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Theorem dchrvmasumiflem1 27559
Description: Lemma for dchrvmasumif 27561. (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 14010 . . 3 ((𝜑𝑚 ∈ ℝ+) → (1...(⌊‘𝑚)) ∈ Fin)
10 simpl 482 . . . . 5 ((𝜑𝑚 ∈ ℝ+) → 𝜑)
11 elfznn 13589 . . . . 5 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
127adantr 480 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑋𝐷)
13 nnz 12631 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
1413adantl 481 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
154, 1, 5, 2, 12, 14dchrzrhcl 27303 . . . . 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 13072 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ+)
19 ifcl 4575 . . . . . . . 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 26679 . . . . . 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 12317 . . . . 5 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
2423recnd 11286 . . . 4 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
2516, 24mulcld 11278 . . 3 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
269, 25fsumcl 15765 . 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 7446 . . 3 (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29 ifeq1 4534 . . . . . . 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 7445 . . . . 5 (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
3231oveq2d 7446 . . . 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 15739 . 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 4576 . 2 (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38 0cn 11250 . . 3 0 ∈ ℂ
39 dchrvmasumif.t . . . 4 (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40 climcl 15531 . . . 4 (seq1( + , 𝐾) ⇝ 𝑇𝑇 ∈ ℂ)
4139, 40syl 17 . . 3 (𝜑𝑇 ∈ ℂ)
42 ifcl 4575 . . 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 12918 . . . . . . . . 9 ℕ = (ℤ‘1)
45 1zzd 12645 . . . . . . . . 9 (𝜑 → 1 ∈ ℤ)
46 nncn 12271 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
4746adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48 nnne0 12297 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ≠ 0)
4948adantl 481 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → 𝑘 ≠ 0)
5015, 47, 49divcld 12040 . . . . . . . . . . 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 7448 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑋‘(𝐿𝑎)) / 𝑎) = ((𝑋‘(𝐿𝑘)) / 𝑘))
5554cbvmptv 5260 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) / 𝑎)) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5651, 55eqtri 2762 . . . . . . . . . . 11 𝐹 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) / 𝑘))
5750, 56fmptd 7133 . . . . . . . . . 10 (𝜑𝐹:ℕ⟶ℂ)
58 ffvelcdm 7100 . . . . . . . . . 10 ((𝐹:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
5957, 58sylan 580 . . . . . . . . 9 ((𝜑𝑘 ∈ ℕ) → (𝐹𝑘) ∈ ℂ)
6044, 45, 59serf 14067 . . . . . . . 8 (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
6160ad2antrr 726 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
62 3re 12343 . . . . . . . . . . 11 3 ∈ ℝ
63 elicopnf 13481 . . . . . . . . . . 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 11259 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
6762a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
68 1le3 12475 . . . . . . . . . . 11 1 ≤ 3
6968a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 3)
7064simplbda 499 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 3 ≤ 𝑚)
7166, 67, 65, 69, 70letrd 11415 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
72 flge1nn 13857 . . . . . . . . 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 7104 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
7675abscld 15471 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
77 simpl 482 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝜑)
78 0red 11261 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
79 3pos 12368 . . . . . . . . . . 11 0 < 3
8079a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 3)
8178, 67, 65, 80, 70ltletrd 11418 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
8265, 81elrpd 13071 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ+)
8377, 82jca 511 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (𝜑𝑚 ∈ ℝ+))
84 elrege0 13490 . . . . . . . . . 10 (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
8584simplbi 497 . . . . . . . . 9 (𝐶 ∈ (0[,)+∞) → 𝐶 ∈ ℝ)
8635, 85syl 17 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
87 rerpdivcl 13062 . . . . . . . 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 26679 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
9265, 71logge0d 26686 . . . . . . 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 7438 . . . . . . . 8 (𝑆 = 0 → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0))
9660adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → seq1( + , 𝐹):ℕ⟶ℂ)
9796, 73ffvelcdmd 7104 . . . . . . . . 9 ((𝜑𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
9897subid1d 11606 . . . . . . . 8 ((𝜑𝑚 ∈ (3[,)+∞)) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
9995, 98sylan9eqr 2796 . . . . . . 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 7452 . . . . . . . . 9 (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
103 oveq2 7438 . . . . . . . . 9 (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
104102, 103breq12d 5160 . . . . . . . 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 11258 . . . . . . . . . 10 1 ∈ ℝ
108 elicopnf 13481 . . . . . . . . . 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 480 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
113100, 112eqbrtrrd 5171 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
114 lemul2a 12119 . . . . 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 1372 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
116 iftrue 4536 . . . . . . . . . . . . . . 15 (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
117116fveq2d 6910 . . . . . . . . . . . . . 14 (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
