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Theorem dchrvmasumiflem1 27545
Description: Lemma for dchrvmasumif 27547. (Contributed by Mario Carneiro, 5-May-2016.)
Hypotheses
Ref Expression
rpvmasum.z 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a (𝜑𝑁 ∈ ℕ)
rpvmasum.g 𝐺 = (DChr‘𝑁)
rpvmasum.d 𝐷 = (Base‘𝐺)
rpvmasum.1 1 = (0g𝐺)
dchrisum.b (𝜑𝑋𝐷)
dchrisum.n1 (𝜑𝑋1 )
dchrvmasumif.f 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))
dchrvmasumif.c (𝜑𝐶 ∈ (0[,)+∞))
dchrvmasumif.s (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
dchrvmasumif.1 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
dchrvmasumif.g 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))
dchrvmasumif.e (𝜑𝐸 ∈ (0[,)+∞))
dchrvmasumif.t (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
dchrvmasumif.2 (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))
Assertion
Ref Expression
dchrvmasumiflem1 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1))
Distinct variable groups:   𝑥,𝑘,𝑦, 1   𝑥,𝑑,𝑦,𝐶   𝑘,𝑑,𝐹,𝑥,𝑦   𝑎,𝑑,𝑘,𝑥,𝑦   𝐸,𝑑,𝑥,𝑦   𝑘,𝐾,𝑦   𝑘,𝑁,𝑥,𝑦   𝜑,𝑑,𝑘,𝑥   𝑇,𝑑,𝑥,𝑦   𝑆,𝑑,𝑘,𝑥,𝑦   𝑘,𝑍,𝑥,𝑦   𝐷,𝑘,𝑥,𝑦   𝐿,𝑎,𝑑,𝑘,𝑥,𝑦   𝑋,𝑎,𝑑,𝑘,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑎)   𝐶(𝑘,𝑎)   𝐷(𝑎,𝑑)   𝑆(𝑎)   𝑇(𝑘,𝑎)   1 (𝑎,𝑑)   𝐸(𝑘,𝑎)   𝐹(𝑎)   𝐺(𝑥,𝑦,𝑘,𝑎,𝑑)   𝐾(𝑥,𝑎,𝑑)   𝑁(𝑎,𝑑)   𝑍(𝑎,𝑑)

Proof of Theorem dchrvmasumiflem1
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 rpvmasum.z . 2 𝑍 = (ℤ/nℤ‘𝑁)
2 rpvmasum.l . 2 𝐿 = (ℤRHom‘𝑍)
3 rpvmasum.a . 2 (𝜑𝑁 ∈ ℕ)
4 rpvmasum.g . 2 𝐺 = (DChr‘𝑁)
5 rpvmasum.d . 2 𝐷 = (Base‘𝐺)
6 rpvmasum.1 . 2 1 = (0g𝐺)
7 dchrisum.b . 2 (𝜑𝑋𝐷)
8 dchrisum.n1 . 2 (𝜑𝑋1 )
9 fzfid 14014 . . 3 ((𝜑𝑚 ∈ ℝ+) → (1...(⌊‘𝑚)) ∈ Fin)
10 simpl 482 . . . . 5 ((𝜑𝑚 ∈ ℝ+) → 𝜑)
11 elfznn 13593 . . . . 5 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
127adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑋𝐷)
13 nnz 12634 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
1413adantl 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
154, 1, 5, 2, 12, 14dchrzrhcl 27289 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
1610, 11, 15syl2an 596 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
17 simpr 484 . . . . . . . 8 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
1811nnrpd 13075 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ+)
19 ifcl 4571 . . . . . . . 8 ((𝑚 ∈ ℝ+𝑘 ∈ ℝ+) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2017, 18, 19syl2an 596 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2120relogcld 26665 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
2211adantl 481 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
2321, 22nndivred 12320 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
2423recnd 11289 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
2516, 24mulcld 11281 . . 3 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
269, 25fsumcl 15769 . 2 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
27 fveq2 6906 . . . 4 (𝑚 = (𝑥 / 𝑑) → (⌊‘𝑚) = (⌊‘(𝑥 / 𝑑)))
2827oveq2d 7447 . . 3 (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29 ifeq1 4529 . . . . . . 7 (𝑚 = (𝑥 / 𝑑) → if(𝑆 = 0, 𝑚, 𝑘) = if(𝑆 = 0, (𝑥 / 𝑑), 𝑘))
3029fveq2d 6910 . . . . . 6 (𝑚 = (𝑥 / 𝑑) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)))
3130oveq1d 7446 . . . . 5 (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
3231oveq2d 7447 . . . 4 (𝑚 = (𝑥 / 𝑑) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
3332adantr 480 . . 3 ((𝑚 = (𝑥 / 𝑑) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
3428, 33sumeq12rdv 15743 . 2 (𝑚 = (𝑥 / 𝑑) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
35 dchrvmasumif.c . . 3 (𝜑𝐶 ∈ (0[,)+∞))
36 dchrvmasumif.e . . 3 (𝜑𝐸 ∈ (0[,)+∞))
3735, 36ifcld 4572 . 2 (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38 0cn 11253 . . 3 0 ∈ ℂ
39 dchrvmasumif.t . . . 4 (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40 climcl 15535 . . . 4 (seq1( + , 𝐾) ⇝ 𝑇𝑇 ∈ ℂ)
4139, 40syl 17 . . 3 (𝜑𝑇 ∈ ℂ)
42 ifcl 4571 . . 3 ((0 ∈ ℂ ∧ 𝑇 ∈ ℂ) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
4338, 41, 42sylancr 587 . 2 (𝜑 → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
44 nnuz 12921 . . . . . . . . 9 ℕ = (ℤ‘1)
45 1zzd 12648 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
46 nncn 12274 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
4746adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48 nnne0 12300 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
4948adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ≠ 0)
5015, 47, 49divcld 12043 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
51 dchrvmasumif.f . . . . . . . . . . . 12 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))
52 2fveq3 6911 . . . . . . . . . . . . . 14 (𝑎 = 𝑘 → (𝑋‘(𝐿𝑎)) = (𝑋‘(𝐿𝑘)))
53 id 22 . . . . . . . . . . . . . 14 (𝑎 = 𝑘𝑎 = 𝑘)
5452, 53oveq12d 7449 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) / 𝑎) = ((𝑋‘(𝐿𝑘)) / 𝑘))
5554cbvmptv 5255 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎)) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5651, 55eqtri 2765 . . . . . . . . . . 11 𝐹 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5750, 56fmptd 7134 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶ℂ)
58 ffvelcdm 7101 . . . . . . . . . 10 ((𝐹:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
5957, 58sylan 580 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
6044, 45, 59serf 14071 . . . . . . . 8 (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
6160ad2antrr 726 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
62 3re 12346 . . . . . . . . . . 11 3 ∈ ℝ
