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Theorem dchrvmasumiflem1 27428

Description: Lemma for dchrvmasumif 27430. (Contributed by Mario Carneiro, 5-May-2016.)

**Hypotheses**
Ref	Expression
rpvmasum.z	⊢ 𝑍 = (ℤ/nℤ‘𝑁)
rpvmasum.l	⊢ 𝐿 = (ℤRHom‘𝑍)
rpvmasum.a	⊢ (𝜑 → 𝑁 ∈ ℕ)
rpvmasum.g	⊢ 𝐺 = (DChr‘𝑁)
rpvmasum.d	⊢ 𝐷 = (Base‘𝐺)
rpvmasum.1	⊢ 1 = (0_g‘𝐺)
dchrisum.b	⊢ (𝜑 → 𝑋 ∈ 𝐷)
dchrisum.n1	⊢ (𝜑 → 𝑋 ≠ 1 )
dchrvmasumif.f	⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
dchrvmasumif.c	⊢ (𝜑 → 𝐶 ∈ (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‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))
dchrvmasumif.e	⊢ (𝜑 → 𝐸 ∈ (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‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((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.

Step	Hyp	Ref	Expression
1		rpvmasum.z	. 2 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
2		rpvmasum.l	. 2 ⊢ 𝐿 = (ℤRHom‘𝑍)
3		rpvmasum.a	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		rpvmasum.g	. 2 ⊢ 𝐺 = (DChr‘𝑁)
5		rpvmasum.d	. 2 ⊢ 𝐷 = (Base‘𝐺)
6		rpvmasum.1	. 2 ⊢ 1 = (0_g‘𝐺)
7		dchrisum.b	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8		dchrisum.n1	. 2 ⊢ (𝜑 → 𝑋 ≠ 1 )
9		fzfid 13898	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (1...(⌊‘𝑚)) ∈ Fin)
10		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝜑)
11		elfznn 13474	. . . . 5 ⊢ (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
12	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑋 ∈ 𝐷)
13		nnz 12510	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
14	13	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
15	4, 1, 5, 2, 12, 14	dchrzrhcl 27172	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
16	10, 11, 15	syl2an 596	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
17		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ⁺)
18	11	nnrpd 12953	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ⁺)
19		ifcl 4524	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ⁺ ∧ 𝑘 ∈ ℝ⁺) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺)
20	17, 18, 19	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺)
21	20	relogcld 26548	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
22	11	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
23	21, 22	nndivred 12200	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
24	23	recnd 11162	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
25	16, 24	mulcld 11154	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
26	9, 25	fsumcl 15658	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
27		fveq2 6826	. . . 4 ⊢ (𝑚 = (𝑥 / 𝑑) → (⌊‘𝑚) = (⌊‘(𝑥 / 𝑑)))
28	27	oveq2d 7369	. . 3 ⊢ (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29		ifeq1 4482	. . . . . . 7 ⊢ (𝑚 = (𝑥 / 𝑑) → if(𝑆 = 0, 𝑚, 𝑘) = if(𝑆 = 0, (𝑥 / 𝑑), 𝑘))
30	29	fveq2d 6830	. . . . . 6 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)))
31	30	oveq1d 7368	. . . . 5 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
32	31	oveq2d 7369	. . . 4 ⊢ (𝑚 = (𝑥 / 𝑑) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
33	32	adantr 480	. . 3 ⊢ ((𝑚 = (𝑥 / 𝑑) ∧ 𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
34	28, 33	sumeq12rdv 15632	. 2 ⊢ (𝑚 = (𝑥 / 𝑑) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘)))
35		dchrvmasumif.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
36		dchrvmasumif.e	. . 3 ⊢ (𝜑 → 𝐸 ∈ (0[,)+∞))
37	35, 36	ifcld 4525	. 2 ⊢ (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38		0cn 11126	. . 3 ⊢ 0 ∈ ℂ
39		dchrvmasumif.t	. . . 4 ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40		climcl 15424	. . . 4 ⊢ (seq1( + , 𝐾) ⇝ 𝑇 → 𝑇 ∈ ℂ)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
42		ifcl 4524	. . 3 ⊢ ((0 ∈ ℂ ∧ 𝑇 ∈ ℂ) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
43	38, 41, 42	sylancr 587	. 2 ⊢ (𝜑 → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
44		nnuz 12796	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
45		1zzd 12524	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
46		nncn 12154	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
47	46	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48		nnne0 12180	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
49	48	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
50	15, 47, 49	divcld 11918	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ ℂ)
51		dchrvmasumif.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
52		2fveq3 6831	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑘 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑘)))
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑘 → 𝑎 = 𝑘)
54	52, 53	oveq12d 7371	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑘 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑘)) / 𝑘))
55	54	cbvmptv 5199	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑘)) / 𝑘))
56	51, 55	eqtri 2752	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑘)) / 𝑘))
57	50, 56	fmptd 7052	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