118117oveq1d 7445 . . . . . . . . . . . . 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 7446 . . . . . . . . . . 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 26631 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+ → (log‘𝑚) ∈ ℝ)
123122adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → (log‘𝑚) ∈ ℝ)
124123recnd 11286 . . . . . . . . . . . . 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 12279 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
128126nnne0d 12313 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
129121, 125, 127, 128div12d 12076 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑚) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
130120, 129eqtrd 2774 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿𝑘)) / 𝑘)))
131130sumeq2dv 15734 . . . . . . . . 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 4536 . . . . . . . . . . 11 (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
133132oveq2d 7446 . . . . . . . . . 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 11606 . . . . . . . . . 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 2796 . . . . . . . . 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 7463 . . . . . . . . . . . . . 14 ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ V
13754, 51, 136fvmpt 7015 . . . . . . . . . . . . 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 2839 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
1429, 124, 141fsummulc2 15816 . . . . . . . . . 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 2784 . . . . . . . 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 2848 . . . . . . . . . 10 ((𝜑𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ (ℤ‘1))
14877, 11, 50syl2an 596 . . . . . . . . . 10 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) / 𝑘) ∈ ℂ)
149146, 147, 148fsumser 15762 . . . . . . . . 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 7446 . . . . . . 7 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) / 𝑘)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
152145, 151eqtrd 2774 . . . . . 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 11286 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
15683, 155sylan 580 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
157156, 75absmuld 15489 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
15891, 92absidd 15457 . . . . . . 7 ((𝜑𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
159158oveq1d 7445 . . . . . 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 2778 . . . 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 4536 . . . . . . . 8 (𝑆 = 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
163162adantl 481 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
164163oveq1d 7445 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
16586recnd 11286 . . . . . . . 8 (𝜑𝐶 ∈ ℂ)
166165ad2antrr 726 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
167 rpcnne0 13050 . . . . . . . 8 (𝑚 ∈ ℝ+ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
168167ad2antlr 727 . . . . . . 7 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
169 div12 11941 . . . . . . 7 ((𝐶 ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
170166, 155, 168, 169syl3anc 1370 . . . . . 6 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑆 = 0) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
171164, 170eqtrd 2774 . . . . 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 5179 . . 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 7452 . . . . . . . 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 7448 . . . . . . . . 9 (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
180179oveq2d 7446 . . . . . . . 8 (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
181176, 180breq12d 5160 . . . . . . 7 (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
182181rspccva 3620 . . . . . 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 7448 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
18752, 186oveq12d 7448 . . . . . . . . . 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 7463 . . . . . . . . . 10 ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
190187, 188, 189fvmpt 7015 . . . . . . . . 9 (𝑘 ∈ ℕ → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
19111, 190syl 17 . . . . . . . 8 (𝑘 ∈ (1...(⌊‘𝑚)) → (𝐾𝑘) = ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
192 ifnefalse 4542 . . . . . . . . . . . . 13 (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
193192fveq2d 6910 . . . . . . . . . . . 12 (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
194193oveq1d 7445 . . . . . . . . . . 11 (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
195194oveq2d 7446 . . . . . . . . . 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 2740 . . . . . . . 8 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) = ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
198191, 197sylan9eqr 2796 . . . . . . 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 13043 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ+)
201200adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑘 ∈ ℕ) → 𝑘 ∈ ℝ+)
202201relogcld 26679 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
203202recnd 11286 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
204203, 47, 49divcld 12040 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
20515, 204mulcld 11278 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ) → ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
206187cbvmptv 5260 . . . . . . . . . . . 12 (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
207188, 206eqtri 2762 . . . . . . . . . . 11 𝐾 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿𝑘)) · ((log‘𝑘) / 𝑘)))
208205, 207fmptd 7133 . . . . . . . . . 10 (𝜑𝐾:ℕ⟶ℂ)
209208ad2antrr 726 . . . . . . . . 9 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
210 ffvelcdm 7100 . . . . . . . . 9 ((𝐾:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐾𝑘) ∈ ℂ)
211209, 11, 210syl2an 596 . . . . . . . 8 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾𝑘) ∈ ℂ)
