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Theorem fsumharmonic 24538
 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 12634 . . . 4 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fsumharmonic.b . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3 elfznn 12241 . . . . . . 7 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 481 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
54nncnd 10913 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
64nnne0d 10942 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
72, 5, 6divcld 10680 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
81, 7fsumcl 14311 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
98abscld 14023 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
102abscld 14023 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
1110, 4nndivred 10946 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
121, 11fsumrecl 14312 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13 fsumharmonic.c . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
141, 13fsumrecl 14312 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15 fsumharmonic.r . . . . 5 (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
1615simpld 474 . . . 4 (𝜑𝑅 ∈ ℝ)
17 fsumharmonic.t . . . . . . . 8 (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
1817simpld 474 . . . . . . 7 (𝜑𝑇 ∈ ℝ)
19 0red 9920 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
20 1red 9934 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
21 0lt1 10429 . . . . . . . . 9 0 < 1
2221a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
2317simprd 478 . . . . . . . 8 (𝜑 → 1 ≤ 𝑇)
2419, 20, 18, 22, 23ltletrd 10076 . . . . . . 7 (𝜑 → 0 < 𝑇)
2518, 24elrpd 11745 . . . . . 6 (𝜑𝑇 ∈ ℝ+)
2625relogcld 24173 . . . . 5 (𝜑 → (log‘𝑇) ∈ ℝ)
2726, 20readdcld 9948 . . . 4 (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
2816, 27remulcld 9949 . . 3 (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
2914, 28readdcld 9948 . 2 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
301, 7fsumabs 14374 . . 3 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
312, 5, 6absdivd 14042 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
324nnrpd 11746 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
3332rprege0d 11755 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34 absid 13884 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
3533, 34syl 17 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
3635oveq2d 6565 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
3731, 36eqtrd 2644 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
3837sumeq2dv 14281 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
3930, 38breqtrd 4609 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40 fsumharmonic.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
4140, 25rpdivcld 11765 . . . . . . . . 9 (𝜑 → (𝐴 / 𝑇) ∈ ℝ+)
4241rprege0d 11755 . . . . . . . 8 (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43 flge0nn0 12483 . . . . . . . 8 (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4442, 43syl 17 . . . . . . 7 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4544nn0red 11229 . . . . . 6 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
4645ltp1d 10833 . . . . 5 (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47 fzdisj 12239 . . . . 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 11209 . . . . . . 7 ((⌊‘(𝐴 / 𝑇)) ∈ ℕ0 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
5044, 49syl 17 . . . . . 6 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51 nnuz 11599 . . . . . 6 ℕ = (ℤ‘1)
5250, 51syl6eleq 2698 . . . . 5 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1))
5341rpred 11748 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
5440rpred 11748 . . . . . 6 (𝜑𝐴 ∈ ℝ)
5518, 24jca 553 . . . . . . . . 9 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
5640rpregt0d 11754 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57 lediv2 10792 . . . . . . . . 9 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5820, 22, 55, 56, 57syl211anc 1324 . . . . . . . 8 (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5923, 58mpbid 221 . . . . . . 7 (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
6054recnd 9947 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
6160div1d 10672 . . . . . . 7 (𝜑 → (𝐴 / 1) = 𝐴)
6259, 61breqtrd 4609 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63 flword2 12476 . . . . . 6 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
6453, 54, 62, 63syl3anc 1318 . . . . 5 (𝜑 → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
65 fzsplit2 12237 . . . . 5 ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6652, 64, 65syl2anc 691 . . . 4 (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6711recnd 9947 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
6848, 66, 1, 67fsumsplit 14318 . . 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 12634 . . . . 5 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70 ssun1 3738 . . . . . . . 8 (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7170, 66syl5sseqr 3617 . . . . . . 7 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
7271sselda 3568 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7372, 11syldan 486 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
7469, 73fsumrecl 14312 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75 fzfid 12634 . . . . 5 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76 ssun2 3739 . . . . . . . 8 (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7776, 66syl5sseqr 3617 . . . . . . 7 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
7877sselda 3568 . . . . . 6 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7978, 11syldan 486 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
8075, 79fsumrecl 14312 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
8172, 13syldan 486 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
