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Theorem fsumharmonic 24455
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 12589 . . . 4 (𝜑 → (1...(⌊‘𝐴)) ∈ Fin)
2 fsumharmonic.b . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐵 ∈ ℂ)
3 elfznn 12196 . . . . . . 7 (𝑛 ∈ (1...(⌊‘𝐴)) → 𝑛 ∈ ℕ)
43adantl 480 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℕ)
54nncnd 10883 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
64nnne0d 10912 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ≠ 0)
72, 5, 6divcld 10650 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝐵 / 𝑛) ∈ ℂ)
81, 7fsumcl 14257 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛) ∈ ℂ)
98abscld 13969 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ∈ ℝ)
102abscld 13969 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
1110, 4nndivred 10916 . . 3 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
121, 11fsumrecl 14258 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
13 fsumharmonic.c . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐶 ∈ ℝ)
141, 13fsumrecl 14258 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 ∈ ℝ)
15 fsumharmonic.r . . . . 5 (𝜑 → (𝑅 ∈ ℝ ∧ 0 ≤ 𝑅))
1615simpld 473 . . . 4 (𝜑𝑅 ∈ ℝ)
17 fsumharmonic.t . . . . . . . 8 (𝜑 → (𝑇 ∈ ℝ ∧ 1 ≤ 𝑇))
1817simpld 473 . . . . . . 7 (𝜑𝑇 ∈ ℝ)
19 0red 9897 . . . . . . . 8 (𝜑 → 0 ∈ ℝ)
20 1red 9911 . . . . . . . 8 (𝜑 → 1 ∈ ℝ)
21 0lt1 10399 . . . . . . . . 9 0 < 1
2221a1i 11 . . . . . . . 8 (𝜑 → 0 < 1)
2317simprd 477 . . . . . . . 8 (𝜑 → 1 ≤ 𝑇)
2419, 20, 18, 22, 23ltletrd 10048 . . . . . . 7 (𝜑 → 0 < 𝑇)
2518, 24elrpd 11701 . . . . . 6 (𝜑𝑇 ∈ ℝ+)
2625relogcld 24090 . . . . 5 (𝜑 → (log‘𝑇) ∈ ℝ)
2726, 20readdcld 9925 . . . 4 (𝜑 → ((log‘𝑇) + 1) ∈ ℝ)
2816, 27remulcld 9926 . . 3 (𝜑 → (𝑅 · ((log‘𝑇) + 1)) ∈ ℝ)
2914, 28readdcld 9925 . 2 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))) ∈ ℝ)
301, 7fsumabs 14320 . . 3 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)))
312, 5, 6absdivd 13988 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / (abs‘𝑛)))
324nnrpd 11702 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
3332rprege0d 11711 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ ℝ ∧ 0 ≤ 𝑛))
34 absid 13830 . . . . . . 7 ((𝑛 ∈ ℝ ∧ 0 ≤ 𝑛) → (abs‘𝑛) = 𝑛)
3533, 34syl 17 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘𝑛) = 𝑛)
3635oveq2d 6543 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / (abs‘𝑛)) = ((abs‘𝐵) / 𝑛))
3731, 36eqtrd 2643 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (abs‘(𝐵 / 𝑛)) = ((abs‘𝐵) / 𝑛))
3837sumeq2dv 14227 . . 3 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(abs‘(𝐵 / 𝑛)) = Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
3930, 38breqtrd 4603 . 2 (𝜑 → (abs‘Σ𝑛 ∈ (1...(⌊‘𝐴))(𝐵 / 𝑛)) ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛))
40 fsumharmonic.a . . . . . . . . . 10 (𝜑𝐴 ∈ ℝ+)
4140, 25rpdivcld 11721 . . . . . . . . 9 (𝜑 → (𝐴 / 𝑇) ∈ ℝ+)
4241rprege0d 11711 . . . . . . . 8 (𝜑 → ((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)))
43 flge0nn0 12438 . . . . . . . 8 (((𝐴 / 𝑇) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝑇)) → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4442, 43syl 17 . . . . . . 7 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℕ0)
4544nn0red 11199 . . . . . 6 (𝜑 → (⌊‘(𝐴 / 𝑇)) ∈ ℝ)
4645ltp1d 10803 . . . . 5 (𝜑 → (⌊‘(𝐴 / 𝑇)) < ((⌊‘(𝐴 / 𝑇)) + 1))
47 fzdisj 12194 . . . . 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 11179 . . . . . . 7 ((⌊‘(𝐴 / 𝑇)) ∈ ℕ0 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
5044, 49syl 17 . . . . . 6 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ ℕ)
51 nnuz 11555 . . . . . 6 ℕ = (ℤ‘1)