63 elicopnf 13485 . . . . . . . . . . 11 (3 ∈ ℝ → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6462, 63mp1i 13 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6564simprbda 498 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ)
66 1red 11262 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
6762a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
68 1le3 12478 . . . . . . . . . . 11 1 ≤ 3
6968a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 3)
7064simplbda 499 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ≤ 𝑚)
7166, 67, 65, 69, 70letrd 11418 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
72 flge1nn 13861 . . . . . . . . 9 ((𝑚 ∈ ℝ ∧ 1 ≤ 𝑚) → (⌊‘𝑚) ∈ ℕ)
7365, 71, 72syl2anc 584 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ ℕ)
7473adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (⌊‘𝑚) ∈ ℕ)
7561, 74ffvelcdmd 7105 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
7675abscld 15475 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
77 simpl 482 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝜑)
78 0red 11264 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
79 3pos 12371 . . . . . . . . . . 11 0 < 3
8079a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 3)
8178, 67, 65, 80, 70ltletrd 11421 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
8265, 81elrpd 13074 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ+)
8377, 82jca 511 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝜑𝑚 ∈ ℝ+))
84 elrege0 13494 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
8584simplbi 497 . . . . . . . . 9 (𝐶 ∈ (0[,)+∞) → 𝐶 ∈ ℝ)
8635, 85syl 17 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
87 rerpdivcl 13065 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝑚 ∈ ℝ+) → (𝐶 / 𝑚) ∈ ℝ)
8886, 87sylan 580 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (𝐶 / 𝑚) ∈ ℝ)
8983, 88syl 17 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝐶 / 𝑚) ∈ ℝ)
9089adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (𝐶 / 𝑚) ∈ ℝ)
9182relogcld 26665 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
9265, 71logge0d 26672 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ≤ (log‘𝑚))
9391, 92jca 511 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
9493adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
95 oveq2 7439 . . . . . . . 8 (𝑆 = 0 → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0))
9660adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → seq1( + , 𝐹):ℕ⟶ℂ)
9796, 73ffvelcdmd 7105 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
9897subid1d 11609 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
9995, 98sylan9eqr 2799 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
10099fveq2d 6910 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) = (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))))
101 2fveq3 6911 . . . . . . . . . 10 (𝑦 = 𝑚 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
102101fvoveq1d 7453 . . . . . . . . 9 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
103 oveq2 7439 . . . . . . . . 9 (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
104102, 103breq12d 5156 . . . . . . . 8 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚)))
105 dchrvmasumif.1 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
106105adantr 480 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
107 1re 11261 . . . . . . . . . 10 1 ∈ ℝ
108 elicopnf 13485 . . . . . . . . . 10 (1 ∈ ℝ → (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚)))
109107, 108ax-mp 5 . . . . . . . . 9 (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚))
11065, 71, 109sylanbrc 583 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ (1[,)+∞))
111104, 106, 110rspcdva 3623 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
112111adantr 480 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
113100, 112eqbrtrrd 5167 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
114 lemul2a 12122 . . . . 5 ((((abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ ∧ (𝐶 / 𝑚) ∈ ℝ ∧ ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚))) ∧ (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚)) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
11576, 90, 94, 113, 114syl31anc 1375 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
116 iftrue 4531 . . . . . . . . . . . . . . 15 (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
117116fveq2d 6910 . . . . . . . . . . . . . 14 (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
118117oveq1d 7446 . . . . . . . . . . . . 13 (𝑆 = 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
119118ad2antlr 727 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
120119oveq2d 7447 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)))
12116adantlr 715 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
122 relogcl 26617 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+ → (log‘𝑚) ∈ ℝ)
123122adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℝ)
124123recnd 11289 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℂ)
125124ad2antrr 726 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘𝑚) ∈ ℂ)
12611adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
127126nncnd 12282 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
128126nnne0d 12316 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
129121, 125, 127, 128div12d 12079 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
130120, 129eqtrd 2777 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
131130sumeq2dv 15738 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
132 iftrue 4531 . . . . . . . . . . 11 (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