8269, 81fsumrecl 14312 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83 fznnfl 12523 . . . . . . . . . . 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 652 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
8632rpred 11748 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
8754adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
8855adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89 lemuldiv2 10783 . . . . . . . . . . . 12 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9086, 87, 88, 89syl3anc 1318 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9118adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
9291, 87, 32lemuldivd 11797 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑇 ≤ (𝐴 / 𝑛)))
9390, 92bitr3d 269 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9472, 93syldan 486 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9585, 94mpbid 221 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96 fsumharmonic.1 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
9796ex 449 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9872, 97syldan 486 . . . . . . . 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 486 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101100abscld 14023 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
10272, 3syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103102nnrpd 11746 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
104101, 81, 103ledivmul2d 11802 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
10599, 104mpbird 246 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
10669, 73, 81, 105fsumle 14372 . . . . 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 14369 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
10974, 82, 14, 106, 108letrd 10073 . . . 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 10943 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
11275, 111fsumrecl 14312 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
11316, 112remulcld 9949 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
11416adantr 480 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115114recnd 9947 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116110nncnd 10913 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117110nnne0d 10942 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118115, 116, 117divrecd 10683 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119114, 110nndivred 10946 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120118, 119eqeltrrd 2689 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
12178, 10syldan 486 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
12278, 32syldan 486 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
123 noel 3878 . . . . . . . . . . . . . . . 16 ¬ 𝑛 ∈ ∅
124 elin 3758 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
12548eleq2d 2673 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126124, 125syl5bbr 273 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127123, 126mtbiri 316 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128 imnan 437 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129127, 128sylibr 223 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130129con2d 128 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131130imp 444 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
13283baibd 946 . . . . . . . . . . . . . . 15 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
13353, 3, 132syl2an 493 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134133, 93bitrd 267 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
13578, 134syldan 486 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136131, 135mtbid 313 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
13754adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138137, 110nndivred 10946 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
13918adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140138, 139ltnled 10063 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141136, 140mpbird 246 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142 fsumharmonic.2 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143142ex 449 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
14478, 143syldan 486 . . . . . . . . . 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 11806 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147146, 118breqtrd 4609 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
14875, 79, 120, 147fsumle 14372 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
14916recnd 9947 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
150111recnd 9947 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15175, 149, 150fsummulc2 14358 . . . . . 6 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152148, 151breqtrrd 4611 . . . . 5 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
1534nnrecred 10943 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
154153recnd 9947 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15548, 66, 1, 154fsumsplit 14318 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
156155oveq1d 6564 . . . . . . . 8 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = ((Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)))
157102nnrecred 10943 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
15869, 157fsumrecl 14312 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
159158recnd 9947 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
160112recnd 9947 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
161159, 160pncan2d 10273 . . . . . . . 8 (𝜑 → ((Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
162156, 161eqtrd 2644 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
1631, 153fsumrecl 14312 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
164163adantr 480 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
165158adantr 480 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
166164, 165resubcld 10337 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
167 0red 9920 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ∈ ℝ)