5250, 51syl6eleq 2697 . . . . 5 (𝜑 → ((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1))
5341rpred 11704 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ∈ ℝ)
5440rpred 11704 . . . . . 6 (𝜑𝐴 ∈ ℝ)
5518, 24jca 552 . . . . . . . . 9 (𝜑 → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
5640rpregt0d 11710 . . . . . . . . 9 (𝜑 → (𝐴 ∈ ℝ ∧ 0 < 𝐴))
57 lediv2 10762 . . . . . . . . 9 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇) ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5820, 22, 55, 56, 57syl211anc 1323 . . . . . . . 8 (𝜑 → (1 ≤ 𝑇 ↔ (𝐴 / 𝑇) ≤ (𝐴 / 1)))
5923, 58mpbid 220 . . . . . . 7 (𝜑 → (𝐴 / 𝑇) ≤ (𝐴 / 1))
6054recnd 9924 . . . . . . . 8 (𝜑𝐴 ∈ ℂ)
6160div1d 10642 . . . . . . 7 (𝜑 → (𝐴 / 1) = 𝐴)
6259, 61breqtrd 4603 . . . . . 6 (𝜑 → (𝐴 / 𝑇) ≤ 𝐴)
63 flword2 12431 . . . . . 6 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐴 / 𝑇) ≤ 𝐴) → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
6453, 54, 62, 63syl3anc 1317 . . . . 5 (𝜑 → (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇))))
65 fzsplit2 12192 . . . . 5 ((((⌊‘(𝐴 / 𝑇)) + 1) ∈ (ℤ‘1) ∧ (⌊‘𝐴) ∈ (ℤ‘(⌊‘(𝐴 / 𝑇)))) → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6652, 64, 65syl2anc 690 . . . 4 (𝜑 → (1...(⌊‘𝐴)) = ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
6711recnd 9924 . . . 4 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℂ)
6848, 66, 1, 67fsumsplit 14264 . . 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 12589 . . . . 5 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
70 ssun1 3737 . . . . . . . 8 (1...(⌊‘(𝐴 / 𝑇))) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7170, 66syl5sseqr 3616 . . . . . . 7 (𝜑 → (1...(⌊‘(𝐴 / 𝑇))) ⊆ (1...(⌊‘𝐴)))
7271sselda 3567 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7372, 11syldan 485 . . . . 5 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
7469, 73fsumrecl 14258 . . . 4 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))((abs‘𝐵) / 𝑛) ∈ ℝ)
75 fzfid 12589 . . . . 5 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ∈ Fin)
76 ssun2 3738 . . . . . . . 8 (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ ((1...(⌊‘(𝐴 / 𝑇))) ∪ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)))
7776, 66syl5sseqr 3616 . . . . . . 7 (𝜑 → (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) ⊆ (1...(⌊‘𝐴)))
7877sselda 3567 . . . . . 6 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ (1...(⌊‘𝐴)))
7978, 11syldan 485 . . . . 5 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ∈ ℝ)
8075, 79fsumrecl 14258 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ∈ ℝ)
8172, 13syldan 485 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐶 ∈ ℝ)
8269, 81fsumrecl 14258 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ∈ ℝ)
83 fznnfl 12478 . . . . . . . . . . 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 651 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ≤ (𝐴 / 𝑇))
8632rpred 11704 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑛 ∈ ℝ)
8754adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
8855adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ∈ ℝ ∧ 0 < 𝑇))
89 lemuldiv2 10753 . . . . . . . . . . . 12 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑇 ∈ ℝ ∧ 0 < 𝑇)) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9086, 87, 88, 89syl3anc 1317 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑇)))
9118adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
9291, 87, 32lemuldivd 11753 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝑇 · 𝑛) ≤ 𝐴𝑇 ≤ (𝐴 / 𝑛)))
9390, 92bitr3d 268 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9472, 93syldan 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (𝑛 ≤ (𝐴 / 𝑇) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
