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Theorem dchrvmasumiflem1 25235
 Description: Lemma for dchrvmasumif 25237. (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 12812 . . 3 ((𝜑𝑚 ∈ ℝ+) → (1...(⌊‘𝑚)) ∈ Fin)
10 simpl 472 . . . . 5 ((𝜑𝑚 ∈ ℝ+) → 𝜑)
11 elfznn 12408 . . . . 5 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
127adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑋𝐷)
13 nnz 11437 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
1413adantl 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
154, 1, 5, 2, 12, 14dchrzrhcl 25015 . . . . 5 ((𝜑𝑘 ∈ ℕ) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
1610, 11, 15syl2an 493 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
17 simpr 476 . . . . . . . 8 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ+)
1811nnrpd 11908 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ+)
19 ifcl 4163 . . . . . . . 8 ((𝑚 ∈ ℝ+𝑘 ∈ ℝ+) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2017, 18, 19syl2an 493 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
2120relogcld 24414 . . . . . 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 11107 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
2423recnd 10106 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
2516, 24mulcld 10098 . . 3 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
269, 25fsumcl 14508 . 2 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
27 fveq2 6229 . . . 4 (𝑚 = (𝑥 / 𝑑) → (⌊‘𝑚) = (⌊‘(𝑥 / 𝑑)))
2827oveq2d 6706 . . 3 (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29 ifeq1 4123 . . . . . . 7 (𝑚 = (𝑥 / 𝑑) → if(𝑆 = 0, 𝑚, 𝑘) = if(𝑆 = 0, (𝑥 / 𝑑), 𝑘))
3029fveq2d 6233 . . . . . 6 (𝑚 = (𝑥 / 𝑑) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)))
3130oveq1d 6705 . . . . 5 (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
3231oveq2d 6706 . . . 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 14482 . 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 4164 . 2 (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38 0cn 10070 . . 3 0 ∈ ℂ
39 dchrvmasumif.t . . . 4 (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40 climcl 14274 . . . 4 (seq1( + , 𝐾) ⇝ 𝑇𝑇 ∈ ℂ)
4139, 40syl 17 . . 3 (𝜑𝑇 ∈ ℂ)
42 ifcl 4163 . . 3 ((0 ∈ ℂ ∧ 𝑇 ∈ ℂ) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
4338, 41, 42sylancr 696 . 2 (𝜑 → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
44 nnuz 11761 . . . . . . . . 9 ℕ = (ℤ‘1)
45 1zzd 11446 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
46 nncn 11066 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
4746adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48 nnne0 11091 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
4948adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ≠ 0)
5015, 47, 49divcld 10839 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
51 dchrvmasumif.f . . . . . . . . . . . 12 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎))
52 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑎 = 𝑘 → (𝐿𝑎) = (𝐿𝑘))
5352fveq2d 6233 . . . . . . . . . . . . . 14 (𝑎 = 𝑘 → (𝑋‘(𝐿𝑎)) = (𝑋‘(𝐿𝑘)))
54 id 22 . . . . . . . . . . . . . 14 (𝑎 = 𝑘𝑎 = 𝑘)
5553, 54oveq12d 6708 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) / 𝑎) = ((𝑋‘(𝐿𝑘)) / 𝑘))
5655cbvmptv 4783 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎)) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5751, 56eqtri 2673 . . . . . . . . . . 11 𝐹 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5850, 57fmptd 6425 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶ℂ)
59 ffvelrn 6397 . . . . . . . . . 10 ((𝐹:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
6058, 59sylan 487 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
6144, 45, 60serf 12869 . . . . . . . 8 (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
6261ad2antrr 762 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
63 3re 11132 . . . . . . . . . . 11 3 ∈ ℝ
64 elicopnf 12307 . . . . . . . . . . 11 (3 ∈ ℝ → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6563, 64mp1i 13 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
6665simprbda 652 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ)
67 1red 10093 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
6863a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
69 1le3 11282 . . . . . . . . . . 11 1 ≤ 3
7069a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 3)
7165simplbda 653 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ≤ 𝑚)
7267, 68, 66, 70, 71letrd 10232 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
73 flge1nn 12662 . . . . . . . . 9 ((𝑚 ∈ ℝ ∧ 1 ≤ 𝑚) → (⌊‘𝑚) ∈ ℕ)
7466, 72, 73syl2anc 694 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ ℕ)
7574adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (⌊‘𝑚) ∈ ℕ)
7662, 75ffvelrnd 6400 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
7776abscld 14219 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
78 simpl 472 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝜑)
79 0red 10079 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
80 3pos 11152 . . . . . . . . . . 11 0 < 3
8180a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 3)
8279, 68, 66, 81, 71ltletrd 10235 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
8366, 82elrpd 11907 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ+)
8478, 83jca 553 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝜑𝑚 ∈ ℝ+))