133132oveq2d 7447 . . . . . . . . . 10 (𝑆 = 0 → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − 0))
13426subid1d 11609 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − 0) = Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
135133, 134sylan9eqr 2799 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
136 ovex 7464 . . . . . . . . . . . . . 14 ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ V
13754, 51, 136fvmpt 7016 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
13822, 137syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
13957adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → 𝐹:ℕ⟶ℂ)
140139, 11, 58syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) ∈ ℂ)
141138, 140eqeltrrd 2842 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
1429, 124, 141fsummulc2 15820 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℝ+) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
143142adantr 480 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
144131, 135, 1433eqtr4d 2787 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)))
14583, 144sylan 580 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)))
14683, 138sylan 580 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
14773, 44eleqtrdi 2851 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ (ℤ‘1))
14877, 11, 50syl2an 596 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
149146, 147, 148fsumser 15766 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
150149adantr 480 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
151150oveq2d 7447 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
152145, 151eqtrd 2777 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
153152fveq2d 6910 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))))
154122ad2antlr 727 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℝ)
155154recnd 11289 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
15683, 155sylan 580 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
157156, 75absmuld 15493 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
15891, 92absidd 15461 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
159158oveq1d 7446 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
160159adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
161153, 157, 1603eqtrd 2781 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
162 iftrue 4531 . . . . . . . 8 (𝑆 = 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
163162adantl 481 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
164163oveq1d 7446 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
16586recnd 11289 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
166165ad2antrr 726 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
167 rpcnne0 13053 . . . . . . . 8 (𝑚 ∈ ℝ+ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
168167ad2antlr 727 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
169 div12 11944 . . . . . . 7 ((𝐶 ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
170166, 155, 168, 169syl3anc 1373 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
171164, 170eqtrd 2777 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
17283, 171sylan 580 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
173115, 161, 1723brtr4d 5175 . . 3 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)))
174 dchrvmasumif.2 . . . . . 6 (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))
175 2fveq3 6911 . . . . . . . . 9 (𝑦 = 𝑚 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
176175fvoveq1d 7453 . . . . . . . 8 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
177 fveq2 6906 . . . . . . . . . 10 (𝑦 = 𝑚 → (log‘𝑦) = (log‘𝑚))
178 id 22 . . . . . . . . . 10 (𝑦 = 𝑚𝑦 = 𝑚)
179177, 178oveq12d 7449 . . . . . . . . 9 (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
180179oveq2d 7447 . . . . . . . 8 (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
181176, 180breq12d 5156 . . . . . . 7 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
182181rspccva 3621 . . . . . 6 ((∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
183174, 182sylan 580 . . . . 5 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
184183adantr 480 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
185 fveq2 6906 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (log‘𝑎) = (log‘𝑘))
186185, 53oveq12d 7449 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
18752, 186oveq12d 7449 . . . . . . . . . 10 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
188 dchrvmasumif.g . . . . . . . . . 10 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))
189 ovex 7464 . . . . . . . . . 10 ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
190187, 188, 189fvmpt 7016 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
19111, 190syl 17 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
192 ifnefalse 4537 . . . . . . . . . . . . 13 (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
193192fveq2d 6910 . . . . . . . . . . . 12 (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
194193oveq1d 7446 . . . . . . . . . . 11 (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
195194oveq2d 7447 . . . . . . . . . 10 (𝑆 ≠ 0 → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
196195adantl 481 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
197196eqcomd 2743 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
198191, 197sylan9eqr 2799 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
199147adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (⌊‘𝑚) ∈ (ℤ‘1))
200 nnrp 13046 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
201200adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
