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Theorem dchrvmasumiflem1 26004
Description: Lemma for dchrvmasumif 26006. (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 13329 . . 3 ((𝜑𝑚 ∈ ℝ+) → (1...(⌊‘𝑚)) ∈ Fin)
10 simpl 483 . . . . 5 ((𝜑𝑚 ∈ ℝ+) → 𝜑)
11 elfznn 12924 . . . . 5 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
127adantr 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑋𝐷)
13 nnz 11992 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
1413adantl 482 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
154, 1, 5, 2, 12, 14dchrzrhcl 25748 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
1610, 11, 15syl2an 595 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
17 simpr 485 . . . . . . . 8 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
1811nnrpd 12417 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ+)
19 ifcl 4507 . . . . . . . 8 ((𝑚 ∈ ℝ+𝑘 ∈ ℝ+) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2017, 18, 19syl2an 595 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2120relogcld 25133 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
2211adantl 482 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
2321, 22nndivred 11679 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
2423recnd 10657 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
2516, 24mulcld 10649 . . 3 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
269, 25fsumcl 15078 . 2 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
27 fveq2 6663 . . . 4 (𝑚 = (𝑥 / 𝑑) → (⌊‘𝑚) = (⌊‘(𝑥 / 𝑑)))
2827oveq2d 7161 . . 3 (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29 ifeq1 4467 . . . . . . 7 (𝑚 = (𝑥 / 𝑑) → if(𝑆 = 0, 𝑚, 𝑘) = if(𝑆 = 0, (𝑥 / 𝑑), 𝑘))
3029fveq2d 6667 . . . . . 6 (𝑚 = (𝑥 / 𝑑) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)))
3130oveq1d 7160 . . . . 5 (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
3231oveq2d 7161 . . . 4 (𝑚 = (𝑥 / 𝑑) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
3332adantr 481 . . 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 15052 . 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 4508 . 2 (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38 0cn 10621 . . 3 0 ∈ ℂ
39 dchrvmasumif.t . . . 4 (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40 climcl 14844 . . . 4 (seq1( + , 𝐾) ⇝ 𝑇𝑇 ∈ ℂ)
4139, 40syl 17 . . 3 (𝜑𝑇 ∈ ℂ)
42 ifcl 4507 . . 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 12269 . . . . . . . . 9 ℕ = (ℤ‘1)
45 1zzd 12001 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
46 nncn 11634 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
4746adantl 482 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48 nnne0 11659 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
4948adantl 482 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ≠ 0)
5015, 47, 49divcld 11404 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
51 dchrvmasumif.f . . . . . . . . . . . 12 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))
52 2fveq3 6668 . . . . . . . . . . . . . 14 (𝑎 = 𝑘 → (𝑋‘(𝐿𝑎)) = (𝑋‘(𝐿𝑘)))
53 id 22 . . . . . . . . . . . . . 14 (𝑎 = 𝑘𝑎 = 𝑘)
5452, 53oveq12d 7163 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) / 𝑎) = ((𝑋‘(𝐿𝑘)) / 𝑘))
5554cbvmptv 5160 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎)) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5651, 55eqtri 2841 . . . . . . . . . . 11 𝐹 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5750, 56fmptd 6870 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶ℂ)
58 ffvelrn 6841 . . . . . . . . . 10 ((𝐹:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
5957, 58sylan 580 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
6044, 45, 59serf 13386 . . . . . . . 8 (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
6160ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
62 3re 11705 . . . . . . . . . . 11 3 ∈ ℝ
63 elicopnf 12821 . . . . . . . . . . 11 (3 ∈ ℝ → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6462, 63mp1i 13 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6564simprbda 499 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ)
66 1red 10630 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
6762a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
68 1le3 11837 . . . . . . . . . . 11 1 ≤ 3
6968a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 3)
7064simplbda 500 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ≤ 𝑚)
7166, 67, 65, 69, 70letrd 10785 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
72 flge1nn 13179 . . . . . . . . 9 ((𝑚 ∈ ℝ ∧ 1 ≤ 𝑚) → (⌊‘𝑚) ∈ ℕ)
7365, 71, 72syl2anc 584 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ ℕ)
7473adantr 481 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (⌊‘𝑚) ∈ ℕ)
7561, 74ffvelrnd 6844 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
7675abscld 14784 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
77 simpl 483 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝜑)
78 0red 10632 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
79 3pos 11730 . . . . . . . . . . 11 0 < 3
8079a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 3)
8178, 67, 65, 80, 70ltletrd 10788 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
8265, 81elrpd 12416 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ+)
8377, 82jca 512 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝜑𝑚 ∈ ℝ+))
84 elrege0 12830 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
8584simplbi 498 . . . . . . . . 9 (𝐶 ∈ (0[,)+∞) → 𝐶 ∈ ℝ)
8635, 85syl 17 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
87 rerpdivcl 12407 . . . . . . . 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 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (𝐶 / 𝑚) ∈ ℝ)
