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Theorem fsumharmonic 26161
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 13693 . . . 4 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fsumharmonic.b . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3 elfznn 13285 . . . . . . 7 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 482 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
54nncnd 11989 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
64nnne0d 12023 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
72, 5, 6divcld 11751 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
81, 7fsumcl 15445 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
98abscld 15148 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
102abscld 15148 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
1110, 4nndivred 12027 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
121, 11fsumrecl 15446 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13 fsumharmonic.c . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
141, 13fsumrecl 15446 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15 fsumharmonic.r . . . . 5 (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
1615simpld 495 . . . 4 (𝜑𝑅 ∈ ℝ)
17 fsumharmonic.t . . . . . . . 8 (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
1817simpld 495 . . . . . . 7 (𝜑𝑇 ∈ ℝ)
19 0red 10978 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
20 1red 10976 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
21 0lt1 11497 . . . . . . . . 9 0 < 1
2221a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
2317simprd 496 . . . . . . . 8 (𝜑 → 1 ≤ 𝑇)
2419, 20, 18, 22, 23ltletrd 11135 . . . . . . 7 (𝜑 → 0 < 𝑇)
2518, 24elrpd 12769 . . . . . 6 (𝜑𝑇 ∈ ℝ+)
2625relogcld 25778 . . . . 5 (𝜑 → (log‘𝑇) ∈ ℝ)
2726, 20readdcld 11004 . . . 4 (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
2816, 27remulcld 11005 . . 3 (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
2914, 28readdcld 11004 . 2 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
301, 7fsumabs 15513 . . 3 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
312, 5, 6absdivd 15167 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
324nnrpd 12770 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
3332rprege0d 12779 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34 absid 15008 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
3533, 34syl 17 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
3635oveq2d 7291 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
3731, 36eqtrd 2778 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
3837sumeq2dv 15415 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
3930, 38breqtrd 5100 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40 fsumharmonic.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
4140, 25rpdivcld 12789 . . . . . . . . 9 (𝜑 → (𝐴 / 𝑇) ∈ ℝ+)
4241rprege0d 12779 . . . . . . . 8 (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43 flge0nn0 13540 . . . . . . . 8 (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4442, 43syl 17 . . . . . . 7 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4544nn0red 12294 . . . . . 6 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
4645ltp1d 11905 . . . . 5 (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47 fzdisj 13283 . . . . 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 12272 . . . . . . 7 ((⌊‘(𝐴 / 𝑇)) ∈ ℕ0 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
5044, 49syl 17 . . . . . 6 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51 nnuz 12621 . . . . . 6 ℕ = (ℤ‘1)
5250, 51eleqtrdi 2849 . . . . 5 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1))
5341rpred 12772 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
5440rpred 12772 . . . . . 6 (𝜑𝐴 ∈ ℝ)
5518, 24jca 512 . . . . . . . . 9 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
5640rpregt0d 12778 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57 lediv2 11865 . . . . . . . . 9 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5820, 22, 55, 56, 57syl211anc 1375 . . . . . . . 8 (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5923, 58mpbid 231 . . . . . . 7 (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
6054recnd 11003 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
6160div1d 11743 . . . . . . 7 (𝜑 → (𝐴 / 1) = 𝐴)
6259, 61breqtrd 5100 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63 flword2 13533 . . . . . 6 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
6453, 54, 62, 63syl3anc 1370 . . . . 5 (𝜑 → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
65 fzsplit2 13281 . . . . 5 ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6652, 64, 65syl2anc 584 . . . 4 (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6711recnd 11003 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
6848, 66, 1, 67fsumsplit 15453 . . 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 13693 . . . . 5 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70 ssun1 4106 . . . . . . . 8 (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7170, 66sseqtrrid 3974 . . . . . . 7 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
7271sselda 3921 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7372, 11syldan 591 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
7469, 73fsumrecl 15446 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75 fzfid 13693 . . . . 5 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76 ssun2 4107 . . . . . . . 8 (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7776, 66sseqtrrid 3974 . . . . . . 7 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
7877sselda 3921 . . . . . 6 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7978, 11syldan 591 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
8075, 79fsumrecl 15446 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
8172, 13syldan 591 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
8269, 81fsumrecl 15446 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83 fznnfl 13582 . . . . . . . . . . 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 500 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