85 elrege0 12316 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
8685simplbi 475 . . . . . . . . 9 (𝐶 ∈ (0[,)+∞) → 𝐶 ∈ ℝ)
8735, 86syl 17 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
88 rerpdivcl 11899 . . . . . . . 8 ((𝐶 ∈ ℝ ∧ 𝑚 ∈ ℝ+) → (𝐶 / 𝑚) ∈ ℝ)
8987, 88sylan 487 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (𝐶 / 𝑚) ∈ ℝ)
9084, 89syl 17 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝐶 / 𝑚) ∈ ℝ)
9190adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (𝐶 / 𝑚) ∈ ℝ)
9283relogcld 24414 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
9366, 72logge0d 24421 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ≤ (log‘𝑚))
9492, 93jca 553 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
9594adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
96 oveq2 6698 . . . . . . . 8 (𝑆 = 0 → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0))
9761adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → seq1( + , 𝐹):ℕ⟶ℂ)
9897, 74ffvelrnd 6400 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
9998subid1d 10419 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
10096, 99sylan9eqr 2707 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
101100fveq2d 6233 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) = (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))))
102 1re 10077 . . . . . . . . . 10 1 ∈ ℝ
103 elicopnf 12307 . . . . . . . . . 10 (1 ∈ ℝ → (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚)))
104102, 103ax-mp 5 . . . . . . . . 9 (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚))
10566, 72, 104sylanbrc 699 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ (1[,)+∞))
106 dchrvmasumif.1 . . . . . . . . 9 (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
107106adantr 480 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
108 fveq2 6229 . . . . . . . . . . . . 13 (𝑦 = 𝑚 → (⌊‘𝑦) = (⌊‘𝑚))
109108fveq2d 6233 . . . . . . . . . . . 12 (𝑦 = 𝑚 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
110109oveq1d 6705 . . . . . . . . . . 11 (𝑦 = 𝑚 → ((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆))
111110fveq2d 6233 . . . . . . . . . 10 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
112 oveq2 6698 . . . . . . . . . 10 (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
113111, 112breq12d 4698 . . . . . . . . 9 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦) ↔ (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚)))
114113rspcv 3336 . . . . . . . 8 (𝑚 ∈ (1[,)+∞) → (∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚)))
115105, 107, 114sylc 65 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
116115adantr 480 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
117101, 116eqbrtrrd 4709 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
118 lemul2a 10916 . . . . 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)))
11977, 91, 95, 117, 118syl31anc 1369 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
120 iftrue 4125 . . . . . . . . . . . . . . 15 (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
121120fveq2d 6233 . . . . . . . . . . . . . 14 (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
122121oveq1d 6705 . . . . . . . . . . . . 13 (𝑆 = 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
123122ad2antlr 763 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
124123oveq2d 6706 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)))
12516adantlr 751 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
126 relogcl 24367 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+ → (log‘𝑚) ∈ ℝ)
127126adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℝ)
128127recnd 10106 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℂ)
129128ad2antrr 762 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘𝑚) ∈ ℂ)
13011adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
131130nncnd 11074 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
132130nnne0d 11103 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
133125, 129, 131, 132div12d 10875 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
134124, 133eqtrd 2685 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
135134sumeq2dv 14477 . . . . . . . . 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)))
136 iftrue 4125 . . . . . . . . . . 11 (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
137136oveq2d 6706 . . . . . . . . . 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))
13826subid1d 10419 . . . . . . . . . 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)))
139137, 138sylan9eqr 2707 . . . . . . . . 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)))
140 ovex 6718 . . . . . . . . . . . . . 14 ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ V
14155, 51, 140fvmpt 6321 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
14222, 141syl 17 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
14358adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → 𝐹:ℕ⟶ℂ)
144143, 11, 59syl2an 493 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) ∈ ℂ)
145142, 144eqeltrrd 2731 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
1469, 128, 145fsummulc2 14560 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℝ+) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
147146adantr 480 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