9182relogcld 25133 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
9265, 71logge0d 25140 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ≤ (log‘𝑚))
9391, 92jca 512 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
9493adantr 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
95 oveq2 7153 . . . . . . . 8 (𝑆 = 0 → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0))
9660adantr 481 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → seq1( + , 𝐹):ℕ⟶ℂ)
9796, 73ffvelrnd 6844 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
9897subid1d 10974 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
9995, 98sylan9eqr 2875 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
10099fveq2d 6667 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) = (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))))
101 2fveq3 6668 . . . . . . . . . 10 (𝑦 = 𝑚 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
102101fvoveq1d 7167 . . . . . . . . 9 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
103 oveq2 7153 . . . . . . . . 9 (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
104102, 103breq12d 5070 . . . . . . . 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 481 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
107 1re 10629 . . . . . . . . . 10 1 ∈ ℝ
108 elicopnf 12821 . . . . . . . . . 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 3622 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
112111adantr 481 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
113100, 112eqbrtrrd 5081 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
114 lemul2a 11483 . . . . 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 1365 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
116 iftrue 4469 . . . . . . . . . . . . . . 15 (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
117116fveq2d 6667 . . . . . . . . . . . . . 14 (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
118117oveq1d 7160 . . . . . . . . . . . . 13 (𝑆 = 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
119118ad2antlr 723 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
120119oveq2d 7161 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)))
12116adantlr 711 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
122 relogcl 25086 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+ → (log‘𝑚) ∈ ℝ)
123122adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℝ)
124123recnd 10657 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℂ)
125124ad2antrr 722 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘𝑚) ∈ ℂ)
12611adantl 482 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
127126nncnd 11642 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
128126nnne0d 11675 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
129121, 125, 127, 128div12d 11440 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
130120, 129eqtrd 2853 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
131130sumeq2dv 15048 . . . . . . . . 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 4469 . . . . . . . . . . 11 (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
133132oveq2d 7161 . . . . . . . . . 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 10974 . . . . . . . . . 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 2875 . . . . . . . . 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 7178 . . . . . . . . . . . . . 14 ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ V
13754, 51, 136fvmpt 6761 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
13822, 137syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
13957adantr 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → 𝐹:ℕ⟶ℂ)
140139, 11, 58syl2an 595 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) ∈ ℂ)
141138, 140eqeltrrd 2911 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
1429, 124, 141fsummulc2 15127 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℝ+) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
143142adantr 481 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
144131, 135, 1433eqtr4d 2863 . . . . . . . 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 2920 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ (ℤ‘1))
14877, 11, 50syl2an 595 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
149146, 147, 148fsumser 15075 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
150149adantr 481 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
151150oveq2d 7161 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
152145, 151eqtrd 2853 . . . . . 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 6667 . . . . 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 723 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℝ)
155154recnd 10657 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
15683, 155sylan 580 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
157156, 75absmuld 14802 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
15891, 92absidd 14770 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
159158oveq1d 7160 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
160159adantr 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
161153, 157, 1603eqtrd 2857 . . . 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 4469 . . . . . . . 8 (𝑆 = 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
163162adantl 482 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
164163oveq1d 7160 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
16586recnd 10657 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
166165ad2antrr 722 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