58		ffvelcdm 7019	. . . . . . . . . 10 ⊢ ((𝐹:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
59	57, 58	sylan 580	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ)
60	44, 45, 59	serf 13955	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
61	60	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
62		3re 12226	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
63		elicopnf 13366	. . . . . . . . . . 11 ⊢ (3 ∈ ℝ → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
64	62, 63	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (3[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 3 ≤ 𝑚)))
65	64	simprbda 498	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ)
66		1red 11135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
67	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
68		1le3 12353	. . . . . . . . . . 11 ⊢ 1 ≤ 3
69	68	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 1 ≤ 3)
70	64	simplbda 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 3 ≤ 𝑚)
71	66, 67, 65, 69, 70	letrd 11291	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
72		flge1nn 13743	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℝ ∧ 1 ≤ 𝑚) → (⌊‘𝑚) ∈ ℕ)
73	65, 71, 72	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ ℕ)
74	73	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (⌊‘𝑚) ∈ ℕ)
75	61, 74	ffvelcdmd 7023	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
76	75	abscld 15364	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
77		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝜑)
78		0red 11137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
79		3pos 12251	. . . . . . . . . . 11 ⊢ 0 < 3
80	79	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 < 3)
81	78, 67, 65, 80, 70	ltletrd 11294	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
82	65, 81	elrpd 12952	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ⁺)
83	77, 82	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (𝜑 ∧ 𝑚 ∈ ℝ⁺))
84		elrege0 13375	. . . . . . . . . 10 ⊢ (𝐶 ∈ (0[,)+∞) ↔ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶))
85	84	simplbi 497	. . . . . . . . 9 ⊢ (𝐶 ∈ (0[,)+∞) → 𝐶 ∈ ℝ)
86	35, 85	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
87		rerpdivcl 12943	. . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ 𝑚 ∈ ℝ⁺) → (𝐶 / 𝑚) ∈ ℝ)
88	86, 87	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (𝐶 / 𝑚) ∈ ℝ)
89	83, 88	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (𝐶 / 𝑚) ∈ ℝ)
90	89	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (𝐶 / 𝑚) ∈ ℝ)
91	82	relogcld 26548	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
92	65, 71	logge0d 26555	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 ≤ (log‘𝑚))
93	91, 92	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
94	93	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) ∈ ℝ ∧ 0 ≤ (log‘𝑚)))
95		oveq2 7361	. . . . . . . 8 ⊢ (𝑆 = 0 → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0))
96	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → seq1( + , 𝐹):ℕ⟶ℂ)
97	96, 73	ffvelcdmd 7023	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
98	97	subid1d 11482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 0) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
99	95, 98	sylan9eqr 2786	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
100	99	fveq2d 6830	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) = (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))))
101		2fveq3 6831	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
102	101	fvoveq1d 7375	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
103		oveq2 7361	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
104	102, 103	breq12d 5108	. . . . . . . 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))
106	105	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
107		1re 11134	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
108		elicopnf 13366	. . . . . . . . . 10 ⊢ (1 ∈ ℝ → (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚)))
109	107, 108	ax-mp 5	. . . . . . . . 9 ⊢ (𝑚 ∈ (1[,)+∞) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚))
110	65, 71, 109	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ (1[,)+∞))
111	104, 106, 110	rspcdva 3580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
112	111	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) ≤ (𝐶 / 𝑚))
113	100, 112	eqbrtrrd 5119	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
114		lemul2a 11997	. . . . 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)))
115	76, 90, 94, 113, 114	syl31anc 1375	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) ≤ ((log‘𝑚) · (𝐶 / 𝑚)))
116		iftrue 4484	. . . . . . . . . . . . . . 15 ⊢ (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
117	116	fveq2d 6830	. . . . . . . . . . . . . 14 ⊢ (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
118	117	oveq1d 7368	. . . . . . . . . . . . 13 ⊢ (𝑆 = 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
119	118	ad2antlr 727	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑚) / 𝑘))
120	119	oveq2d 7369	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑚) / 𝑘)))
121	16	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
122		relogcl 26500	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ⁺ → (log‘𝑚) ∈ ℝ)
123	122	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (log‘𝑚) ∈ ℝ)
124	123	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (log‘𝑚) ∈ ℂ)
125	124	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (log‘𝑚) ∈ ℂ)
126	11	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℕ)
127	126	nncnd 12162	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
128	126	nnne0d 12196	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
129	121, 125, 127, 128	div12d 11954	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑚) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿‘𝑘)) / 𝑘)))
130	120, 129	eqtrd 2764	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘𝑚) · ((𝑋‘(𝐿‘𝑘)) / 𝑘)))
131	130	sumeq2dv 15627	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿‘𝑘)) / 𝑘)))
132		iftrue 4484	. . . . . . . . . . 11 ⊢ (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
133	132	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑆 = 0 → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − 0))
134	26	subid1d 11482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − 0) = Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
135	133, 134	sylan9eqr 2786	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
136		ovex 7386	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ V
137	54, 51, 136	fvmpt 6934	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) = ((𝑋‘(𝐿‘𝑘)) / 𝑘))
138	22, 137	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹‘𝑘) = ((𝑋‘(𝐿‘𝑘)) / 𝑘))
139	57	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝐹:ℕ⟶ℂ)
140	139, 11, 58	syl2an 596	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹‘𝑘) ∈ ℂ)
141	138, 140	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ ℂ)
142	9, 124, 141	fsummulc2 15709	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿‘𝑘)) / 𝑘)))
143	142	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘)) = Σ𝑘 ∈ (1...(⌊‘𝑚))((log‘𝑚) · ((𝑋‘(𝐿‘𝑘)) / 𝑘)))
144	131, 135, 143	3eqtr4d 2774	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘)))
145	83, 144	sylan 580	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘)))
146	83, 138	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐹‘𝑘) = ((𝑋‘(𝐿‘𝑘)) / 𝑘))
147	73, 44	eleqtrdi 2838	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (⌊‘𝑚) ∈ (ℤ_≥‘1))
148	77, 11, 50	syl2an 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ ℂ)
149	146, 147, 148	fsumser 15655	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
150	149	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
151	150	oveq2d 7369	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((log‘𝑚) · Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) / 𝑘)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
152	145, 151	eqtrd 2764	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚))))
153	152	fveq2d 6830	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))))
154	122	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℝ)
155	154	recnd 11162	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
156	83, 155	sylan 580	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (log‘𝑚) ∈ ℂ)
157	156, 75	absmuld 15382	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
158	91, 92	absidd 15348	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
159	158	oveq1d 7368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
160	159	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
161	153, 157, 160	3eqtrd 2768	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = ((log‘𝑚) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
162		iftrue 4484	. . . . . . . 8 ⊢ (𝑆 = 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
163	162	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐶)
164	163	oveq1d 7368	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
165	86	recnd 11162	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
166	165	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
167		rpcnne0 12930	. . . . . . . 8 ⊢ (𝑚 ∈ ℝ⁺ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
168	167	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
169		div12 11819	. . . . . . 7 ⊢ ((𝐶 ∈ ℂ ∧ (log‘𝑚) ∈ ℂ ∧ (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0)) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
170	166, 155, 168, 169	syl3anc 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (𝐶 · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
171	164, 170	eqtrd 2764	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
172	83, 171	sylan 580	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = ((log‘𝑚) · (𝐶 / 𝑚)))
173	115, 161, 172	3brtr4d 5127	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((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 6831	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
176	175	fvoveq1d 7375	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