202201relogcld 26665 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
203202recnd 11289 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
204203, 47, 49divcld 12043 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
20515, 204mulcld 11281 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
206187cbvmptv 5255 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
207188, 206eqtri 2765 . . . . . . . . . . 11 𝐾 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
208205, 207fmptd 7134 . . . . . . . . . 10 (𝜑𝐾:ℕ⟶ℂ)
209208ad2antrr 726 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
210 ffvelcdm 7101 . . . . . . . . 9 ((𝐾:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐾𝑘) ∈ ℂ)
211209, 11, 210syl2an 596 . . . . . . . 8 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) ∈ ℂ)
212198, 211eqeltrrd 2842 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
213198, 199, 212fsumser 15766 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
214 ifnefalse 4537 . . . . . . 7 (𝑆 ≠ 0 → if(𝑆 = 0, 0, 𝑇) = 𝑇)
215214adantl 481 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 0, 𝑇) = 𝑇)
216213, 215oveq12d 7449 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇))
217216fveq2d 6910 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
218 ifnefalse 4537 . . . . . 6 (𝑆 ≠ 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
219218adantl 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
220219oveq1d 7446 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
221184, 217, 2203brtr4d 5175 . . 3 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)))
222173, 221pm2.61dane 3029 . 2 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)))
223 fzfid 14014 . . . 4 (𝜑 → (1...2) ∈ Fin)
2247adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑋𝐷)
225 elfzelz 13564 . . . . . . . 8 (𝑘 ∈ (1...2) → 𝑘 ∈ ℤ)
226225adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℤ)
2274, 1, 5, 2, 224, 226dchrzrhcl 27289 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
228227abscld 15475 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
229 3rp 13040 . . . . . . 7 3 ∈ ℝ+
230 relogcl 26617 . . . . . . 7 (3 ∈ ℝ+ → (log‘3) ∈ ℝ)
231229, 230ax-mp 5 . . . . . 6 (log‘3) ∈ ℝ
232 elfznn 13593 . . . . . . 7 (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
233232adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
234 nndivre 12307 . . . . . 6 (((log‘3) ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((log‘3) / 𝑘) ∈ ℝ)
235231, 233, 234sylancr 587 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
236228, 235remulcld 11291 . . . 4 ((𝜑𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
237223, 236fsumrecl 15770 . . 3 (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
23843abscld 15475 . . 3 (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
239237, 238readdcld 11290 . 2 (𝜑 → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
240 simpl 482 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝜑)
24162rexri 11319 . . . . . . . . . . 11 3 ∈ ℝ*
242 elico2 13451 . . . . . . . . . . 11 ((1 ∈ ℝ ∧ 3 ∈ ℝ*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3)))
243107, 241, 242mp2an 692 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3))
244243simp1bi 1146 . . . . . . . . 9 (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
245244adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ)
246 0red 11264 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
247 1red 11262 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
248 0lt1 11785 . . . . . . . . . 10 0 < 1
249248a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 1)
250243simp2bi 1147 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
251250adantl 481 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ≤ 𝑚)
252246, 247, 245, 249, 251ltletrd 11421 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 𝑚)
253245, 252elrpd 13074 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ+)
254240, 253jca 511 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → (𝜑𝑚 ∈ ℝ+))
25543adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
25626, 255subcld 11620 . . . . . 6 ((𝜑𝑚 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
257254, 256syl 17 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
258257abscld 15475 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
259254, 26syl 17 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
260259abscld 15475 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
261238adantr 480 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
262260, 261readdcld 11290 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
263237adantr 480 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
264263, 261readdcld 11290 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
26526, 255abs2dif2d 15497 . . . . 5 ((𝜑𝑚 ∈ ℝ+) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))))
266254, 265syl 17 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))))
26725abscld 15475 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2689, 267fsumrecl 15770 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
269254, 268syl 17 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2709, 25fsumabs 15837 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
271254, 270syl 17 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
272 fzfid 14014 . . . . . . . . 9 ((𝜑𝑚 ∈ ℝ+) → (1...2) ∈ Fin)
273227adantlr 715 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
27417adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑚 ∈ ℝ+)
275232adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
276275nnrpd 13075 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ+)