9585, 94mpbid 220 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑇 ≤ (𝐴 / 𝑛))
96 fsumharmonic.1 . . . . . . . . . 10 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ 𝑇 ≤ (𝐴 / 𝑛)) → (abs‘𝐵) ≤ (𝐶 · 𝑛))
9796ex 448 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑇 ≤ (𝐴 / 𝑛) → (abs‘𝐵) ≤ (𝐶 · 𝑛)))
9872, 97syldan 485 . . . . . . . 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 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝐵 ∈ ℂ)
101100abscld 13969 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (abs‘𝐵) ∈ ℝ)
10272, 3syl 17 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℕ)
103102nnrpd 11702 . . . . . . . 8 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
104101, 81, 103ledivmul2d 11758 . . . . . . 7 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (((abs‘𝐵) / 𝑛) ≤ 𝐶 ↔ (abs‘𝐵) ≤ (𝐶 · 𝑛)))
10599, 104mpbird 245 . . . . . 6 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → ((abs‘𝐵) / 𝑛) ≤ 𝐶)
10669, 73, 81, 105fsumle 14318 . . . . 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 14315 . . . . 5 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))𝐶 ≤ Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶)
10974, 82, 14, 106, 108letrd 10045 . . . 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 10913 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
11275, 111fsumrecl 14258 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
11316, 112remulcld 9926 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ∈ ℝ)
11416adantr 479 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℝ)
115114recnd 9924 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑅 ∈ ℂ)
116110nncnd 10883 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℂ)
117110nnne0d 10912 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ≠ 0)
118115, 116, 117divrecd 10653 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) = (𝑅 · (1 / 𝑛)))
119114, 110nndivred 10916 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 / 𝑛) ∈ ℝ)
120118, 119eqeltrrd 2688 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑅 · (1 / 𝑛)) ∈ ℝ)
12178, 10syldan 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (abs‘𝐵) ∈ ℝ)
12278, 32syldan 485 . . . . . . . . 9 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑛 ∈ ℝ+)
123 noel 3877 . . . . . . . . . . . . . . . 16 ¬ 𝑛 ∈ ∅
124 elin 3757 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
12548eleq2d 2672 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1...(⌊‘(𝐴 / 𝑇))) ∩ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
126124, 125syl5bbr 272 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ 𝑛 ∈ ∅))
127123, 126mtbiri 315 . . . . . . . . . . . . . . 15 (𝜑 → ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
128 imnan 436 . . . . . . . . . . . . . . 15 ((𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) ↔ ¬ (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ∧ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
129127, 128sylibr 222 . . . . . . . . . . . . . 14 (𝜑 → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) → ¬ 𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))))
130129con2d 127 . . . . . . . . . . . . 13 (𝜑 → (𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴)) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))))
131130imp 443 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))))
13283baibd 945 . . . . . . . . . . . . . . 15 (((𝐴 / 𝑇) ∈ ℝ ∧ 𝑛 ∈ ℕ) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
13353, 3, 132syl2an 492 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑛 ≤ (𝐴 / 𝑇)))