177		fveq2 6826	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → (log‘𝑦) = (log‘𝑚))
178		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → 𝑦 = 𝑚)
179	177, 178	oveq12d 7371	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
180	179	oveq2d 7369	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
181	176, 180	breq12d 5108	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
182	181	rspccva 3578	. . . . . 6 ⊢ ((∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
183	174, 182	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
184	183	adantr 480	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚)))
185		fveq2 6826	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑘 → (log‘𝑎) = (log‘𝑘))
186	185, 53	oveq12d 7371	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
187	52, 186	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
188		dchrvmasumif.g	. . . . . . . . . 10 ⊢ 𝐾 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))
189		ovex 7386	. . . . . . . . . 10 ⊢ ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
190	187, 188, 189	fvmpt 6934	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ → (𝐾‘𝑘) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
191	11, 190	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝑚)) → (𝐾‘𝑘) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
192		ifnefalse 4490	. . . . . . . . . . . . 13 ⊢ (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
193	192	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
194	193	oveq1d 7368	. . . . . . . . . . 11 ⊢ (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
195	194	oveq2d 7369	. . . . . . . . . 10 ⊢ (𝑆 ≠ 0 → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
196	195	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
197	196	eqcomd 2735	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)) = ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
198	191, 197	sylan9eqr 2786	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾‘𝑘) = ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)))
199	147	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (⌊‘𝑚) ∈ (ℤ_≥‘1))
200		nnrp 12923	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
201	200	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
202	201	relogcld 26548	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
203	202	recnd 11162	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
204	203, 47, 49	divcld 11918	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
205	15, 204	mulcld 11154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
206	187	cbvmptv 5199	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
207	188, 206	eqtri 2752	. . . . . . . . . . 11 ⊢ 𝐾 = (𝑘 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)))
208	205, 207	fmptd 7052	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:ℕ⟶ℂ)
209	208	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
210		ffvelcdm 7019	. . . . . . . . 9 ⊢ ((𝐾:ℕ⟶ℂ ∧ 𝑘 ∈ ℕ) → (𝐾‘𝑘) ∈ ℂ)
211	209, 11, 210	syl2an 596	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (𝐾‘𝑘) ∈ ℂ)
212	198, 211	eqeltrrd 2829	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
213	198, 199, 212	fsumser 15655	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
214		ifnefalse 4490	. . . . . . 7 ⊢ (𝑆 ≠ 0 → if(𝑆 = 0, 0, 𝑇) = 𝑇)
215	214	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 0, 𝑇) = 𝑇)
216	213, 215	oveq12d 7371	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) = ((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇))
217	216	fveq2d 6830	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
218		ifnefalse 4490	. . . . . 6 ⊢ (𝑆 ≠ 0 → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
219	218	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → if(𝑆 = 0, 𝐶, 𝐸) = 𝐸)
220	219	oveq1d 7368	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
221	184, 217, 220	3brtr4d 5127	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)))
222	173, 221	pm2.61dane 3012	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)))
223		fzfid 13898	. . . 4 ⊢ (𝜑 → (1...2) ∈ Fin)
224	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → 𝑋 ∈ 𝐷)
225		elfzelz 13445	. . . . . . . 8 ⊢ (𝑘 ∈ (1...2) → 𝑘 ∈ ℤ)
226	225	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℤ)
227	4, 1, 5, 2, 224, 226	dchrzrhcl 27172	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
228	227	abscld 15364	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ)
229		3rp 12917	. . . . . . 7 ⊢ 3 ∈ ℝ⁺
230		relogcl 26500	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
231	229, 230	ax-mp 5	. . . . . 6 ⊢ (log‘3) ∈ ℝ
232		elfznn 13474	. . . . . . 7 ⊢ (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
233	232	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
234		nndivre 12187	. . . . . 6 ⊢ (((log‘3) ∈ ℝ ∧ 𝑘 ∈ ℕ) → ((log‘3) / 𝑘) ∈ ℝ)
235	231, 233, 234	sylancr 587	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
236	228, 235	remulcld 11164	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
237	223, 236	fsumrecl 15659	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