277274, 276ifcld 4572 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
278277relogcld 26665 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
279278, 275nndivred 12320 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
280279recnd 11289 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
281273, 280mulcld 11281 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
282281abscld 15475 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
283272, 282fsumrecl 15770 . . . . . . . 8 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
284254, 283syl 17 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
285 fzfid 14014 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...2) ∈ Fin)
286254, 281sylan 580 . . . . . . . . 9 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
287286abscld 15475 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288286absge0d 15483 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
289245flcld 13838 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
290 2z 12649 . . . . . . . . . . 11 2 ∈ ℤ
291290a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ ℤ)
292243simp3bi 1148 . . . . . . . . . . . . . 14 (𝑚 ∈ (1[,)3) → 𝑚 < 3)
293292adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 < 3)
294 3z 12650 . . . . . . . . . . . . . 14 3 ∈ ℤ
295 fllt 13846 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ ∧ 3 ∈ ℤ) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
296245, 294, 295sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
297293, 296mpbid 232 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < 3)
298 df-3 12330 . . . . . . . . . . . 12 3 = (2 + 1)
299297, 298breqtrdi 5184 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
300 rpre 13043 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+𝑚 ∈ ℝ)
301300adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ)
302301flcld 13838 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (⌊‘𝑚) ∈ ℤ)
303 zleltp1 12668 . . . . . . . . . . . . 13 (((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
304302, 290, 303sylancl 586 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℝ+) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
305254, 304syl 17 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
306299, 305mpbird 257 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ≤ 2)
307 eluz2 12884 . . . . . . . . . 10 (2 ∈ (ℤ‘(⌊‘𝑚)) ↔ ((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘𝑚) ≤ 2))
308289, 291, 306, 307syl3anbrc 1344 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ (ℤ‘(⌊‘𝑚)))
309 fzss2 13604 . . . . . . . . 9 (2 ∈ (ℤ‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...2))
310308, 309syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...(⌊‘𝑚)) ⊆ (1...2))
311285, 287, 288, 310fsumless 15832 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
312236adantlr 715 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
313273, 280absmuld 15493 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) = ((abs‘(𝑋‘(𝐿𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
314254, 313sylan 580 . . . . . . . . 9 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) = ((abs‘(𝑋‘(𝐿𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
315254, 279sylan 580 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
316254, 278sylan 580 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
317 log1 26627 . . . . . . . . . . . . . 14 (log‘1) = 0
318 elfzle1 13567 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
319 breq2 5147 . . . . . . . . . . . . . . . . 17 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
320 breq2 5147 . . . . . . . . . . . . . . . . 17 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
321319, 320ifboth 4565 . . . . . . . . . . . . . . . 16 ((1 ≤ 𝑚 ∧ 1 ≤ 𝑘) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
322251, 318, 321syl2an 596 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
323 1rp 13038 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ+
324 logleb 26645 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℝ+ ∧ if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
325323, 277, 324sylancr 587 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
326254, 325sylan 580 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
327322, 326mpbid 232 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
328317, 327eqbrtrrid 5179 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
329276rpregt0d 13083 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
330254, 329sylan 580 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
331 divge0 12137 . . . . . . . . . . . . 13 ((((log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ ∧ 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))) ∧ (𝑘 ∈ ℝ ∧ 0 < 𝑘)) → 0 ≤ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
332316, 328, 330, 331syl21anc 838 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
333315, 332absidd 15461 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
334333, 315eqeltrd 2841 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ)
335235adantlr 715 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
336228adantlr 715 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
337273absge0d 15483 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘(𝑋‘(𝐿𝑘))))
338336, 337jca 511 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿𝑘)))))
339254, 338sylan 580 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿𝑘)))))
340292ad2antlr 727 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑚 < 3)
341275nnred 12281 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
