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Theorem dchrvmasumiflem1 27412

Description: Lemma for dchrvmasumif 27414. (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 13938	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (1...(⌊‘𝑚)) ∈ Fin)
10		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝜑)
11		elfznn 13514	. . . . 5 ⊢ (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℕ)
12	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑋 ∈ 𝐷)
13		nnz 12550	. . . . . . 7 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
14	13	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
15	4, 1, 5, 2, 12, 14	dchrzrhcl 27156	. . . . 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 12993	. . . . . . . 8 ⊢ (𝑘 ∈ (1...(⌊‘𝑚)) → 𝑘 ∈ ℝ⁺)
19		ifcl 4534	. . . . . . . 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 26532	. . . . . 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 12240	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
24	23	recnd 11202	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
25	16, 24	mulcld 11194	. . 3 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
26	9, 25	fsumcl 15699	. 2 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
27		fveq2 6858	. . . 4 ⊢ (𝑚 = (𝑥 / 𝑑) → (⌊‘𝑚) = (⌊‘(𝑥 / 𝑑)))
28	27	oveq2d 7403	. . 3 ⊢ (𝑚 = (𝑥 / 𝑑) → (1...(⌊‘𝑚)) = (1...(⌊‘(𝑥 / 𝑑))))
29		ifeq1 4492	. . . . . . 7 ⊢ (𝑚 = (𝑥 / 𝑑) → if(𝑆 = 0, 𝑚, 𝑘) = if(𝑆 = 0, (𝑥 / 𝑑), 𝑘))
30	29	fveq2d 6862	. . . . . 6 ⊢ (𝑚 = (𝑥 / 𝑑) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)))
31	30	oveq1d 7402	. . . . 5 ⊢ (𝑚 = (𝑥 / 𝑑) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘if(𝑆 = 0, (𝑥 / 𝑑), 𝑘)) / 𝑘))
32	31	oveq2d 7403	. . . 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 15673	. 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 4535	. 2 ⊢ (𝜑 → if(𝑆 = 0, 𝐶, 𝐸) ∈ (0[,)+∞))
38		0cn 11166	. . 3 ⊢ 0 ∈ ℂ
39		dchrvmasumif.t	. . . 4 ⊢ (𝜑 → seq1( + , 𝐾) ⇝ 𝑇)
40		climcl 15465	. . . 4 ⊢ (seq1( + , 𝐾) ⇝ 𝑇 → 𝑇 ∈ ℂ)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ∈ ℂ)
42		ifcl 4534	. . 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 12836	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
45		1zzd 12564	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
46		nncn 12194	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℂ)
47	46	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℂ)
48		nnne0 12220	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ≠ 0)
49	48	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ≠ 0)
50	15, 47, 49	divcld 11958	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ ℂ)
51		dchrvmasumif.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
52		2fveq3 6863	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑘 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑘)))
53		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑘 → 𝑎 = 𝑘)
54	52, 53	oveq12d 7405	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑘 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑘)) / 𝑘))
55	54	cbvmptv 5211	. . . . . . . . . . . 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 7086	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:ℕ⟶ℂ)
58		ffvelcdm 7053	. . . . . . . . . 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 13995	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹):ℕ⟶ℂ)
61	60	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → seq1( + , 𝐹):ℕ⟶ℂ)
62		3re 12266	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ
63		elicopnf 13406	. . . . . . . . . . 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 11175	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 1 ∈ ℝ)
67	62	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 3 ∈ ℝ)
68		1le3 12393	. . . . . . . . . . 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 11331	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 1 ≤ 𝑚)
72		flge1nn 13783	. . . . . . . . 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 7057	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
76	75	abscld 15405	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ∈ ℝ)
77		simpl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝜑)
78		0red 11177	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 ∈ ℝ)
79		3pos 12291	. . . . . . . . . . 11 ⊢ 0 < 3
80	79	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 < 3)
81	78, 67, 65, 80, 70	ltletrd 11334	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 0 < 𝑚)
82	65, 81	elrpd 12992	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → 𝑚 ∈ ℝ⁺)
