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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 (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 13095 . . . 4 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fsumharmonic.b . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3 elfznn 12691 . . . . . . 7 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 475 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
54nncnd 11396 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
64nnne0d 11429 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
72, 5, 6divcld 11153 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
81, 7fsumcl 14875 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
98abscld 14587 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
102abscld 14587 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
1110, 4nndivred 11433 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
121, 11fsumrecl 14876 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13 fsumharmonic.c . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
141, 13fsumrecl 14876 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15 fsumharmonic.r . . . . 5 (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
1615simpld 490 . . . 4 (𝜑𝑅 ∈ ℝ)
17 fsumharmonic.t . . . . . . . 8 (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
1817simpld 490 . . . . . . 7 (𝜑𝑇 ∈ ℝ)
19 0red 10382 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
20 1red 10379 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
21 0lt1 10899 . . . . . . . . 9 0 < 1
2221a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
2317simprd 491 . . . . . . . 8 (𝜑 → 1 ≤ 𝑇)
2419, 20, 18, 22, 23ltletrd 10538 . . . . . . 7 (𝜑 → 0 < 𝑇)
2518, 24elrpd 12182 . . . . . 6 (𝜑𝑇 ∈ ℝ+)
2625relogcld 24810 . . . . 5 (𝜑 → (log‘𝑇) ∈ ℝ)
2726, 20readdcld 10408 . . . 4 (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
2816, 27remulcld 10409 . . 3 (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
2914, 28readdcld 10408 . 2 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
301, 7fsumabs 14941 . . 3 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
312, 5, 6absdivd 14606 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
324nnrpd 12183 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
3332rprege0d 12192 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34 absid 14447 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
3533, 34syl 17 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
3635oveq2d 6940 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
3731, 36eqtrd 2814 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
3837sumeq2dv 14845 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
3930, 38breqtrd 4914 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40 fsumharmonic.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
4140, 25rpdivcld 12202 . . . . . . . . 9 (𝜑 → (𝐴 / 𝑇) ∈ ℝ+)
4241rprege0d 12192 . . . . . . . 8 (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43 flge0nn0 12944 . . . . . . . 8 (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4442, 43syl 17 . . . . . . 7 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4544nn0red 11707 . . . . . 6 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
4645ltp1d 11310 . . . . 5 (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47 fzdisj 12689 . . . . 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 11687 . . . . . . 7 ((⌊‘(𝐴 / 𝑇)) ∈ ℕ0 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
5044, 49syl 17 . . . . . 6 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51 nnuz 12033 . . . . . 6 ℕ = (ℤ‘1)
5250, 51syl6eleq 2869 . . . . 5 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1))
5341rpred 12185 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
5440rpred 12185 . . . . . 6 (𝜑𝐴 ∈ ℝ)
5518, 24jca 507 . . . . . . . . 9 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
5640rpregt0d 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)))
5820, 22, 55, 56, 57syl211anc 1444 . . . . . . . 8 (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5923, 58mpbid 224 . . . . . . 7 (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
6054recnd 10407 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
6160div1d 11145 . . . . . . 7 (𝜑 → (𝐴 / 1) = 𝐴)
6259, 61breqtrd 4914 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63 flword2 12937 . . . . . 6 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
6453, 54, 62, 63syl3anc 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))))
6652, 64, 65syl2anc 579 . . . 4 (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6711recnd 10407 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
6848, 66, 1, 67fsumsplit 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)))
7170, 66syl5sseqr 3873 . . . . . . 7 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
7271sselda 3821 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7372, 11syldan 585 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
7469, 73fsumrecl 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)))
7776, 66syl5sseqr 3873 . . . . . . 7 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
7877sselda 3821 . . . . . 6 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7978, 11syldan 585 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
8075, 79fsumrecl 14876 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
8172, 13syldan 585 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
8269, 81fsumrecl 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))))
8453, 83syl 17 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑇))))
8584simplbda 495 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
8632rpred 12185 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
8754adantr 474 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
8855adantr 474 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89 lemuldiv2 11260 . . . . . . . . . . . 12 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9086, 87, 88, 89syl3anc 1439 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9118adantr 474 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
9291, 87, 32lemuldivd 12234 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑇 ≤ (𝐴 / 𝑛)))
9390, 92bitr3d 273 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9472, 93syldan 585 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9585, 94mpbid 224 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96 fsumharmonic.1 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
9796ex 403 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9872, 97syldan 585 . . . . . . . 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 585 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101100abscld 14587 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
10272, 3syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103102nnrpd 12183 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
104101, 81, 103ledivmul2d 12239 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
10599, 104mpbird 249 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
10669, 73, 81, 105fsumle 14939 . . . . 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 14936 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
10974, 82, 14, 106, 108letrd 10535 . . . 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 11430 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
11275, 111fsumrecl 14876 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
11316, 112remulcld 10409 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
11416adantr 474 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115114recnd 10407 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116110nncnd 11396 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117110nnne0d 11429 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118115, 116, 117divrecd 11156 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119114, 110nndivred 11433 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120118, 119eqeltrrd 2860 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