148135, 139, 1473eqtr4d 2695 . . . . . . . 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)))
14984, 148sylan 487 . . . . . . 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)))
15084, 142sylan 487 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹𝑘) = ((𝑋‘(𝐿𝑘)) / 𝑘))
15174, 44syl6eleq 2740 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ (ℤ‘1))
15278, 11, 50syl2an 493 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
153150, 151, 152fsumser 14505 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
154153adantr 480 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
155154oveq2d 6706 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
156149, 155eqtrd 2685 . . . . . 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))))
157156fveq2d 6233 . . . . 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)))))
158126ad2antlr 763 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℝ)
159158recnd 10106 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
16084, 159sylan 487 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
161160, 76absmuld 14237 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
16292, 93absidd 14205 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
163162oveq1d 6705 . . . . . 6 ((𝜑𝑚 ∈ (3[,)+∞)) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
164163adantr 480 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
165157, 161, 1643eqtrd 2689 . . . 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)))))
166 iftrue 4125 . . . . . . . 8 (𝑆 = 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
167166adantl 481 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
168167oveq1d 6705 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
16987recnd 10106 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
170169ad2antrr 762 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
171 rpcnne0 11888 . . . . . . . 8 (𝑚 ∈ ℝ+ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
172171ad2antlr 763 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
173 div12 10745 . . . . . . 7 ((𝐶 ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
174170, 159, 172, 173syl3anc 1366 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
175168, 174eqtrd 2685 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
17684, 175sylan 487 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
177119, 165, 1763brtr4d 4717 . . 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)))
178 dchrvmasumif.2 . . . . . 6 (𝜑 → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)))
179108fveq2d 6233 . . . . . . . . . 10 (𝑦 = 𝑚 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
180179oveq1d 6705 . . . . . . . . 9 (𝑦 = 𝑚 → ((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇) = ((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇))
181180fveq2d 6233 . . . . . . . 8 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
182 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝑚 → (log‘𝑦) = (log‘𝑚))
183 id 22 . . . . . . . . . 10 (𝑦 = 𝑚𝑦 = 𝑚)
184182, 183oveq12d 6708 . . . . . . . . 9 (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
185184oveq2d 6706 . . . . . . . 8 (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
186181, 185breq12d 4698 . . . . . . 7 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
187186rspccva 3339 . . . . . 6 ((∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
188178, 187sylan 487 . . . . 5 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
189188adantr 480 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
190 fveq2 6229 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (log‘𝑎) = (log‘𝑘))
191190, 54oveq12d 6708 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
19253, 191oveq12d 6708 . . . . . . . . . 10 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
193 dchrvmasumif.g . . . . . . . . . 10 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎)))
194 ovex 6718 . . . . . . . . . 10 ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
195192, 193, 194fvmpt 6321 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
19611, 195syl 17 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
197 ifnefalse 4131 . . . . . . . . . . . . 13 (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
198197fveq2d 6233 . . . . . . . . . . . 12 (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
199198oveq1d 6705 . . . . . . . . . . 11 (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
200199oveq2d 6706 . . . . . . . . . 10 (𝑆 ≠ 0 → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
201200adantl 481 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
202201eqcomd 2657 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
203196, 202sylan9eqr 2707 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
204151adantr 480 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (⌊‘𝑚) ∈ (ℤ‘1))
205 nnrp 11880 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
206205adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
207206relogcld 24414 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
208207recnd 10106 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
209208, 47, 49divcld 10839 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
21015, 209mulcld 10098 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
211192cbvmptv 4783 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