134133, 93bitrd 266 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
13578, 134syldan 485 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇))) ↔ 𝑇 ≤ (𝐴 / 𝑛)))
136131, 135mtbid 312 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ¬ 𝑇 ≤ (𝐴 / 𝑛))
13754adantr 479 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝐴 ∈ ℝ)
138137, 110nndivred 10916 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) ∈ ℝ)
13918adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → 𝑇 ∈ ℝ)
140138, 139ltnled 10035 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 ↔ ¬ 𝑇 ≤ (𝐴 / 𝑛)))
141136, 140mpbird 245 . . . . . . . . . 10 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (𝐴 / 𝑛) < 𝑇)
142 fsumharmonic.2 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ (1...(⌊‘𝐴))) ∧ (𝐴 / 𝑛) < 𝑇) → (abs‘𝐵) ≤ 𝑅)
143142ex 448 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → ((𝐴 / 𝑛) < 𝑇 → (abs‘𝐵) ≤ 𝑅))
14478, 143syldan 485 . . . . . . . . . 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 11762 . . . . . . . 8 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 / 𝑛))
147146, 118breqtrd 4603 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → ((abs‘𝐵) / 𝑛) ≤ (𝑅 · (1 / 𝑛)))
14875, 79, 120, 147fsumle 14318 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
14916recnd 9924 . . . . . . 7 (𝜑𝑅 ∈ ℂ)
150111recnd 9924 . . . . . . 7 ((𝜑𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15175, 149, 150fsummulc2 14304 . . . . . 6 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(𝑅 · (1 / 𝑛)))
152148, 151breqtrrd 4605 . . . . 5 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
1534nnrecred 10913 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℝ)
154153recnd 9924 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...(⌊‘𝐴))) → (1 / 𝑛) ∈ ℂ)
15548, 66, 1, 154fsumsplit 14264 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) = (Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) + Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)))
156155oveq1d 6542 . . . . . . . 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 10913 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
15869, 157fsumrecl 14258 . . . . . . . . . 10 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
159158recnd 9924 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℂ)
160112recnd 9924 . . . . . . . . 9 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ∈ ℂ)
161159, 160pncan2d 10245 . . . . . . . 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 2643 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) = Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛))
1631, 153fsumrecl 14258 . . . . . . . . . . 11 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
164163adantr 479 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ∈ ℝ)
165158adantr 479 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛) ∈ ℝ)
166164, 165resubcld 10309 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ∈ ℝ)
167 0red 9897 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ∈ ℝ)
16827adantr 479 . . . . . . . . 9 ((𝜑𝐴 < 1) → ((log‘𝑇) + 1) ∈ ℝ)
169 fzfid 12589 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘(𝐴 / 𝑇))) ∈ Fin)
170103adantlr 746 . . . . . . . . . . . . 13 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 𝑛 ∈ ℝ+)
171170rpreccld 11714 . . . . . . . . . . . 12 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ+)
172171rpred 11704 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → (1 / 𝑛) ∈ ℝ)
173171rpge0d 11708 . . . . . . . . . . 11 (((𝜑𝐴 < 1) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))) → 0 ≤ (1 / 𝑛))
17440adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ+)