238	43	abscld 15364	. . 3 ⊢ (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
239	237, 238	readdcld 11163	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
240		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝜑)
241	62	rexri 11192	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ^*
242		elico2 13331	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 3 ∈ ℝ^*) → (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3)))
243	107, 241, 242	mp2an 692	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1[,)3) ↔ (𝑚 ∈ ℝ ∧ 1 ≤ 𝑚 ∧ 𝑚 < 3))
244	243	simp1bi 1145	. . . . . . . . 9 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 ∈ ℝ)
245	244	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ)
246		0red 11137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
247		1red 11135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
248		0lt1 11660	. . . . . . . . . 10 ⊢ 0 < 1
249	248	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 < 1)
250	243	simp2bi 1146	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1[,)3) → 1 ≤ 𝑚)
251	250	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 1 ≤ 𝑚)
252	246, 247, 245, 249, 251	ltletrd 11294	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 < 𝑚)
253	245, 252	elrpd 12952	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑚 ∈ ℝ⁺)
254	240, 253	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (𝜑 ∧ 𝑚 ∈ ℝ⁺))
255	43	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → if(𝑆 = 0, 0, 𝑇) ∈ ℂ)
256	26, 255	subcld 11493	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
257	254, 256	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇)) ∈ ℂ)
258	257	abscld 15364	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
259	254, 26	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
260	259	abscld 15364	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
261	238	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
262	260, 261	readdcld 11163	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
263	237	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
264	263, 261	readdcld 11163	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))) ∈ ℝ)
265	26, 255	abs2dif2d 15386	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))))
266	254, 265	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))))
267	25	abscld 15364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
268	9, 267	fsumrecl 15659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
269	254, 268	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
270	9, 25	fsumabs 15726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
271	254, 270	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
272		fzfid 13898	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (1...2) ∈ Fin)
273	227	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
274	17	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑚 ∈ ℝ⁺)
275	232	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
276	275	nnrpd 12953	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ⁺)
277	274, 276	ifcld 4525	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺)
278	277	relogcld 26548	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
279	278, 275	nndivred 12200	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
280	279	recnd 11162	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
281	273, 280	mulcld 11154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
282	281	abscld 15364	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
283	272, 282	fsumrecl 15659	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
284	254, 283	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
285		fzfid 13898	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (1...2) ∈ Fin)
286	254, 281	sylan 580	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
287	286	abscld 15364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288	286	absge0d 15372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
289	245	flcld 13720	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
290		2z 12525	. . . . . . . . . . 11 ⊢ 2 ∈ ℤ
291	290	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 2 ∈ ℤ)
292	243	simp3bi 1147	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ (1[,)3) → 𝑚 < 3)
293	292	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 𝑚 < 3)
294		3z 12526	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
295		fllt 13728	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ ∧ 3 ∈ ℤ) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
296	245, 294, 295	sylancl 586	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (𝑚 < 3 ↔ (⌊‘𝑚) < 3))
297	293, 296	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < 3)
298		df-3 12210	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
299	297, 298	breqtrdi 5136	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
300		rpre 12920	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
301	300	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
302	301	flcld 13720	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (⌊‘𝑚) ∈ ℤ)