83	77, 82	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (𝜑 ∧ 𝑚 ∈ ℝ⁺))
84		elrege0 13415	. . . . . . . . . 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 12983	. . . . . . . 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 26532	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (log‘𝑚) ∈ ℝ)
92	65, 71	logge0d 26539	. . . . . . 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 7395	. . . . . . . 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 7057	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (seq1( + , 𝐹)‘(⌊‘𝑚)) ∈ ℂ)
98	97	subid1d 11522	. . . . . . . 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 6862	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)) = (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))))
101		2fveq3 6863	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → (seq1( + , 𝐹)‘(⌊‘𝑦)) = (seq1( + , 𝐹)‘(⌊‘𝑚)))
102	101	fvoveq1d 7409	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) = (abs‘((seq1( + , 𝐹)‘(⌊‘𝑚)) − 𝑆)))
103		oveq2 7395	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (𝐶 / 𝑦) = (𝐶 / 𝑚))
104	102, 103	breq12d 5120	. . . . . . . 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 11174	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
108		elicopnf 13406	. . . . . . . . . 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 3589	. . . . . . 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 5131	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚))) ≤ (𝐶 / 𝑚))
114		lemul2a 12037	. . . . 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 4494	. . . . . . . . . . . . . . 15 ⊢ (𝑆 = 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑚)
117	116	fveq2d 6862	. . . . . . . . . . . . . 14 ⊢ (𝑆 = 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑚))
118	117	oveq1d 7402	. . . . . . . . . . . . 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 7403	. . . . . . . . . . 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 26484	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ⁺ → (log‘𝑚) ∈ ℝ)
123	122	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (log‘𝑚) ∈ ℝ)
124	123	recnd 11202	. . . . . . . . . . . . 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 12202	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ∈ ℂ)
128	126	nnne0d 12236	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → 𝑘 ≠ 0)
129	121, 125, 127, 128	div12d 11994	. . . . . . . . . . 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 15668	. . . . . . . . 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 4494	. . . . . . . . . . 11 ⊢ (𝑆 = 0 → if(𝑆 = 0, 0, 𝑇) = 0)
133	132	oveq2d 7403	. . . . . . . . . 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 11522	. . . . . . . . . 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 7420	. . . . . . . . . . . . . 14 ⊢ ((𝑋‘(𝐿‘𝑘)) / 𝑘) ∈ V
137	54, 51, 136	fvmpt 6968	. . . . . . . . . . . . 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 15750	. . . . . . . . . 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 15696	. . . . . . . . 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 7403	. . . . . . 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 6862	. . . . 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 11202	. . . . . . 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 15423	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 = 0) → (abs‘((log‘𝑚) · (seq1( + , 𝐹)‘(⌊‘𝑚)))) = ((abs‘(log‘𝑚)) · (abs‘(seq1( + , 𝐹)‘(⌊‘𝑚)))))
158	91, 92	absidd 15389	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) → (abs‘(log‘𝑚)) = (log‘𝑚))
159	158	oveq1d 7402	. . . . . 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 4494	. . . . . . . 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 7402	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐶 · ((log‘𝑚) / 𝑚)))
165	86	recnd 11202	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
166	165	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → 𝐶 ∈ ℂ)
167		rpcnne0 12970	. . . . . . . 8 ⊢ (𝑚 ∈ ℝ⁺ → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
168	167	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑆 = 0) → (𝑚 ∈ ℂ ∧ 𝑚 ≠ 0))
169		div12 11859	. . . . . . 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 5139	. . 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 6863	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (seq1( + , 𝐾)‘(⌊‘𝑦)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