167 rpcnne0 12395 . . . . . . . 8 (𝑚 ∈ ℝ+ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
168167ad2antlr 723 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
169 div12 11308 . . . . . . 7 ((𝐶 ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
170166, 155, 168, 169syl3anc 1363 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
171164, 170eqtrd 2853 . . . . 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 5089 . . 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 6668 . . . . . . . . 9 (𝑦 = 𝑚 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
176175fvoveq1d 7167 . . . . . . . 8 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
177 fveq2 6663 . . . . . . . . . 10 (𝑦 = 𝑚 → (log‘𝑦) = (log‘𝑚))
178 id 22 . . . . . . . . . 10 (𝑦 = 𝑚𝑦 = 𝑚)
179177, 178oveq12d 7163 . . . . . . . . 9 (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
180179oveq2d 7161 . . . . . . . 8 (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
181176, 180breq12d 5070 . . . . . . 7 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
182181rspccva 3619 . . . . . 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 481 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
185 fveq2 6663 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (log‘𝑎) = (log‘𝑘))
186185, 53oveq12d 7163 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
18752, 186oveq12d 7163 . . . . . . . . . 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 7178 . . . . . . . . . 10 ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
190187, 188, 189fvmpt 6761 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
19111, 190syl 17 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
192 ifnefalse 4475 . . . . . . . . . . . . 13 (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
193192fveq2d 6667 . . . . . . . . . . . 12 (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
194193oveq1d 7160 . . . . . . . . . . 11 (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
195194oveq2d 7161 . . . . . . . . . 10 (𝑆 ≠ 0 → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
196195adantl 482 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
197196eqcomd 2824 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
198191, 197sylan9eqr 2875 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
199147adantr 481 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (⌊‘𝑚) ∈ (ℤ‘1))
200 nnrp 12388 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
201200adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
202201relogcld 25133 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
203202recnd 10657 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
204203, 47, 49divcld 11404 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
20515, 204mulcld 10649 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
206187cbvmptv 5160 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
207188, 206eqtri 2841 . . . . . . . . . . 11 𝐾 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
208205, 207fmptd 6870 . . . . . . . . . 10 (𝜑𝐾:ℕ⟶ℂ)
209208ad2antrr 722 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
210 ffvelrn 6841 . . . . . . . . 9 ((𝐾:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐾𝑘) ∈ ℂ)
211209, 11, 210syl2an 595 . . . . . . . 8 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) ∈ ℂ)
212198, 211eqeltrrd 2911 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
213198, 199, 212fsumser 15075 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
214 ifnefalse 4475 . . . . . . 7 (𝑆 ≠ 0 → if(𝑆 = 0, 0, 𝑇) = 𝑇)
215214adantl 482 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 0, 𝑇) = 𝑇)
216213, 215oveq12d 7163 . . . . 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 6667 . . . 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 4475 . . . . . 6 (𝑆 ≠ 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
219218adantl 482 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
220219oveq1d 7160 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
221184, 217, 2203brtr4d 5089 . . 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 3101 . 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 13329 . . . 4 (𝜑 → (1...2) ∈ Fin)
2247adantr 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑋𝐷)
225 elfzelz 12896 . . . . . . . 8 (𝑘 ∈ (1...2) → 𝑘 ∈ ℤ)
226225adantl 482 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℤ)
2274, 1, 5, 2, 224, 226dchrzrhcl 25748 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
228227abscld 14784 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
229 3rp 12383 . . . . . . 7 3 ∈ ℝ+
230 relogcl 25086 . . . . . . 7 (3 ∈ ℝ+ → (log‘3) ∈ ℝ)
231229, 230ax-mp 5 . . . . . 6 (log‘3) ∈ ℝ
232 elfznn 12924 . . . . . . 7 (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
233232adantl 482 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
234 nndivre 11666 . . . . . 6 (((log‘3) ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((log‘3) / 𝑘) ∈ ℝ)
235231, 233, 234sylancr 587 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
236228, 235remulcld 10659 . . . 4 ((𝜑𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
237223, 236fsumrecl 15079 . . 3 (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
23843abscld 14784 . . 3 (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
239237, 238readdcld 10658 . 2 (𝜑 → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
240 simpl 483 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝜑)
24162rexri 10687 . . . . . . . . . . 11 3 ∈ ℝ*
242 elico2 12788 . . . . . . . . . . 11 ((1 ∈ ℝ ∧ 3 ∈ ℝ*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3)))
243107, 241, 242mp2an 688 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3))