342 2re 12340 . . . . . . . . . . . . . . . . . 18 2 ∈ ℝ
343342a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 ∈ ℝ)
34462a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 3 ∈ ℝ)
345 elfzle2 13568 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
346345adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
347 2lt3 12438 . . . . . . . . . . . . . . . . . 18 2 < 3
348347a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
349341, 343, 344, 346, 348lelttrd 11419 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
350254, 349sylan 580 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
351 breq1 5146 . . . . . . . . . . . . . . . 16 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
352 breq1 5146 . . . . . . . . . . . . . . . 16 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
353351, 352ifboth 4565 . . . . . . . . . . . . . . 15 ((𝑚 < 3 ∧ 𝑘 < 3) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
354340, 350, 353syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
355277rpred 13077 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
356 ltle 11349 . . . . . . . . . . . . . . . 16 ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ ∧ 3 ∈ ℝ) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
357355, 62, 356sylancl 586 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
358254, 357sylan 580 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
359354, 358mpd 15 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3)
360 logleb 26645 . . . . . . . . . . . . . . 15 ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+ ∧ 3 ∈ ℝ+) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
361277, 229, 360sylancl 586 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
362254, 361sylan 580 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
363359, 362mpbid 232 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3))
364231a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘3) ∈ ℝ)
365278, 364, 276lediv1d 13123 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3) ↔ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘)))
366254, 365sylan 580 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3) ↔ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘)))
367363, 366mpbid 232 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘))
368333, 367eqbrtrd 5165 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
369 lemul2a 12122 . . . . . . . . . 10 ((((abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ ∧ ((log‘3) / 𝑘) ∈ ℝ ∧ ((abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿𝑘))))) ∧ (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘)) → ((abs‘(𝑋‘(𝐿𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
370334, 335, 339, 368, 369syl31anc 1375 . . . . . . . . 9 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
371314, 370eqbrtrd 5165 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
372285, 287, 312, 371fsumle 15835 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
373269, 284, 263, 311, 372letrd 11418 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
374260, 269, 263, 271, 373letrd 11418 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
37526abscld 15475 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
376237adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
377255abscld 15475 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
378375, 376, 377leadd1d 11857 . . . . . 6 ((𝜑𝑚 ∈ ℝ+) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ↔ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇)))))
379254, 378syl 17 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ↔ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇)))))
380374, 379mpbid 232 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
381258, 262, 264, 266, 380letrd 11418 . . 3 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
382381ralrimiva 3146 . 2 (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
3831, 2, 3, 4, 5, 6, 7, 8, 26, 34, 37, 43, 222, 239, 382dchrvmasumlem3 27543 1 (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)))) ∈ 𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  ifcif 4525   class class class wbr 5143  cmpt 5225  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cmin 11492   / cdiv 11920  cn 12266  2c2 12321  3c3 12322  cz 12613  cuz 12878  +crp 13034  [,)cico 13389  ...cfz 13547  cfl 13830  seqcseq 14042  abscabs 15273  cli 15520  𝑂(1)co1 15522  Σcsu 15722  Basecbs 17247  0gc0g 17484  ℤRHomczrh 21510  ℤ/nczn 21513  logclog 26596  μcmu 27138  DChrcdchr 27276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-shft 15106  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15507  df-clim 15524  df-rlim 15525  df-o1 15526  df-lo1 15527  df-sum 15723  df-ef 16103  df-e 16104  df-sin 16105  df-cos 16106  df-tan 16107  df-pi 16108  df-dvds 16291  df-prm 16709  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-qus 17554  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-rhm 20472  df-subrng 20546  df-subrg 20570  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-fbas 21361  df-fg 21362  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-zn 21517  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-lp 23144  df-perf 23145  df-cn 23235  df-cnp 23236  df-haus 23323  df-cmp 23395  df-tx 23570  df-hmeo 23763  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948  df-xms 24330  df-ms 24331  df-tms 24332  df-cncf 24904  df-limc 25901  df-dv 25902  df-ulm 26420  df-log 26598  df-cxp 26599  df-atan 26910  df-em 27036  df-mu 27144  df-dchr 27277
This theorem is referenced by:  dchrvmasumiflem2  27546
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