176	175	fvoveq1d 7409	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) = (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)))
177		fveq2 6858	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → (log‘𝑦) = (log‘𝑚))
178		id 22	. . . . . . . . . 10 ⊢ (𝑦 = 𝑚 → 𝑦 = 𝑚)
179	177, 178	oveq12d 7405	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → ((log‘𝑦) / 𝑦) = ((log‘𝑚) / 𝑚))
180	179	oveq2d 7403	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (𝐸 · ((log‘𝑦) / 𝑦)) = (𝐸 · ((log‘𝑚) / 𝑚)))
181	176, 180	breq12d 5120	. . . . . . 7 ⊢ (𝑦 = 𝑚 → ((abs‘((seq1( + , 𝐾)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐸 · ((log‘𝑦) / 𝑦)) ↔ (abs‘((seq1( + , 𝐾)‘(⌊‘𝑚)) − 𝑇)) ≤ (𝐸 · ((log‘𝑚) / 𝑚))))
182	181	rspccva 3587	. . . . . 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 6858	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑘 → (log‘𝑎) = (log‘𝑘))
186	185, 53	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((log‘𝑎) / 𝑎) = ((log‘𝑘) / 𝑘))
187	52, 186	oveq12d 7405	. . . . . . . . . 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 7420	. . . . . . . . . 10 ⊢ ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)) ∈ V
190	187, 188, 189	fvmpt 6968	. . . . . . . . 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 4500	. . . . . . . . . . . . 13 ⊢ (𝑆 ≠ 0 → if(𝑆 = 0, 𝑚, 𝑘) = 𝑘)
193	192	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (𝑆 ≠ 0 → (log‘if(𝑆 = 0, 𝑚, 𝑘)) = (log‘𝑘))
194	193	oveq1d 7402	. . . . . . . . . . 11 ⊢ (𝑆 ≠ 0 → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) = ((log‘𝑘) / 𝑘))
195	194	oveq2d 7403	. . . . . . . . . 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 12963	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ⁺)
201	200	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ⁺)
202	201	relogcld 26532	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℝ)
203	202	recnd 11202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (log‘𝑘) ∈ ℂ)
204	203, 47, 49	divcld 11958	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((log‘𝑘) / 𝑘) ∈ ℂ)
205	15, 204	mulcld 11194	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑋‘(𝐿‘𝑘)) · ((log‘𝑘) / 𝑘)) ∈ ℂ)
206	187	cbvmptv 5211	. . . . . . . . . . . 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 7086	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:ℕ⟶ℂ)
209	208	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → 𝐾:ℕ⟶ℂ)
210		ffvelcdm 7053	. . . . . . . . 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 15696	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → Σ𝑘 ∈ (1...(⌊‘𝑚))((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) = (seq1( + , 𝐾)‘(⌊‘𝑚)))
214		ifnefalse 4500	. . . . . . 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 7405	. . . . 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 6862	. . . 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 4500	. . . . . 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 7402	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ (3[,)+∞)) ∧ 𝑆 ≠ 0) → (if(𝑆 = 0, 𝐶, 𝐸) · ((log‘𝑚) / 𝑚)) = (𝐸 · ((log‘𝑚) / 𝑚)))
221	184, 217, 220	3brtr4d 5139	. . 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 13938	. . . 4 ⊢ (𝜑 → (1...2) ∈ Fin)
224	7	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → 𝑋 ∈ 𝐷)
225		elfzelz 13485	. . . . . . . 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 27156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → (𝑋‘(𝐿‘𝑘)) ∈ ℂ)
228	227	abscld 15405	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → (abs‘(𝑋‘(𝐿‘𝑘))) ∈ ℝ)
229		3rp 12957	. . . . . . 7 ⊢ 3 ∈ ℝ⁺
230		relogcl 26484	. . . . . . 7 ⊢ (3 ∈ ℝ⁺ → (log‘3) ∈ ℝ)
231	229, 230	ax-mp 5	. . . . . 6 ⊢ (log‘3) ∈ ℝ
232		elfznn 13514	. . . . . . 7 ⊢ (𝑘 ∈ (1...2) → 𝑘 ∈ ℕ)
233	232	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℕ)
234		nndivre 12227	. . . . . 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 11204	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...2)) → ((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
237	223, 236	fsumrecl 15700	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (1...2)((abs‘(𝑋‘(𝐿‘𝑘))) · ((log‘3) / 𝑘)) ∈ ℝ)
238	43	abscld 15405	. . 3 ⊢ (𝜑 → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
239	237, 238	readdcld 11203	. 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 11232	. . . . . . . . . . 11 ⊢ 3 ∈ ℝ^*
242		elico2 13371	. . . . . . . . . . 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 11177	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 ∈ ℝ)
247		1red 11175	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 1 ∈ ℝ)