244243simp1bi 1137 . . . . . . . . 9 (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
245244adantl 482 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ)
246 0red 10632 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
247 1red 10630 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
248 0lt1 11150 . . . . . . . . . 10 0 < 1
249248a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 1)
250243simp2bi 1138 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
251250adantl 482 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ≤ 𝑚)
252246, 247, 245, 249, 251ltletrd 10788 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 𝑚)
253245, 252elrpd 12416 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ+)
254240, 253jca 512 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → (𝜑𝑚 ∈ ℝ+))
25543adantr 481 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
25626, 255subcld 10985 . . . . . 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 14784 . . . 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 14784 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
261238adantr 481 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
262260, 261readdcld 10658 . . . 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 481 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
264263, 261readdcld 10658 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
26526, 255abs2dif2d 14806 . . . . 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 14784 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2689, 267fsumrecl 15079 . . . . . . 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 15144 . . . . . . 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 13329 . . . . . . . . 9 ((𝜑𝑚 ∈ ℝ+) → (1...2) ∈ Fin)
273227adantlr 711 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
27417adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑚 ∈ ℝ+)
275232adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
276275nnrpd 12417 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ+)
277274, 276ifcld 4508 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
278277relogcld 25133 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
279278, 275nndivred 11679 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
280279recnd 10657 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
281273, 280mulcld 10649 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
282281abscld 14784 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
283272, 282fsumrecl 15079 . . . . . . . 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 13329 . . . . . . . 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 14784 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288286absge0d 14792 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
289245flcld 13156 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
290 2z 12002 . . . . . . . . . . 11 2 ∈ ℤ
291290a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ ℤ)
292243simp3bi 1139 . . . . . . . . . . . . . 14 (𝑚 ∈ (1[,)3) → 𝑚 < 3)
293292adantl 482 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 < 3)
294 3z 12003 . . . . . . . . . . . . . 14 3 ∈ ℤ
295 fllt 13164 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ ∧ 3 ∈ ℤ) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
296245, 294, 295sylancl 586 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
297293, 296mpbid 233 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < 3)
298 df-3 11689 . . . . . . . . . . . 12 3 = (2 + 1)
299297, 298breqtrdi 5098 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
300 rpre 12385 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+𝑚 ∈ ℝ)
301300adantl 482 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ)
302301flcld 13156 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (⌊‘𝑚) ∈ ℤ)
303 zleltp1 12021 . . . . . . . . . . . . 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 258 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ≤ 2)
307 eluz2 12237 . . . . . . . . . 10 (2 ∈ (ℤ‘(⌊‘𝑚)) ↔ ((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘𝑚) ≤ 2))
308289, 291, 306, 307syl3anbrc 1335 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ (ℤ‘(⌊‘𝑚)))
309 fzss2 12935 . . . . . . . . 9 (2 ∈ (ℤ‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...2))
310308, 309syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...(⌊‘𝑚)) ⊆ (1...2))
311285, 287, 288, 310fsumless 15139 . . . . . . 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 711 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
313273, 280absmuld 14802 . . . . . . . . . 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 25096 . . . . . . . . . . . . . 14 (log‘1) = 0
318 elfzle1 12898 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
319 breq2 5061 . . . . . . . . . . . . . . . . 17 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
320 breq2 5061 . . . . . . . . . . . . . . . . 17 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
321319, 320ifboth 4501 . . . . . . . . . . . . . . . 16 ((1 ≤ 𝑚 ∧ 1 ≤ 𝑘) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
322251, 318, 321syl2an 595 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
323 1rp 12381 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ+
324 logleb 25113 . . . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
328317, 327eqbrtrrid 5093 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
329276rpregt0d 12425 . . . . . . . . . . . . . 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 11497 . . . . . . . . . . . . 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 833 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
333315, 332absidd 14770 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