303		zleltp1 12544	. . . . . . . . . . . . 13 ⊢ (((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
304	302, 290, 303	sylancl 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
305	254, 304	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → ((⌊‘𝑚) ≤ 2 ↔ (⌊‘𝑚) < (2 + 1)))
306	299, 305	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ≤ 2)
307		eluz2 12759	. . . . . . . . . 10 ⊢ (2 ∈ (ℤ_≥‘(⌊‘𝑚)) ↔ ((⌊‘𝑚) ∈ ℤ ∧ 2 ∈ ℤ ∧ (⌊‘𝑚) ≤ 2))
308	289, 291, 306, 307	syl3anbrc 1344	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 2 ∈ (ℤ_≥‘(⌊‘𝑚)))
309		fzss2 13485	. . . . . . . . 9 ⊢ (2 ∈ (ℤ_≥‘(⌊‘𝑚)) → (1...(⌊‘𝑚)) ⊆ (1...2))
310	308, 309	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (1...(⌊‘𝑚)) ⊆ (1...2))
311	285, 287, 288, 310	fsumless 15721	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
312	236	adantlr 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
313	273, 280	absmuld 15382	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) = ((abs‘(𝑋‘(𝐿‘𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
314	254, 313	sylan 580	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) = ((abs‘(𝑋‘(𝐿‘𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
315	254, 279	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
316	254, 278	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
317		log1 26510	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
318		elfzle1 13448	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
319		breq2 5099	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
320		breq2 5099	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
321	319, 320	ifboth 4518	. . . . . . . . . . . . . . . 16 ⊢ ((1 ≤ 𝑚 ∧ 1 ≤ 𝑘) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
322	251, 318, 321	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 1 ≤ if(𝑆 = 0, 𝑚, 𝑘))
323		1rp 12915	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ⁺
324		logleb 26528	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ⁺ ∧ if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
325	323, 277, 324	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
326	254, 325	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (1 ≤ if(𝑆 = 0, 𝑚, 𝑘) ↔ (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘))))
327	322, 326	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘1) ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
328	317, 327	eqbrtrrid 5131	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
329	276	rpregt0d 12961	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
330	254, 329	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (𝑘 ∈ ℝ ∧ 0 < 𝑘))
331		divge0 12012	. . . . . . . . . . . . 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))
332	316, 328, 330, 331	syl21anc 837	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
333	315, 332	absidd 15348	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))
334	333, 315	eqeltrd 2828	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ)
335	235	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘3) / 𝑘) ∈ ℝ)
336	228	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ)
337	273	absge0d 15372	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘(𝑋‘(𝐿‘𝑘))))
338	336, 337	jca 511	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑘)))))
339	254, 338	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑘)))))
340	292	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑚 < 3)
341	275	nnred 12161	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
342		2re 12220	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
343	342	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 2 ∈ ℝ)
344	62	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 3 ∈ ℝ)
345		elfzle2 13449	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
346	345	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
347		2lt3 12313	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 3
348	347	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
349	341, 343, 344, 346, 348	lelttrd 11292	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
350	254, 349	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 𝑘 < 3)
351		breq1 5098	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
352		breq1 5098	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
353	351, 352	ifboth 4518	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 < 3 ∧ 𝑘 < 3) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
354	340, 350, 353	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) < 3)
355	277	rpred 12955	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
356		ltle 11222	. . . . . . . . . . . . . . . 16 ⊢ ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ ∧ 3 ∈ ℝ) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
357	355, 62, 356	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
358	254, 357	sylan 580	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) < 3 → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3))
359	354, 358	mpd 15	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ≤ 3)