212198, 211eqeltrrd 2839 . . . . . . 7 ((((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
213198, 199, 212fsumser 15762 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
214 ifnefalse 4542 . . . . . . 7 (𝑆 ≠ 0 → if(𝑆 = 0, 0, 𝑇) = 𝑇)
215214adantl 481 . . . . . 6 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 0, 𝑇) = 𝑇)
216213, 215oveq12d 7448 . . . . 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 4542 . . . . . 6 (𝑆 ≠ 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
219218adantl 481 . . . . 5 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
220219oveq1d 7445 . . . 4 (((𝜑𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
221184, 217, 2203brtr4d 5179 . . 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 3026 . 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 14010 . . . 4 (𝜑 → (1...2) ∈ Fin)
2247adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑋𝐷)
225 elfzelz 13560 . . . . . . . 8 (𝑘 ∈ (1...2) → 𝑘 ∈ ℤ)
226225adantl 481 . . . . . . 7 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℤ)
2274, 1, 5, 2, 224, 226dchrzrhcl 27303 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → (𝑋‘(𝐿𝑘)) ∈ ℂ)
228227abscld 15471 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿𝑘))) ∈ ℝ)
229 3rp 13037 . . . . . . 7 3 ∈ ℝ+
230 relogcl 26631 . . . . . . 7 (3 ∈ ℝ+ → (log‘3) ∈ ℝ)
231229, 230ax-mp 5 . . . . . 6 (log‘3) ∈ ℝ
232 elfznn 13589 . . . . . . 7 (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
233232adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
234 nndivre 12304 . . . . . 6 (((log‘3) ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((log‘3) / 𝑘) ∈ ℝ)
235231, 233, 234sylancr 587 . . . . 5 ((𝜑𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
236228, 235remulcld 11288 . . . 4 ((𝜑𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
237223, 236fsumrecl 15766 . . 3 (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
23843abscld 15471 . . 3 (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
239237, 238readdcld 11287 . 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 11316 . . . . . . . . . . 11 3 ∈ ℝ*
242 elico2 13447 . . . . . . . . . . 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 1144 . . . . . . . . 9 (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
245244adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ)
246 0red 11261 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
247 1red 11259 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
248 0lt1 11782 . . . . . . . . . 10 0 < 1
249248a1i 11 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 1)
250243simp2bi 1145 . . . . . . . . . 10 (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
251250adantl 481 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 1 ≤ 𝑚)
252246, 247, 245, 249, 251ltletrd 11418 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → 0 < 𝑚)
253245, 252elrpd 13071 . . . . . . 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 11617 . . . . . 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 15471 . . . 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 15471 . . . . 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 11287 . . . 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 11287 . . . 4 ((𝜑𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
26526, 255abs2dif2d 15493 . . . . 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 15471 . . . . . . . 8 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
2689, 267fsumrecl 15766 . . . . . . 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 15833 . . . . . . 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 14010 . . . . . . . . 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 13072 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ+)
277274, 276ifcld 4576 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ+)
278277relogcld 26679 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
279278, 275nndivred 12317 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
280279recnd 11286 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
281273, 280mulcld 11278 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
282281abscld 15471 . . . . . . . . 9 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
283272, 282fsumrecl 15766 . . . . . . . 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 14010 . . . . . . . 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 15471 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288286absge0d 15479 . . . . . . . 8 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
289245flcld 13834 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
290 2z 12646 . . . . . . . . . . 11 2 ∈ ℤ
291290a1i 11 . . . . . . . . . 10 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ ℤ)
292243simp3bi 1146 . . . . . . . . . . . . . 14 (𝑚 ∈ (1[,)3) → 𝑚 < 3)
293292adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ (1[,)3)) → 𝑚 < 3)
294 3z 12647 . . . . . . . . . . . . . 14 3 ∈ ℤ
295 fllt 13842 . . . . . . . . . . . . . 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 12327 . . . . . . . . . . . 12 3 = (2 + 1)
299297, 298breqtrdi 5188 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
300 rpre 13040 . . . . . . . . . . . . . . 15 (𝑚 ∈ ℝ+𝑚 ∈ ℝ)
301300adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℝ+) → 𝑚 ∈ ℝ)
302301flcld 13834 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℝ+) → (⌊‘𝑚) ∈ ℤ)
303 zleltp1 12665 . . . . . . . . . . . . 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 12881 . . . . . . . . . 10 (2 ∈ (ℤ‘(⌊‘𝑚)) ↔ ((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘𝑚) ≤ 2))
308289, 291, 306, 307syl3anbrc 1342 . . . . . . . . 9 ((𝜑𝑚 ∈ (1[,)3)) → 2 ∈ (ℤ‘(⌊‘𝑚)))
309 fzss2 13600 . . . . . . . . 9 (2 ∈ (ℤ‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...2))
310308, 309syl 17 . . . . . . . 8 ((𝜑𝑚 ∈ (1[,)3)) → (1...(⌊‘𝑚)) ⊆ (1...2))
311285, 287, 288, 310fsumless 15828 . . . . . . 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 15489 . . . . . . . . . 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 26641 . . . . . . . . . . . . . 14 (log‘1) = 0
