knoppndvlem15 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem15 37004

Description: Lemma for knoppndv 37012. (Contributed by Asger C. Ipsen, 6-Jul-2021.)

**Hypotheses**
Ref	Expression
knoppndvlem15.t	⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
knoppndvlem15.f	⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
knoppndvlem15.w	⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
knoppndvlem15.a	⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
knoppndvlem15.b	⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))
knoppndvlem15.c	⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
knoppndvlem15.j	⊢ (𝜑 → 𝐽 ∈ ℕ₀)
knoppndvlem15.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
knoppndvlem15.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
knoppndvlem15.1	⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))

**Assertion**
Ref	Expression
knoppndvlem15	⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))))

Distinct variable groups: 𝐴,𝑖,𝑛,𝑤,𝑦 𝑥,𝐴,𝑖,𝑤 𝐵,𝑖,𝑛,𝑤,𝑦 𝑥,𝐵 𝐶,𝑖,𝑛,𝑦 𝑖,𝐹,𝑤 𝑖,𝐽,𝑛,𝑦 𝑥,𝐽 𝑛,𝑀,𝑦 𝑥,𝑀 𝑖,𝑁,𝑛,𝑦 𝑥,𝑁 𝑇,𝑛,𝑦 𝜑,𝑖,𝑛,𝑤,𝑦 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥,𝑤) 𝑇(𝑥,𝑤,𝑖) 𝐹(𝑥,𝑦,𝑛) 𝐽(𝑤) 𝑀(𝑤,𝑖) 𝑁(𝑤) 𝑊(𝑥,𝑦,𝑤,𝑖,𝑛)