360		logleb 26528	. . . . . . . . . . . . . . 15 ⊢ ((if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺ ∧ 3 ∈ ℝ⁺) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
361	277, 229, 360	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
362	254, 361	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (if(𝑆 = 0, 𝑚, 𝑘) ≤ 3 ↔ (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3)))
363	359, 362	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3))
364	231	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (log‘3) ∈ ℝ)
365	278, 364, 276	lediv1d 13001	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3) ↔ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘)))
366	254, 365	sylan 580	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) ≤ (log‘3) ↔ ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘)))
367	363, 366	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ≤ ((log‘3) / 𝑘))
368	333, 367	eqbrtrd 5117	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
369		lemul2a 11997	. . . . . . . . . 10 ⊢ ((((abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℝ ∧ ((log‘3) / 𝑘) ∈ ℝ ∧ ((abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ ∧ 0 ≤ (abs‘(𝑋‘(𝐿‘𝑘))))) ∧ (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘)) → ((abs‘(𝑋‘(𝐿‘𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
370	334, 335, 339, 368, 369	syl31anc 1375	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) · (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
371	314, 370	eqbrtrd 5117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
372	285, 287, 312, 371	fsumle 15724	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...2)(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
373	269, 284, 263, 311, 372	letrd 11291	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → Σ𝑘 ∈ (1...(⌊‘𝑚))(abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
374	260, 269, 263, 271, 373	letrd 11291	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)))
375	26	abscld 15364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
376	237	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
377	255	abscld 15364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
378	375, 376, 377	leadd1d 11732	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ↔ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇)))))
379	254, 378	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ≤ Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ↔ ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇)))))
380	374, 379	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → ((abs‘Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) + (abs‘if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
381	258, 262, 264, 266, 380	letrd 11291	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
382	381	ralrimiva 3121	. 2 ⊢ (𝜑 → ∀𝑚 ∈ (1[,)3)(abs‘(Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) − if(𝑆 = 0, 0, 𝑇))) ≤ (Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) + (abs‘if(𝑆 = 0, 0, 𝑇))))
383	1, 2, 3, 4, 5, 6, 7, 8, 26, 34, 37, 43, 222, 239, 382	dchrvmasumlem3 27426	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑑 ∈ (1...(⌊‘𝑥))(((𝑋‘(𝐿‘𝑑)) · ((μ‘𝑑) / 𝑑)) · (Σ𝑘 ∈ (1...(⌊‘(𝑥 / 𝑑)))((𝑋‘(𝐿‘𝑘)) · ((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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ⊆ wss 3905 ifcif 4478 class class class wbr 5095 ↦ cmpt 5176 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 +∞cpnf 11165 ℝ^*cxr 11167 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 ℕcn 12146 2c2 12201 3c3 12202 ℤcz 12489 ℤ_≥cuz 12753 ℝ⁺crp 12911 [,)cico 13268 ...cfz 13428 ⌊cfl 13712 seqcseq 13926 abscabs 15159 ⇝ cli 15409 𝑂(1)co1 15411 Σcsu 15611 Basecbs 17138 0_gc0g 17361 ℤRHomczrh 21424 ℤ/nℤczn 21427 logclog 26479 μcmu 27021 DChrcdchr 27159

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8632 df-ec 8634 df-qs 8638 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-acn 9857 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-o1 15415 df-lo1 15416 df-sum 15612 df-ef 15992 df-e 15993 df-sin 15994 df-cos 15995 df-tan 15996 df-pi 15997 df-dvds 16182 df-prm 16601 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-qus 17431 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-mhm 18675 df-submnd 18676 df-grp 18833 df-minusg 18834 df-sbg 18835 df-mulg 18965 df-subg 19020 df-nsg 19021 df-eqg 19022 df-ghm 19110 df-cntz 19214 df-od 19425 df-cmn 19679 df-abl 19680 df-mgp 20044 df-rng 20056 df-ur 20085 df-ring 20138 df-cring 20139 df-oppr 20240 df-dvdsr 20260 df-unit 20261 df-invr 20291 df-dvr 20304 df-rhm 20375 df-subrng 20449 df-subrg 20473 df-drng 20634 df-lmod 20783 df-lss 20853 df-lsp 20893 df-sra 21095 df-rgmod 21096 df-lidl 21133 df-rsp 21134 df-2idl 21175 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-zring 21372 df-zrh 21428 df-zn 21431 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-limc 25783 df-dv 25784 df-ulm 26302 df-log 26481 df-cxp 26482 df-atan 26793 df-em 26919 df-mu 27027 df-dchr 27160

This theorem is referenced by: dchrvmasumiflem2 27429

W3C validator