 21477  df-topon 21494  df-topsp 21516  df-bases 21529  df-cld 21602  df-ntr 21603  df-cls 21604  df-nei 21681  df-lp 21719  df-perf 21720  df-cn 21810  df-cnp 21811  df-haus 21898  df-cmp 21970  df-tx 22145  df-hmeo 22338  df-fil 22429  df-fm 22521  df-flim 22522  df-flf 22523  df-xms 22905  df-ms 22906  df-tms 22907  df-cncf 23461  df-limc 24447  df-dv 24448  df-ulm 24950  df-log 25126  df-atan 25431  df-em 25556 This theorem is referenced by:  dchrvmasumlem2  26060  mulog2sumlem2  26097
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