248		0lt1 11700	. . . . . . . . . 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 11334	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → 0 < 𝑚)
253	245, 252	elrpd 12992	. . . . . . 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 11533	. . . . . 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 15405	. . . 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 15405	. . . . 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 11203	. . . 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 11203	. . . 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 15427	. . . . 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 15405	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...(⌊‘𝑚))) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
268	9, 267	fsumrecl 15700	. . . . . . 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 15767	. . . . . . 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 13938	. . . . . . . . 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 12993	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ⁺)
277	274, 276	ifcld 4535	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ⁺)
278	277	relogcld 26532	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (log‘if(𝑆 = 0, 𝑚, 𝑘)) ∈ ℝ)
279	278, 275	nndivred 12240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℝ)
280	279	recnd 11202	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘) ∈ ℂ)
281	273, 280	mulcld 11194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → ((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ∈ ℂ)
282	281	abscld 15405	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
283	272, 282	fsumrecl 15700	. . . . . . . 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 13938	. . . . . . . 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 15405	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))) ∈ ℝ)
288	286	absge0d 15413	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (abs‘((𝑋‘(𝐿‘𝑘)) · ((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘))))
289	245	flcld 13760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) ∈ ℤ)
290		2z 12565	. . . . . . . . . . 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 12566	. . . . . . . . . . . . . 14 ⊢ 3 ∈ ℤ
295		fllt 13768	. . . . . . . . . . . . . 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 12250	. . . . . . . . . . . 12 ⊢ 3 = (2 + 1)
299	297, 298	breqtrdi 5148	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1[,)3)) → (⌊‘𝑚) < (2 + 1))
300		rpre 12960	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ ℝ⁺ → 𝑚 ∈ ℝ)
301	300	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → 𝑚 ∈ ℝ)
302	301	flcld 13760	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (⌊‘𝑚) ∈ ℤ)
303		zleltp1 12584	. . . . . . . . . . . . 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 12799	. . . . . . . . . 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 13525	. . . . . . . . 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 15762	. . . . . . 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 15423	. . . . . . . . . 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 26494	. . . . . . . . . . . . . 14 ⊢ (log‘1) = 0
318		elfzle1 13488	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...2) → 1 ≤ 𝑘)
319		breq2 5111	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑚 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
320		breq2 5111	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (1 ≤ 𝑘 ↔ 1 ≤ if(𝑆 = 0, 𝑚, 𝑘)))
321	319, 320	ifboth 4528	. . . . . . . . . . . . . . . 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 12955	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ⁺
324		logleb 26512	. . . . . . . . . . . . . . . . 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 5143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → 0 ≤ (log‘if(𝑆 = 0, 𝑚, 𝑘)))
329	276	rpregt0d 13001	. . . . . . . . . . . . . 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 12052	. . . . . . . . . . . . 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 15389	. . . . . . . . . . 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 15413	. . . . . . . . . . . 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 12201	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ∈ ℝ)
342		2re 12260	. . . . . . . . . . . . . . . . . 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 13489	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...2) → 𝑘 ≤ 2)