12178, 10syldan 585 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
12278, 32syldan 585 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(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))))
12548eleq2d 2845 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126124, 125syl5bbr 277 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127123, 126mtbiri 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))))
129127, 128sylibr 226 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130129con2d 132 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131130imp 397 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
13283baibd 535 . . . . . . . . . . . . . . 15 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
13353, 3, 132syl2an 589 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134133, 93bitrd 271 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
13578, 134syldan 585 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136131, 135mtbid 316 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
13754adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138137, 110nndivred 11433 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
13918adantr 474 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140138, 139ltnled 10525 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141136, 140mpbird 249 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142 fsumharmonic.2 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143142ex 403 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
14478, 143syldan 585 . . . . . . . . . 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 12243 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147146, 118breqtrd 4914 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
14875, 79, 120, 147fsumle 14939 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
14916recnd 10407 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
150111recnd 10407 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15175, 149, 150fsummulc2 14924 . . . . . 6 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152148, 151breqtrrd 4916 . . . . 5 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
153102nnrecred 11430 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
15469, 153fsumrecl 14876 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
155154recnd 10407 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
156112recnd 10407 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
1574nnrecred 11430 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
158157recnd 10407 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15948, 66, 1, 158fsumsplit 14882 . . . . . . . 8 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
160155, 156, 159mvrladdd 10790 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
1611, 157fsumrecl 14876 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
162161adantr 474 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
163154adantr 474 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
164162, 163resubcld 10805 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
165 0red 10382 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ∈ ℝ)
16627adantr 474 . . . . . . . . 9 ((𝜑𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
167 fzfid 13095 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
168103adantlr 705 . . . . . . . . . . . . 13 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
169168rpreccld 12195 . . . . . . . . . . . 12 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ+)
170169rpred 12185 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
171169rpge0d 12189 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
17240adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ+)
173172rpge0d 12189 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 0 ≤ 𝐴)
174 simpr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 < 1)
175 0p1e1 11508 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
176174, 175syl6breqr 4930 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 𝐴 < (0 + 1))
17754adantr 474 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ)
178 0z 11743 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
179 flbi 12940 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
180177, 178, 179sylancl 580 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
181173, 176, 180mpbir2and 703 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 1) → (⌊‘𝐴) = 0)
182181oveq2d 6940 . . . . . . . . . . . . 13 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
183 fz10 12683 . . . . . . . . . . . . 13 (1...0) = ∅
184182, 183syl6eq 2830 . . . . . . . . . . . 12 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
185 0ss 4198 . . . . . . . . . . . 12 ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
186184, 185syl6eqss 3874 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
187167, 170, 171, 186fsumless 14936 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
188162, 163suble0d 10968 . . . . . . . . . 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 249 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
19018, 23logge0d 24817 . . . . . . . . . . 11 (𝜑 → 0 ≤ (log‘𝑇))
191 0le1 10900 . . . . . . . . . . . 12 0 ≤ 1
192191a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 1)
19326, 20, 190, 192addge0d 10953 . . . . . . . . . 10 (𝜑 → 0 ≤ ((log‘𝑇) + 1))
194193adantr 474 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
195164, 165, 166, 189, 194letrd 10535 . . . . . . . 8 ((𝜑𝐴 < 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))
19754, 196sylan 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))
19941, 198syl 17 . . . . . . . . . . 11 (𝜑 → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
200199adantr 474 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20140relogcld 24810 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐴) ∈ ℝ)
202 peano2re 10551 . . . . . . . . . . . . 13 ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
203201, 202syl 17 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
20441relogcld 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)))))
206161, 154, 203, 204, 205syl22anc 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)))))
207206adantr 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)))))
208197, 200, 207mp2and 689 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
209201recnd 10407 . . . . . . . . . . . 12 (𝜑 → (log‘𝐴) ∈ ℂ)
21020recnd 10407 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
21126recnd 10407 . . . . . . . . . . . 12 (𝜑 → (log‘𝑇) ∈ ℂ)
212209, 210, 211pnncand 10775 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
21340, 25relogdivd 24813 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
214213oveq2d 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)))
217211, 215, 216sylancl 580 . . . . . . . . . . 11 (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
218212, 214, 2173eqtr4d 2824 . . . . . . . . . 10 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
219218adantr 474 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
220208, 219breqtrd 4914 . . . . . . . 8 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
221195, 220, 54, 20ltlecasei 10486 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
222160, 221eqbrtrrd 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)))
224112, 27, 15, 222, 223syl31anc 1441 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
22580, 113, 28, 152, 224letrd 10535 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
22674, 80, 14, 28, 109, 225le2addd 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))))
22768, 226eqbrtrd 4910 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
2289, 12, 29, 39, 227letrd 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  cc 10272  cr 10273  0cc0 10274  1c1 10275   + caddc 10277   · cmul 10279   < clt 10413  cle 10414  cmin 10608   / cdiv 11034  cn 11378  0cn0 11646  cz 11732  cuz 11996  +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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