212193, 211eqtri 2673 . . . . . . . . . . 11 𝐾 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
213210, 212fmptd 6425 . . . . . . . . . 10 (𝜑𝐾:ℕ⟶ℂ)
214213ad2antrr 762 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
215 ffvelrn 6397 . . . . . . . . 9 ((𝐾:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐾𝑘) ∈ ℂ)
216214, 11, 215syl2an 493 . . . . . . . 8 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) ∈ ℂ)
217203, 216eqeltrrd 2731 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
218203, 204, 217fsumser 14505 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
219 ifnefalse 4131 . . . . . . 7 (𝑆 ≠ 0 → if(𝑆 = 0, 0, 𝑇) = 𝑇)
220219adantl 481 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 0, 𝑇) = 𝑇)
221218, 220oveq12d 6708 . . . . 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))
222221fveq2d 6233 . . . 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)))
223 ifnefalse 4131 . . . . . 6 (𝑆 ≠ 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
224223adantl 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
225224oveq1d 6705 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
226189, 222, 2253brtr4d 4717 . . 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)))
227177, 226pm2.61dane 2910 . 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)))
228 fzfid 12812 . . . 4 (𝜑 → (1...2) ∈ Fin)
2297adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑋𝐷)
230 elfzelz 12380 . . . . . . . 8 (𝑘 ∈ (1...2) → 𝑘 ∈ ℤ)
231230adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℤ)
2324, 1, 5, 2, 229, 231dchrzrhcl 25015 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
233232abscld 14219 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
23463, 80elrpii 11873 . . . . . . 7 3 ∈ ℝ+
235 relogcl 24367 . . . . . . 7 (3 ∈ ℝ+ → (log‘3) ∈ ℝ)
236234, 235ax-mp 5 . . . . . 6 (log‘3) ∈ ℝ
237 elfznn 12408 . . . . . . 7 (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
238237adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
239 nndivre 11094 . . . . . 6 (((log‘3) ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((log‘3) / 𝑘) ∈ ℝ)
240236, 238, 239sylancr 696 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
241233, 240remulcld 10108 . . . 4 ((𝜑𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
242228, 241fsumrecl 14509 . . 3 (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
24343abscld 14219 . . 3 (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
244242, 243readdcld 10107 . 2 (𝜑 → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
245 simpl 472 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝜑)
24663rexri 10135 . . . . . . . . . . 11 3 ∈ ℝ*
247 elico2 12275 . . . . . . . . . . 11 ((1 ∈ ℝ ∧ 3 ∈ ℝ*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3)))
248102, 246, 247mp2an 708 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚𝑚 < 3))
249248simp1bi 1096 . . . . . . . . 9 (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
250249adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ)
251 0red 10079 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
252 1red 10093 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
253 0lt1 10588 . . . . . . . . . 10 0 < 1
254253a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 1)
255248simp2bi 1097 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
256255adantl 481 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ≤ 𝑚)
257251, 252, 250, 254, 256ltletrd 10235 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 𝑚)
258250, 257elrpd 11907 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ+)
259245, 258jca 553 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → (𝜑𝑚 ∈ ℝ+))
26043adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
26126, 260subcld 10430 . . . . . 6 ((𝜑𝑚 ∈ ℝ+) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
262259, 261syl 17 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
263262abscld 14219 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
264259, 26syl 17 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
265264abscld 14219 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
266243adantr 480 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
267265, 266readdcld 10107 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
268242adantr 480 . . . . 5 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
269268, 266readdcld 10107 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
27026, 260abs2dif2d 14241 . . . . 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))))
271259, 270syl 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))))
27225abscld 14219 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2739, 272fsumrecl 14509 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
274259, 273syl 17 . . . . . 6 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2759, 25fsumabs 14577 . . . . . . 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))))
276259, 275syl 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))))
277 fzfid 12812 . . . . . . . . 9 ((𝜑𝑚 ∈ ℝ+) → (1...2) ∈ Fin)
278232adantlr 751 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
27917adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑚 ∈ ℝ+)
280237adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
281280nnrpd 11908 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ+)
282279, 281ifcld 4164 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