175174rpge0d 11708 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 0 ≤ 𝐴)
176 simpr 475 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 < 1)
177 0p1e1 10979 . . . . . . . . . . . . . . . 16 (0 + 1) = 1
178176, 177syl6breqr 4619 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → 𝐴 < (0 + 1))
17954adantr 479 . . . . . . . . . . . . . . . 16 ((𝜑𝐴 < 1) → 𝐴 ∈ ℝ)
180 0z 11221 . . . . . . . . . . . . . . . 16 0 ∈ ℤ
181 flbi 12434 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℝ ∧ 0 ∈ ℤ) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
182179, 180, 181sylancl 692 . . . . . . . . . . . . . . 15 ((𝜑𝐴 < 1) → ((⌊‘𝐴) = 0 ↔ (0 ≤ 𝐴𝐴 < (0 + 1))))
183175, 178, 182mpbir2and 958 . . . . . . . . . . . . . 14 ((𝜑𝐴 < 1) → (⌊‘𝐴) = 0)
184183oveq2d 6543 . . . . . . . . . . . . 13 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = (1...0))
185 fz10 12188 . . . . . . . . . . . . 13 (1...0) = ∅
186184, 185syl6eq 2659 . . . . . . . . . . . 12 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) = ∅)
187 0ss 3923 . . . . . . . . . . . 12 ∅ ⊆ (1...(⌊‘(𝐴 / 𝑇)))
188186, 187syl6eqss 3617 . . . . . . . . . . 11 ((𝜑𝐴 < 1) → (1...(⌊‘𝐴)) ⊆ (1...(⌊‘(𝐴 / 𝑇))))
189169, 172, 173, 188fsumless 14315 . . . . . . . . . 10 ((𝜑𝐴 < 1) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
190164, 165suble0d 10467 . . . . . . . . . 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 245 . . . . . . . . 9 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ 0)
19218, 23logge0d 24097 . . . . . . . . . . 11 (𝜑 → 0 ≤ (log‘𝑇))
193 0le1 10400 . . . . . . . . . . . 12 0 ≤ 1
194193a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 1)
19526, 20, 192, 194addge0d 10452 . . . . . . . . . 10 (𝜑 → 0 ≤ ((log‘𝑇) + 1))
196195adantr 479 . . . . . . . . 9 ((𝜑𝐴 < 1) → 0 ≤ ((log‘𝑇) + 1))
197166, 167, 168, 191, 196letrd 10045 . . . . . . . 8 ((𝜑𝐴 < 1) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
198 harmonicubnd 24453 . . . . . . . . . . 11 ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
19954, 198sylan 486 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝐴) + 1))
200 harmoniclbnd 24452 . . . . . . . . . . . 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 479 . . . . . . . . . 10 ((𝜑 ∧ 1 ≤ 𝐴) → (log‘(𝐴 / 𝑇)) ≤ Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛))
20340relogcld 24090 . . . . . . . . . . . . 13 (𝜑 → (log‘𝐴) ∈ ℝ)
204 peano2re 10060 . . . . . . . . . . . . 13 ((log‘𝐴) ∈ ℝ → ((log‘𝐴) + 1) ∈ ℝ)
205203, 204syl 17 . . . . . . . . . . . 12 (𝜑 → ((log‘𝐴) + 1) ∈ ℝ)
20641relogcld 24090 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) ∈ ℝ)
207 le2sub 10376 . . . . . . . . . . . 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 1318 . . . . . . . . . . 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 479 . . . . . . . . . 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 710 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))))
211203recnd 9924 . . . . . . . . . . . 12 (𝜑 → (log‘𝐴) ∈ ℂ)
21220recnd 9924 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℂ)
21326recnd 9924 . . . . . . . . . . . 12 (𝜑 → (log‘𝑇) ∈ ℂ)
214211, 212, 213pnncand 10282 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))) = (1 + (log‘𝑇)))
21540, 25relogdivd 24093 . . . . . . . . . . . 12 (𝜑 → (log‘(𝐴 / 𝑇)) = ((log‘𝐴) − (log‘𝑇)))
216215oveq2d 6543 . . . . . . . . . . 11 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = (((log‘𝐴) + 1) − ((log‘𝐴) − (log‘𝑇))))
217 ax-1cn 9850 . . . . . . . . . . . 12 1 ∈ ℂ
218 addcom 10073 . . . . . . . . . . . 12 (((log‘𝑇) ∈ ℂ ∧ 1 ∈ ℂ) → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