**Proof of Theorem knoppndvlem15**
Step	Hyp	Ref	Expression
1		knoppndvlem15.c	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1))
2	1	knoppndvlem3 36992	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
3	2	simpld 499	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11237	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5	4	abscld 15490	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
6		knoppndvlem15.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	5, 6	reexpcld 14199	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶)↑𝐽) ∈ ℝ)
8		2re 12315	. . . . . . 7 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
10		2ne0 12347	. . . . . . 7 ⊢ 2 ≠ 0
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
12	7, 9, 11	redivcld 12043	. . . . 5 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℝ)
13		1red 11209	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
14		knoppndvlem15.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
15	14	nnred 12248	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
16	9, 15	remulcld 11239	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
17	16, 5	remulcld 11239	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ)
18	17, 13	resubcld 11642	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ)
19		0red 11211	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
20		0lt1 11736	. . . . . . . . . . 11 ⊢ 0 < 1
21	20	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 0 < 1)
22		knoppndvlem15.1	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < (𝑁 · (abs‘𝐶)))
23	1, 14, 22	knoppndvlem12 37001	. . . . . . . . . . 11 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
24	23	simprd 500	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
25	19, 13, 18, 21, 24	lttrd 11371	. . . . . . . . 9 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
26	18, 25	jca 520	. . . . . . . 8 ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
27		gt0ne0 11679	. . . . . . . 8 ⊢ (((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) → (((2 · 𝑁) · (abs‘𝐶)) − 1) ≠ 0)
28	26, 27	syl 18	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ≠ 0)
29	13, 18, 28	redivcld 12043	. . . . . 6 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ)
30	13, 29	resubcld 11642	. . . . 5 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
31	12, 30	remulcld 11239	. . . 4 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ∈ ℝ)
32		knoppndvlem15.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥)))
33		knoppndvlem15.f	. . . . . . . . 9 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ₀ ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦)))))
34		knoppndvlem15.a	. . . . . . . . . . 11 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀)
35	34	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀))
36	6	nn0zd 12616	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
37		knoppndvlem15.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38	14, 36, 37	knoppndvlem1 36990	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
39	35, 38	eqeltrd 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
40	32, 33, 14, 3, 39, 6	knoppcnlem3 36973	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
41	40	recnd 11237	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℂ)
42		knoppndvlem15.b	. . . . . . . . . . 11 ⊢ 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1))
43	42	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)))
44	37	peano2zd 12703	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
45	14, 36, 44	knoppndvlem1 36990	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
46	43, 45	eqeltrd 2869	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
47	32, 33, 14, 3, 46, 6	knoppcnlem3 36973	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
48	47	recnd 11237	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
49	41, 48	subcld 11569	. . . . . 6 ⊢ (𝜑 → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) ∈ ℂ)
50	49	abscld 15490	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) ∈ ℝ)
51	32, 33, 46, 3, 14	knoppndvlem5 36994	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℝ)
52	51	recnd 11237	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℂ)
53	32, 33, 39, 3, 14	knoppndvlem5 36994	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℝ)
54	53	recnd 11237	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℂ)
55	52, 54	subcld 11569	. . . . . 6 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) ∈ ℂ)
56	55	abscld 15490	. . . . 5 ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ∈ ℝ)
57	50, 56	resubcld 11642	. . . 4 ⊢ (𝜑 → ((abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) − (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))) ∈ ℝ)
58	49, 55	subcld 11569	. . . . 5 ⊢ (𝜑 → ((((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ∈ ℂ)
59	58	abscld 15490	. . . 4 ⊢ (𝜑 → (abs‘((((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))) ∈ ℝ)
60	12, 29	jca 520	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ))
61		remulcl 11185	. . . . . . . 8 ⊢ (((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ) → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
62	60, 61	syl 18	. . . . . . 7 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
63	12, 62	jca 520	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ))
64		resubcl 11522	. . . . . 6 ⊢ (((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ) → ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ∈ ℝ)
65	63, 64	syl 18	. . . . 5 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ∈ ℝ)
66	12	recnd 11237	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℂ)
67		1cnd 11202	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
68	29	recnd 11237	. . . . . . 7 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℂ)
69	66, 67, 68	subdid 11670	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) = (((((abs‘𝐶)↑𝐽) / 2) · 1) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
70	66	mulridd 11226	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · 1) = (((abs‘𝐶)↑𝐽) / 2))
71	70	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → (((((abs‘𝐶)↑𝐽) / 2) · 1) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) = ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
72	65	leidd 11780	. . . . . . 7 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
73	71, 72	eqbrtrd 5137	. . . . . 6 ⊢ (𝜑 → (((((abs‘𝐶)↑𝐽) / 2) · 1) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
74	69, 73	eqbrtrd 5137	. . . . 5 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))))
75	12, 29	remulcld 11239	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
76	12	leidd 11780	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (((abs‘𝐶)↑𝐽) / 2))
77	41, 48	abssubd 15507	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))))
78	32, 33, 34, 42, 1, 6, 37, 14	knoppndvlem10 36999	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
79	77, 78	eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
80	79	eqcomd 2775	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
81	76, 80	breqtrd 5141	. . . . . 6 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
82	32, 33, 34, 42, 1, 6, 37, 14, 22	knoppndvlem14 37003	. . . . . 6 ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ≤ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))))
83	12, 56, 50, 75, 81, 82	le2subd 11834	. . . . 5 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) − (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))))
84	31, 65, 57, 74, 83	letrd 11367	. . . 4 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ ((abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) − (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))))
85	49, 55	abs2difd 15511	. . . 4 ⊢ (𝜑 → ((abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) − (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))) ≤ (abs‘((((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))))
86	31, 57, 59, 84, 85	letrd 11367	. . 3 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))))
87	49, 55	abssubd 15507	. . 3 ⊢ (𝜑 → (abs‘((((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)))) = (abs‘((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)))))
88	86, 87	breqtrd 5141	. 2 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)))))
89		knoppndvlem15.w	. . . . . . . 8 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ₀ ((𝐹‘𝑤)‘𝑖))
90	32, 33, 89, 42, 1, 6, 44, 14	knoppndvlem6 36995	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐵) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖))
91		elnn0uz 12903	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ₀ ↔ 𝐽 ∈ (ℤ_≥‘0))
92	6, 91	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘0))
93	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑁 ∈ ℕ)
94	3	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐶 ∈ ℝ)
95	46	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐵 ∈ ℝ)
96		elfznn0 13648	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝐽) → 𝑖 ∈ ℕ₀)
97	96	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑖 ∈ ℕ₀)
98	32, 33, 93, 94, 95, 97	knoppcnlem3 36973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℝ)
99	98	recnd 11237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℂ)
100		fveq2 6882	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐵)‘𝑖) = ((𝐹‘𝐵)‘𝐽))
101	92, 99, 100	fsumm1 15802	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
102	90, 101	eqtrd 2804	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐵) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
103	32, 33, 89, 34, 1, 6, 37, 14	knoppndvlem6 36995	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
104	39	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐴 ∈ ℝ)
105	32, 33, 93, 94, 104, 97	knoppcnlem3 36973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ)
106	105	recnd 11237	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℂ)
107		fveq2 6882	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐴)‘𝑖) = ((𝐹‘𝐴)‘𝐽))
108	92, 106, 107	fsumm1 15802	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
109	103, 108	eqtrd 2804	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐴) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
110	102, 109	oveq12d 7429	. . . . 5 ⊢ (𝜑 → ((𝑊‘𝐵) − (𝑊‘𝐴)) = ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽))))
111	52, 54, 41, 48	subadd4d 11617	. . . . . 6 ⊢ (𝜑 → ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽))))
112	111	eqcomd 2775	. . . . 5 ⊢ (𝜑 → ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)) − (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽))) = ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
113	110, 112	eqtrd 2804	. . . 4 ⊢ (𝜑 → ((𝑊‘𝐵) − (𝑊‘𝐴)) = ((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
114	113	fveq2d 6886	. . 3 ⊢ (𝜑 → (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))) = (abs‘((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)))))
115	114	eqcomd 2775	. 2 ⊢ (𝜑 → (abs‘((Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) − (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)))) = (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))))
116	88, 115	breqtrd 5141	1 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 class class class wbr 5113 ↦ cmpt 5196 ‘cfv 6537 (class class class)co 7411 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 · cmul 11105 < clt 11243 ≤ cle 11244 − cmin 11441 -cneg 11442 / cdiv 11871 ℕcn 12233 2c2 12295 ℕ₀cn0 12504 ℤcz 12591 ℤ_≥cuz 12862 (,)cioo 13372 ...cfz 13535 ⌊cfl 13823 ↑cexp 14097 abscabs 15285 Σcsu 15737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9402 df-inf 9403 df-oi 9472 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-n0 12505 df-z 12592 df-uz 12863 df-rp 13017 df-ioo 13376 df-ico 13378 df-fz 13536 df-fzo 13683 df-fl 13825 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-limsup 15522 df-clim 15539 df-rlim 15540 df-sum 15738 df-dvds 16311 df-ulm 26506

This theorem is referenced by: knoppndvlem17 37006

W3C validator