283282relogcld 24414 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
284283, 280nndivred 11107 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
285284recnd 10106 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
286278, 285mulcld 10098 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
287286abscld 14219 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288277, 287fsumrecl 14509 . . . . . . . 8 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
289259, 288syl 17 . . . . . . 7 ((𝜑𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
290 fzfid 12812 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...2) ∈ Fin)
291259, 286sylan 487 . . . . . . . . 9 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
292291abscld 14219 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
293291absge0d 14227 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
294250flcld 12639 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
295 2z 11447 . . . . . . . . . . 11 2 ∈ ℤ
296295a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ ℤ)
297248simp3bi 1098 . . . . . . . . . . . . . 14 (𝑚 ∈ (1[,)3) → 𝑚 < 3)
298297adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 < 3)
299 3z 11448 . . . . . . . . . . . . . 14 3 ∈ ℤ
300 fllt 12647 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ ∧ 3 ∈ ℤ) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
301250, 299, 300sylancl 695 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
302298, 301mpbid 222 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < 3)
303 df-3 11118 . . . . . . . . . . . 12 3 = (2 + 1)
304302, 303syl6breq 4726 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
305 rpre 11877 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+𝑚 ∈ ℝ)
306305adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ)
307306flcld 12639 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (⌊‘𝑚) ∈ ℤ)
308 zleltp1 11466 . . . . . . . . . . . . 13 (((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
309307, 295, 308sylancl 695 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℝ+) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
310259, 309syl 17 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
311304, 310mpbird 247 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ≤ 2)
312 eluz2 11731 . . . . . . . . . 10 (2 ∈ (ℤ‘(⌊‘𝑚)) ↔ ((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘𝑚) ≤ 2))
313294, 296, 311, 312syl3anbrc 1265 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ (ℤ‘(⌊‘𝑚)))
314 fzss2 12419 . . . . . . . . 9 (2 ∈ (ℤ‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...2))
315313, 314syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...(⌊‘𝑚)) ⊆ (1...2))
316290, 292, 293, 315fsumless 14572 . . . . . . 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))))
317241adantlr 751 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
318278, 285absmuld 14237 . . . . . . . . . 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))))
319259, 318sylan 487 . . . . . . . . 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))))
320259, 284sylan 487 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
321259, 283sylan 487 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
322 log1 24377 . . . . . . . . . . . . . 14 (log‘1) = 0
323 elfzle1 12382 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
324 breq2 4689 . . . . . . . . . . . . . . . . 17 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
325 breq2 4689 . . . . . . . . . . . . . . . . 17 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
326324, 325ifboth 4157 . . . . . . . . . . . . . . . 16 ((1 ≤ 𝑚 ∧ 1 ≤ 𝑘) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
327256, 323, 326syl2an 493 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
328 1rp 11874 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ+
329 logleb 24394 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℝ+ ∧ if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
330328, 282, 329sylancr 696 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
331259, 330sylan 487 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
332327, 331mpbid 222 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
333322, 332syl5eqbrr 4721 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
334281rpregt0d 11916 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
335259, 334sylan 487 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
336 divge0 10930 . . . . . . . . . . . . 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))
337321, 333, 335, 336syl21anc 1365 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
338320, 337absidd 14205 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
339338, 320eqeltrd 2730 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ)
340240adantlr 751 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
341233adantlr 751 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
342278absge0d 14227 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘(𝑋‘(𝐿𝑘))))
343341, 342jca 553 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿𝑘)))))
344259, 343sylan 487 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿𝑘)))))
345297ad2antlr 763 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑚 < 3)
346280nnred 11073 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
347 2re 11128 . . . . . . . . . . . . . . . . . 18 2 ∈ ℝ