8632rpred 12772 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
8754adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
8855adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89 lemuldiv2 11856 . . . . . . . . . . . 12 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9086, 87, 88, 89syl3anc 1370 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9118adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
9291, 87, 32lemuldivd 12821 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑇 ≤ (𝐴 / 𝑛)))
9390, 92bitr3d 280 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9472, 93syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9585, 94mpbid 231 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96 fsumharmonic.1 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
9796ex 413 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9872, 97syldan 591 . . . . . . . 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 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101100abscld 15148 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
10272, 3syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103102nnrpd 12770 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
104101, 81, 103ledivmul2d 12826 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
10599, 104mpbird 256 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
10669, 73, 81, 105fsumle 15511 . . . . 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 15508 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
10974, 82, 14, 106, 108letrd 11132 . . . 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 12024 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
11275, 111fsumrecl 15446 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
11316, 112remulcld 11005 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
11416adantr 481 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115114recnd 11003 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116110nncnd 11989 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117110nnne0d 12023 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118115, 116, 117divrecd 11754 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119114, 110nndivred 12027 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120118, 119eqeltrrd 2840 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
12178, 10syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
12278, 32syldan 591 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
123 noel 4264 . . . . . . . . . . . . . . . 16 ¬ 𝑛 ∈ ∅
124 elin 3903 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
12548eleq2d 2824 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126124, 125bitr3id 285 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127123, 126mtbiri 327 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128 imnan 400 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129127, 128sylibr 233 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130129con2d 134 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131130imp 407 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
13283baibd 540 . . . . . . . . . . . . . . 15 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
13353, 3, 132syl2an 596 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134133, 93bitrd 278 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
13578, 134syldan 591 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136131, 135mtbid 324 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
13754adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138137, 110nndivred 12027 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
13918adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140138, 139ltnled 11122 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141136, 140mpbird 256 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142 fsumharmonic.2 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143142ex 413 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
14478, 143syldan 591 . . . . . . . . . 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 12830 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147146, 118breqtrd 5100 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
14875, 79, 120, 147fsumle 15511 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
14916recnd 11003 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
150111recnd 11003 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15175, 149, 150fsummulc2 15496 . . . . . 6 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152148, 151breqtrrd 5102 . . . . 5 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153102nnrecred 12024 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
15469, 153fsumrecl 15446 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155154recnd 11003 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156112recnd 11003 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
1574nnrecred 12024 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158157recnd 11003 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15948, 66, 1, 158fsumsplit 15453 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
160155, 156, 159mvrladdd 11388 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
1611, 157fsumrecl 15446 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162161adantr 481 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163154adantr 481 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164162, 163resubcld 11403 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165 0red 10978 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ∈ ℝ)
16627adantr 481 . . . . . . . . 9 ((𝜑𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167 fzfid 13693 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168103adantlr 712 . . . . . . . . . . . . 13 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
169168rpreccld 12782 . . . . . . . . . . . 12 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ+)
170169rpred 12772 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171169rpge0d 12776 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
17240adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ+)