318 elfzle1 13563 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
319 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
320 breq2 5151 . . . . . . . . . . . . . . . . 17 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
321319, 320ifboth 4569 . . . . . . . . . . . . . . . 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 13035 . . . . . . . . . . . . . . . . 17 1 ∈ ℝ+
324 logleb 26659 . . . . . . . . . . . . . . . . 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 5183 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
329276rpregt0d 13080 . . . . . . . . . . . . . 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 12134 . . . . . . . . . . . . 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 15457 . . . . . . . . . . 11 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
334333, 315eqeltrd 2838 . . . . . . . . . 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 15479 . . . . . . . . . . . 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 12278 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
342 2re 12337 . . . . . . . . . . . . . . . . . 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 13564 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
346345adantl 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
347 2lt3 12435 . . . . . . . . . . . . . . . . . 18 2 < 3
348347a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
349341, 343, 344, 346, 348lelttrd 11416 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
350254, 349sylan 580 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
351 breq1 5150 . . . . . . . . . . . . . . . 16 (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
352 breq1 5150 . . . . . . . . . . . . . . . 16 (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
353351, 352ifboth 4569 . . . . . . . . . . . . . . 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 13074 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℝ+) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
356 ltle 11346 . . . . . . . . . . . . . . . 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 26659 . . . . . . . . . . . . . . 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 13120 . . . . . . . . . . . . 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 5169 . . . . . . . . . 10 (((𝜑𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
369 lemul2a 12119 . . . . . . . . . 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 1372 . . . . . . . . 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 5169 . . . . . . . 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 15831 . . . . . . 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 11415 . . . . . 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 11415 . . . . 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 15471 . . . . . . 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 15471 . . . . . . 7 ((𝜑𝑚 ∈ ℝ+) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
378375, 376, 377leadd1d 11854 . . . . . 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 11415 . . 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 3143 . 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 27557 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 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  wss 3962  ifcif 4530   class class class wbr 5147  cmpt 5230  wf 6558  cfv 6562  (class class class)co 7430  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  cn 12263  2c2 12318  3c3 12319  cz 12610  cuz 12875  +crp 13031  [,)cico 13385  ...cfz 13543  cfl 13826  seqcseq 14038  abscabs 15269  cli 15516  𝑂(1)co1 15518  Σcsu 15718  Basecbs 17244  0gc0g 17485  ℤRHomczrh 21527  ℤ/nczn 21530  logclog 26610  μcmu 27152  DChrcdchr 27290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231  ax-mulf 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-disj 5115  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-omul 8509  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-fi 9448  df-sup 9479  df-inf 9480  df-oi 9547  df-card 9976  df-acn 9979  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-q 12988  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-ioo 13387  df-ioc 13388  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-seq 14039  df-exp 14099  df-fac 14309  df-bc 14338  df-hash 14366  df-shft 15102  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521  df-o1 15522  df-lo1 15523  df-sum 15719  df-ef 16099  df-e 16100  df-sin 16101  df-cos 16102  df-tan 16103  df-pi 16104  df-dvds 16287  df-prm 16705  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  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 17468  df-topn 17469  df-0g 17487  df-gsum 17488  df-topgen 17489  df-pt 17490  df-prds 17493  df-xrs 17548  df-qtop 17553  df-imas 17554  df-qus 17555  df-xps 17556  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-nsg 19154  df-eqg 19155  df-ghm 19243  df-cntz 19347  df-od 19560  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-rhm 20488  df-subrng 20562  df-subrg 20586  df-drng 20747  df-lmod 20876  df-lss 20947  df-lsp 20987  df-sra 21189  df-rgmod 21190  df-lidl 21235  df-rsp 21236  df-2idl 21277  df-psmet 21373  df-xmet 21374  df-met 21375  df-bl 21376  df-mopn 21377  df-fbas 21378  df-fg 21379  df-cnfld 21382  df-zring 21475  df-zrh 21531  df-zn 21534  df-top 22915  df-topon 22932  df-topsp 22954  df-bases 22968  df-cld 23042  df-ntr 23043  df-cls 23044  df-nei 23121  df-lp 23159  df-perf 23160  df-cn 23250  df-cnp 23251  df-haus 23338  df-cmp 23410  df-tx 23585  df-hmeo 23778  df-fil 23869  df-fm 23961  df-flim 23962  df-flf 23963  df-xms 24345  df-ms 24346  df-tms 24347  df-cncf 24917  df-limc 25915  df-dv 25916  df-ulm 26434  df-log 26612  df-cxp 26613  df-atan 26924  df-em 27050  df-mu 27158  df-dchr 27291
This theorem is referenced by:  dchrvmasumiflem2  27560
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