348347a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 ∈ ℝ)
34963a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 3 ∈ ℝ)
350 elfzle2 12383 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
351350adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
352 2lt3 11233 . . . . . . . . . . . . . . . . . 18 2 < 3
353352a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
354346, 348, 349, 351, 353lelttrd 10233 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
355259, 354sylan 487 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
356 breq1 4688 . . . . . . . . . . . . . . . 16 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
357 breq1 4688 . . . . . . . . . . . . . . . 16 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
358356, 357ifboth 4157 . . . . . . . . . . . . . . 15 ((𝑚 < 3 ∧ 𝑘 < 3) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
359345, 355, 358syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
360282rpred 11910 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
361 ltle 10164 . . . . . . . . . . . . . . . 16 ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ ∧ 3 ∈ ℝ) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
362360, 63, 361sylancl 695 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
363259, 362sylan 487 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
364359, 363mpd 15 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3)
365 logleb 24394 . . . . . . . . . . . . . . 15 ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+ ∧ 3 ∈ ℝ+) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
366282, 234, 365sylancl 695 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
367259, 366sylan 487 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
368364, 367mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3))
369236a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘3) ∈ ℝ)
370283, 369, 281lediv1d 11956 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3) ↔ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘)))
371259, 370sylan 487 . . . . . . . . . . . 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)))
372368, 371mpbid 222 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘))
373338, 372eqbrtrd 4707 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
374 lemul2a 10916 . . . . . . . . . 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)))
375339, 340, 344, 373, 374syl31anc 1369 . . . . . . . . 9 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
376319, 375eqbrtrd 4707 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)))
377290, 292, 317, 376fsumle 14575 . . . . . . 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)))
378274, 289, 268, 316, 377letrd 10232 . . . . . 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)))
379265, 274, 268, 276, 378letrd 10232 . . . . 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)))
38026abscld 14219 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
381242adantr 480 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
382260abscld 14219 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
383380, 381, 382leadd1d 10659 . . . . . 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)))))
384259, 383syl 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)))))
385379, 384mpbid 222 . . . 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))))
386263, 267, 269, 271, 385letrd 10232 . . 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))))
387386ralrimiva 2995 . 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))))
3881, 2, 3, 4, 5, 6, 7, 8, 26, 34, 37, 43, 227, 244, 387dchrvmasumlem3 25233 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941   ⊆ wss 3607  ifcif 4119   class class class wbr 4685   ↦ cmpt 4762  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113   − cmin 10304   / cdiv 10722  ℕcn 11058  2c2 11108  3c3 11109  ℤcz 11415  ℤ≥cuz 11725  ℝ+crp 11870  [,)cico 12215  ...cfz 12364  ⌊cfl 12631  seqcseq 12841  abscabs 14018   ⇝ cli 14259  𝑂(1)co1 14261  Σcsu 14460  Basecbs 15904  0gc0g 16147  ℤRHomczrh 19896  ℤ/nℤczn 19899  logclog 24346  μcmu 24866  DChrcdchr 25002 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-disj 4653  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-er 7787  df-ec 7789  df-qs 7793  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-seq 12842  df-exp 12901  df-fac 13101  df-bc 13130  df-hash 13158  df-shft 13851  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-o1 14265  df-lo1 14266  df-sum 14461  df-ef 14842  df-e 14843  df-sin 14844  df-cos 14845  df-pi 14847  df-dvds 15028  df-prm 15433  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-qus 16216  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-nsg 17639  df-eqg 17640  df-ghm 17705  df-cntz 17796  df-od 17994  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-rnghom 18763  df-drng 18797  df-subrg 18826  df-lmod 18913  df-lss 18981  df-lsp 19020  df-sra 19220  df-rgmod 19221  df-lidl 19222  df-rsp 19223  df-2idl 19280  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-zring 19867  df-zrh 19900  df-zn 19903  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-cmp 21238  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676  df-log 24348  df-cxp 24349  df-em 24764  df-mu 24872  df-dchr 25003 This theorem is referenced by:  dchrvmasumiflem2  25236
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