346	345	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 𝑘 ≤ 2)
347		2lt3 12353	. . . . . . . . . . . . . . . . . 18 ⊢ 2 < 3
348	347	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → 2 < 3)
349	341, 343, 344, 346, 348	lelttrd 11332	. . . . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑚 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
352		breq1 5110	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = if(𝑆 = 0, 𝑚, 𝑘) → (𝑘 < 3 ↔ if(𝑆 = 0, 𝑚, 𝑘) < 3))
353	351, 352	ifboth 4528	. . . . . . . . . . . . . . 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 12995	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑚 ∈ ℝ⁺) ∧ 𝑘 ∈ (1...2)) → if(𝑆 = 0, 𝑚, 𝑘) ∈ ℝ)
356		ltle 11262	. . . . . . . . . . . . . . . 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 26512	. . . . . . . . . . . . . . 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 13041	. . . . . . . . . . . . 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 5129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (1[,)3)) ∧ 𝑘 ∈ (1...2)) → (abs‘((log‘if(𝑆 = 0, 𝑚, 𝑘)) / 𝑘)) ≤ ((log‘3) / 𝑘))
369		lemul2a 12037	. . . . . . . . . 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 5129	. . . . . . . 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 15765	. . . . . . 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 11331	. . . . . 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 11331	. . . . 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 15405	. . . . . . 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 15405	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℝ⁺) → (abs‘if(𝑆 = 0, 0, 𝑇)) ∈ ℝ)
378	375, 376, 377	leadd1d 11772	. . . . . 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 11331	. . 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 3125	. 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 27410	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 3914 ifcif 4488 class class class wbr 5107 ↦ cmpt 5188 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 0cc0 11068 1c1 11069 + caddc 11071 · cmul 11073 +∞cpnf 11205 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 − cmin 11405 / cdiv 11835 ℕcn 12186 2c2 12241 3c3 12242 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 [,)cico 13308 ...cfz 13468 ⌊cfl 13752 seqcseq 13966 abscabs 15200 ⇝ cli 15450 𝑂(1)co1 15452 Σcsu 15652 Basecbs 17179 0_gc0g 17402 ℤRHomczrh 21409 ℤ/nℤczn 21412 logclog 26463 μcmu 27005 DChrcdchr 27143

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148

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 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-disj 5075 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-tpos 8205 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-oadd 8438 df-omul 8439 df-er 8671 df-ec 8673 df-qs 8677 df-map 8801 df-pm 8802 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-acn 9895 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-xnn0 12516 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-fl 13754 df-mod 13832 df-seq 13967 df-exp 14027 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15033 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-limsup 15437 df-clim 15454 df-rlim 15455 df-o1 15456 df-lo1 15457 df-sum 15653 df-ef 16033 df-e 16034 df-sin 16035 df-cos 16036 df-tan 16037 df-pi 16038 df-dvds 16223 df-prm 16642 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-qus 17472 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-mhm 18710 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-mulg 19000 df-subg 19055 df-nsg 19056 df-eqg 19057 df-ghm 19145 df-cntz 19249 df-od 19458 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-cring 20145 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-rhm 20381 df-subrng 20455 df-subrg 20479 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-sra 21080 df-rgmod 21081 df-lidl 21118 df-rsp 21119 df-2idl 21160 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-fbas 21261 df-fg 21262 df-cnfld 21265 df-zring 21357 df-zrh 21413 df-zn 21416 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-perf 23024 df-cn 23114 df-cnp 23115 df-haus 23202 df-cmp 23274 df-tx 23449 df-hmeo 23642 df-fil 23733 df-fm 23825 df-flim 23826 df-flf 23827 df-xms 24208 df-ms 24209 df-tms 24210 df-cncf 24771 df-limc 25767 df-dv 25768 df-ulm 26286 df-log 26465 df-cxp 26466 df-atan 26777 df-em 26903 df-mu 27011 df-dchr 27144

This theorem is referenced by: dchrvmasumiflem2 27413

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