219213, 217, 218sylancl 692 . . . . . . . . . . 11 (𝜑 → ((log‘𝑇) + 1) = (1 + (log‘𝑇)))
220214, 216, 2193eqtr4d 2653 . . . . . . . . . 10 (𝜑 → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
221220adantr 479 . . . . . . . . 9 ((𝜑 ∧ 1 ≤ 𝐴) → (((log‘𝐴) + 1) − (log‘(𝐴 / 𝑇))) = ((log‘𝑇) + 1))
222210, 221breqtrd 4603 . . . . . . . 8 ((𝜑 ∧ 1 ≤ 𝐴) → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
223197, 222, 54, 20ltlecasei 9996 . . . . . . 7 (𝜑 → (Σ𝑛 ∈ (1...(⌊‘𝐴))(1 / 𝑛) − Σ𝑛 ∈ (1...(⌊‘(𝐴 / 𝑇)))(1 / 𝑛)) ≤ ((log‘𝑇) + 1))
224162, 223eqbrtrrd 4601 . . . . . 6 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛) ≤ ((log‘𝑇) + 1))
225 lemul2a 10727 . . . . . 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 1320 . . . . 5 (𝜑 → (𝑅 · Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))(1 / 𝑛)) ≤ (𝑅 · ((log‘𝑇) + 1)))
22780, 113, 28, 152, 226letrd 10045 . . . 4 (𝜑 → Σ𝑛 ∈ (((⌊‘(𝐴 / 𝑇)) + 1)...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (𝑅 · ((log‘𝑇) + 1)))
22874, 80, 14, 28, 109, 227le2addd 10495 . . 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 4599 . 2 (𝜑 → Σ𝑛 ∈ (1...(⌊‘𝐴))((abs‘𝐵) / 𝑛) ≤ (Σ𝑛 ∈ (1...(⌊‘𝐴))𝐶 + (𝑅 · ((log‘𝑇) + 1))))
2309, 12, 29, 39, 229letrd 10045 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 194  wa 382   = wceq 1474  wcel 1976  cun 3537  cin 3538  c0 3873   class class class wbr 4577  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   · cmul 9797   < clt 9930  cle 9931  cmin 10117   / cdiv 10533  cn 10867  0cn0 11139  cz 11210  cuz 11519  +crp 11664  ...cfz 12152  cfl 12408  abscabs 13768  Σcsu 14210  logclog 24022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870  ax-addf 9871  ax-mulf 9872
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-fal 1480  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-2o 7425  df-oadd 7428  df-er 7606  df-map 7723  df-pm 7724  df-ixp 7772  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-fi 8177  df-sup 8208  df-inf 8209  df-oi 8275  df-card 8625  df-cda 8850  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-dec 11326  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-ioo 12006  df-ioc 12007  df-ico 12008  df-icc 12009  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-seq 12619  df-exp 12678  df-fac 12878  df-bc 12907  df-hash 12935  df-shft 13601  df-cj 13633  df-re 13634  df-im 13635  df-sqrt 13769  df-abs 13770  df-limsup 13996  df-clim 14013  df-rlim 14014  df-sum 14211  df-ef 14583  df-e 14584  df-sin 14585  df-cos 14586  df-pi 14588  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-sets 15647  df-ress 15648  df-plusg 15727  df-mulr 15728  df-starv 15729  df-sca 15730  df-vsca 15731  df-ip 15732  df-tset 15733  df-ple 15734  df-ds 15737  df-unif 15738  df-hom 15739  df-cco 15740  df-rest 15852  df-topn 15853  df-0g 15871  df-gsum 15872  df-topgen 15873  df-pt 15874  df-prds 15877  df-xrs 15931  df-qtop 15936  df-imas 15937  df-xps 15939  df-mre 16015  df-mrc 16016  df-acs 16018  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-submnd 17105  df-mulg 17310  df-cntz 17519  df-cmn 17964  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-fbas 19510  df-fg 19511  df-cnfld 19514  df-top 20463  df-bases 20464  df-topon 20465  df-topsp 20466  df-cld 20575  df-ntr 20576  df-cls 20577  df-nei 20654  df-lp 20692  df-perf 20693  df-cn 20783  df-cnp 20784  df-haus 20871  df-tx 21117  df-hmeo 21310  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496  df-xms 21876  df-ms 21877  df-tms 21878  df-cncf 22420  df-limc 23353  df-dv 23354  df-log 24024  df-em 24436
This theorem is referenced by:  dchrvmasumlem2  24904  mulog2sumlem2  24941
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