16827adantr 480 . . . . . . . . 9 ((𝜑𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
169 fzfid 12634 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
170103adantlr 747 . . . . . . . . . . . . 13 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
171170rpreccld 11758 . . . . . . . . . . . 12 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ+)
172171rpred 11748 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
173171rpge0d 11752 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
17440adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ+)
175174rpge0d 11752 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 0 ≤ 𝐴)
176 simpr 476 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 < 1)
177 0p1e1 11009 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
178176, 177syl6breqr 4625 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 𝐴 < (0 + 1))
17954adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ)
180 0z 11265 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
181 flbi 12479 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
182179, 180, 181sylancl 693 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
183175, 178, 182mpbir2and 959 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 1) → (⌊‘𝐴) = 0)
184183oveq2d 6565 . . . . . . . . . . . . 13 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
185 fz10 12233 . . . . . . . . . . . . 13 (1...0) = ∅
186184, 185syl6eq 2660 . . . . . . . . . . . 12 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
187 0ss 3924 . . . . . . . . . . . 12 ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
188186, 187syl6eqss 3618 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
189169, 172, 173, 188fsumless 14369 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
190164, 165suble0d 10497 . . . . . . . . . 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)))
191189, 190mpbird 246 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
19218, 23logge0d 24180 . . . . . . . . . . 11 (𝜑 → 0 ≤ (log‘𝑇))
193 0le1 10430 . . . . . . . . . . . 12 0 ≤ 1
194193a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 1)
19526, 20, 192, 194addge0d 10482 . . . . . . . . . 10 (𝜑 → 0 ≤ ((log‘𝑇) + 1))
196195adantr 480 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
197166, 167, 168, 191, 196letrd 10073 . . . . . . . 8 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
198 harmonicubnd 24536 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
19954, 198sylan 487 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
200 harmoniclbnd 24535 . . . . . . . . . . . 12 ((𝐴 / 𝑇) ∈ ℝ+ → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20141, 200syl 17 . . . . . . . . . . 11 (𝜑 → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
202201adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20340relogcld 24173 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐴) ∈ ℝ)
204 peano2re 10088 . . . . . . . . . . . . 13 ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
205203, 204syl 17 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
20641relogcld 24173 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
207 le2sub 10406 . . . . . . . . . . . 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)))))
208163, 158, 205, 206, 207syl22anc 1319 . . . . . . . . . . 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)))))
209208adantr 480 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → ((Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1) ∧ (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇)))))
210199, 202, 209mp2and 711 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
211203recnd 9947 . . . . . . . . . . . 12 (𝜑 → (log‘𝐴) ∈ ℂ)
21220recnd 9947 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
21326recnd 9947 . . . . . . . . . . . 12 (𝜑 → (log‘𝑇) ∈ ℂ)
214211, 212, 213pnncand 10310 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
21540, 25relogdivd 24176 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
216215oveq2d 6565 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
217 ax-1cn 9873 . . . . . . . . . . . 12 1 ∈ ℂ
218 addcom 10101 . . . . . . . . . . . 12 (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
219213, 217, 218sylancl 693 . . . . . . . . . . 11 (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
220214, 216, 2193eqtr4d 2654 . . . . . . . . . 10 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
221220adantr 480 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
222210, 221breqtrd 4609 . . . . . . . 8 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
223197, 222, 54, 20ltlecasei 10024 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
224162, 223eqbrtrrd 4607 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
225 lemul2a 10757 . . . . . 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)))
226112, 27, 15, 224, 225syl31anc 1321 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
22780, 113, 28, 152, 226letrd 10073 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
22874, 80, 14, 28, 109, 227le2addd 10525 . . 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))))
22968, 228eqbrtrd 4605 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
2309, 12, 29, 39, 229letrd 10073 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ℝ+crp 11708  ...cfz 12197  ⌊cfl 12453  abscabs 13822  Σcsu 14264  logclog 24105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-shft 13655  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068  df-sum 14265  df-ef 14637  df-e 14638  df-sin 14639  df-cos 14640  df-pi 14642  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-pt 15928  df-prds 15931  df-xrs 15985  df-qtop 15990  df-imas 15991  df-xps 15993  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-perf 20751  df-cn 20841  df-cnp 20842  df-haus 20929  df-tx 21175  df-hmeo 21368  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-tms 21937  df-cncf 22489  df-limc 23436  df-dv 23437  df-log 24107  df-em 24519 This theorem is referenced by:  dchrvmasumlem2  24987  mulog2sumlem2  25024
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