334333, 315eqeltrd 2910 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ)
335235adantlr 711 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
336228adantlr 711 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
337273absge0d 14792 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘(𝑋‘(𝐿𝑘))))
338336, 337jca 512 . . . . . . . . . . 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 723 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑚 < 3)
341275nnred 11641 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
342 2re 11699 . . . . . . . . . . . . . . . . . 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 12899 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
346345adantl 482 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
347 2lt3 11797 . . . . . . . . . . . . . . . . . 18 2 < 3
348347a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
349341, 343, 344, 346, 348lelttrd 10786 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
350254, 349sylan 580 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
351 breq1 5060 . . . . . . . . . . . . . . . 16 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
352 breq1 5060 . . . . . . . . . . . . . . . 16 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
353351, 352ifboth 4501 . . . . . . . . . . . . . . 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 12419 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
356 ltle 10717 . . . . . . . . . . . . . . . 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 25113 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . 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 12465 . . . . . . . . . . . . 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 233 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘))
368333, 367eqbrtrd 5079 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
369 lemul2a 11483 . . . . . . . . . 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 1365 . . . . . . . . 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 5079 . . . . . . . 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 15142 . . . . . . 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 10785 . . . . . 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 10785 . . . . 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 14784 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
376237adantr 481 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
377255abscld 14784 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
378375, 376, 377leadd1d 11222 . . . . . 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 233 . . . 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 10785 . . 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 3179 . 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 26002 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wss 3933  ifcif 4463   class class class wbr 5057  cmpt 5137  wf 6344  cfv 6348  (class class class)co 7145  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  +∞cpnf 10660  *cxr 10662   < clt 10663  cle 10664  cmin 10858   / cdiv 11285  cn 11626  2c2 11680  3c3 11681  cz 11969  cuz 12231  +crp 12377  [,)cico 12728  ...cfz 12880  cfl 13148  seqcseq 13357  abscabs 14581  cli 14829  𝑂(1)co1 14831  Σcsu 15030  Basecbs 16471  0gc0g 16701  ℤRHomczrh 20575  ℤ/nczn 20578  logclog 25065  μcmu 25599  DChrcdchr 25735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-disj 5023  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-tpos 7881  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-omul 8096  df-er 8278  df-ec 8280  df-qs 8284  df-map 8397  df-pm 8398  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-fi 8863  df-sup 8894  df-inf 8895  df-oi 8962  df-card 9356  df-acn 9359  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ioo 12730  df-ioc 12731  df-ico 12732  df-icc 12733  df-fz 12881  df-fzo 13022  df-fl 13150  df-mod 13226  df-seq 13358  df-exp 13418  df-fac 13622  df-bc 13651  df-hash 13679  df-shft 14414  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-limsup 14816  df-clim 14833  df-rlim 14834  df-o1 14835  df-lo1 14836  df-sum 15031  df-ef 15409  df-e 15410  df-sin 15411  df-cos 15412  df-tan 15413  df-pi 15414  df-dvds 15596  df-prm 16004  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-starv 16568  df-sca 16569  df-vsca 16570  df-ip 16571  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-hom 16577  df-cco 16578  df-rest 16684  df-topn 16685  df-0g 16703  df-gsum 16704  df-topgen 16705  df-pt 16706  df-prds 16709  df-xrs 16763  df-qtop 16768  df-imas 16769  df-qus 16770  df-xps 16771  df-mre 16845  df-mrc 16846  df-acs 16848  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-mulg 18163  df-subg 18214  df-nsg 18215  df-eqg 18216  df-ghm 18294  df-cntz 18385  df-od 18585  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-cring 19229  df-oppr 19302  df-dvdsr 19320  df-unit 19321  df-invr 19351  df-dvr 19362  df-rnghom 19396  df-drng 19433  df-subrg 19462  df-lmod 19565  df-lss 19633  df-lsp 19673  df-sra 19873  df-rgmod 19874  df-lidl 19875  df-rsp 19876  df-2idl 19933  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-zring 20546  df-zrh 20579  df-zn 20582  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cn 21763  df-cnp 21764  df-haus 21851  df-cmp 21923  df-tx 22098  df-hmeo 22291  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-tms 22859  df-cncf 23413  df-limc 24391  df-dv 24392  df-ulm 24892  df-log 25067  df-cxp 25068  df-atan 25372  df-em 25497  df-mu 25605  df-dchr 25736
This theorem is referenced by:  dchrvmasumiflem2  26005
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