173172rpge0d 12776 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 0 ≤ 𝐴)
174 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 < 1)
175 0p1e1 12095 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
176174, 175breqtrrdi 5116 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 𝐴 < (0 + 1))
17754adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ)
178 0z 12330 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
179 flbi 13536 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
180177, 178, 179sylancl 586 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
181173, 176, 180mpbir2and 710 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 1) → (⌊‘𝐴) = 0)
182181oveq2d 7291 . . . . . . . . . . . . 13 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183 fz10 13277 . . . . . . . . . . . . 13 (1...0) = ∅
184182, 183eqtrdi 2794 . . . . . . . . . . . 12 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185 0ss 4330 . . . . . . . . . . . 12 ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186184, 185eqsstrdi 3975 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187167, 170, 171, 186fsumless 15508 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188162, 163suble0d 11566 . . . . . . . . . 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 256 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
19018, 23logge0d 25785 . . . . . . . . . . 11 (𝜑 → 0 ≤ (log‘𝑇))
191 0le1 11498 . . . . . . . . . . . 12 0 ≤ 1
192191a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 1)
19326, 20, 190, 192addge0d 11551 . . . . . . . . . 10 (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194193adantr 481 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195164, 165, 166, 189, 194letrd 11132 . . . . . . . 8 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
196 harmonicubnd 26159 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
19754, 196sylan 580 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
198 harmoniclbnd 26158 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20140relogcld 25778 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐴) ∈ ℝ)
202 peano2re 11148 . . . . . . . . . . . . 13 ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203201, 202syl 17 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
20441relogcld 25778 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
205 le2sub 11474 . . . . . . . . . . . 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 836 . . . . . . . . . . 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 481 . . . . . . . . . 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 696 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209201recnd 11003 . . . . . . . . . . . 12 (𝜑 → (log‘𝐴) ∈ ℂ)
21020recnd 11003 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
21126recnd 11003 . . . . . . . . . . . 12 (𝜑 → (log‘𝑇) ∈ ℂ)
212209, 210, 211pnncand 11371 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
21340, 25relogdivd 25781 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214213oveq2d 7291 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
215 ax-1cn 10929 . . . . . . . . . . . 12 1 ∈ ℂ
216 addcom 11161 . . . . . . . . . . . 12 (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
217211, 215, 216sylancl 586 . . . . . . . . . . 11 (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218212, 214, 2173eqtr4d 2788 . . . . . . . . . 10 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219218adantr 481 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220208, 219breqtrd 5100 . . . . . . . 8 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221195, 220, 54, 20ltlecasei 11083 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222160, 221eqbrtrrd 5098 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
223 lemul2a 11830 . . . . . 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 1372 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
22580, 113, 28, 152, 224letrd 11132 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
22674, 80, 14, 28, 109, 225le2addd 11594 . . 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 5096 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
2289, 12, 29, 39, 227letrd 11132 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 205  wa 396   = wceq 1539  wcel 2106  cun 3885  cin 3886  c0 4256   class class class wbr 5074  cfv 6433  (class class class)co 7275  cc 10869  cr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876   < clt 11009  cle 11010  cmin 11205   / cdiv 11632  cn 11973  0cn0 12233  cz 12319  cuz 12582  +crp 12730  ...cfz 13239  cfl 13510  abscabs 14945  Σcsu 15397  logclog 25710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-fi 9170  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-q 12689  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ioo 13083  df-ioc 13084  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-fac 13988  df-bc 14017  df-hash 14045  df-shft 14778  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198  df-sum 15398  df-ef 15777  df-e 15778  df-sin 15779  df-cos 15780  df-tan 15781  df-pi 15782  df-dvds 15964  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-sca 16978  df-vsca 16979  df-ip 16980  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-hom 16986  df-cco 16987  df-rest 17133  df-topn 17134  df-0g 17152  df-gsum 17153  df-topgen 17154  df-pt 17155  df-prds 17158  df-xrs 17213  df-qtop 17218  df-imas 17219  df-xps 17221  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-mulg 18701  df-cntz 18923  df-cmn 19388  df-psmet 20589  df-xmet 20590  df-met 20591  df-bl 20592  df-mopn 20593  df-fbas 20594  df-fg 20595  df-cnfld 20598  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cld 22170  df-ntr 22171  df-cls 22172  df-nei 22249  df-lp 22287  df-perf 22288  df-cn 22378  df-cnp 22379  df-haus 22466  df-cmp 22538  df-tx 22713  df-hmeo 22906  df-fil 22997  df-fm 23089  df-flim 23090  df-flf 23091  df-xms 23473  df-ms 23474  df-tms 23475  df-cncf 24041  df-limc 25030  df-dv 25031  df-ulm 25536  df-log 25712  df-atan 26017  df-em 26142
This theorem is referenced by:  dchrvmasumlem2  26646  mulog2sumlem2  26683
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