knoppndvlem15 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem15 34706

Description: Lemma for knoppndv 34714. (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 34694	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
3	2	simpld 495	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11003	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5	4	abscld 15148	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
6		knoppndvlem15.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	5, 6	reexpcld 13881	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶)↑𝐽) ∈ ℝ)
8		2re 12047	. . . . . . 7 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
10		2ne0 12077	. . . . . . 7 ⊢ 2 ≠ 0
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
12	7, 9, 11	redivcld 11803	. . . . 5 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℝ)
13		1red 10976	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
14		knoppndvlem15.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
15	14	nnred 11988	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
16	9, 15	remulcld 11005	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
17	16, 5	remulcld 11005	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ)
18	17, 13	resubcld 11403	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ)
19		0red 10978	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
20		0lt1 11497	. . . . . . . . . . 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 34703	. . . . . . . . . . 11 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
24	23	simprd 496	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
25	19, 13, 18, 21, 24	lttrd 11136	. . . . . . . . 9 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
26	18, 25	jca 512	. . . . . . . 8 ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
27		gt0ne0 11440	. . . . . . . 8 ⊢ (((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)) → (((2 · 𝑁) · (abs‘𝐶)) − 1) ≠ 0)
28	26, 27	syl 17	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ≠ 0)
29	13, 18, 28	redivcld 11803	. . . . . 6 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ)
30	13, 29	resubcld 11403	. . . . 5 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
31	12, 30	remulcld 11005	. . . 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 12424	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
37		knoppndvlem15.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38	14, 36, 37	knoppndvlem1 34692	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
39	35, 38	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
40	32, 33, 14, 3, 39, 6	knoppcnlem3 34675	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
41	40	recnd 11003	. . . . . . 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 12429	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
45	14, 36, 44	knoppndvlem1 34692	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
46	43, 45	eqeltrd 2839	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
47	32, 33, 14, 3, 46, 6	knoppcnlem3 34675	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
48	47	recnd 11003	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
49	41, 48	subcld 11332	. . . . . 6 ⊢ (𝜑 → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) ∈ ℂ)
50	49	abscld 15148	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) ∈ ℝ)
51	32, 33, 46, 3, 14	knoppndvlem5 34696	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℝ)
52	51	recnd 11003	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℂ)
53	32, 33, 39, 3, 14	knoppndvlem5 34696	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℝ)
54	53	recnd 11003	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℂ)
55	52, 54	subcld 11332	. . . . . 6 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) ∈ ℂ)
56	55	abscld 15148	. . . . 5 ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ∈ ℝ)
57	50, 56	resubcld 11403	. . . 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 11332	. . . . 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 15148	. . . 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 512	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ))
61		remulcl 10956	. . . . . . . 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 17	. . . . . . 7 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
63	12, 62	jca 512	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ))
64		resubcl 11285	. . . . . 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 17	. . . . 5 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) − ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ∈ ℝ)
66	12	recnd 11003	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℂ)
67		1cnd 10970	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
68	29	recnd 11003	. . . . . . 7 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℂ)
69	66, 67, 68	subdid 11431	. . . . . 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	mulid1d 10992	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · 1) = (((abs‘𝐶)↑𝐽) / 2))
71	70	oveq1d 7290	. . . . . . 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 11541	. . . . . . 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 5096	. . . . . 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 5096	. . . . 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 11005	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
76	12	leidd 11541	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (((abs‘𝐶)↑𝐽) / 2))
77	41, 48	abssubd 15165	. . . . . . . . 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 34701	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
79	77, 78	eqtrd 2778	. . . . . . . 8 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
80	79	eqcomd 2744	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
81	76, 80	breqtrd 5100	. . . . . 6 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
82	32, 33, 34, 42, 1, 6, 37, 14, 22	knoppndvlem14 34705	. . . . . 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 11595	. . . . 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 11132	. . . 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 15169	. . . 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 11132	. . 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 15165	. . 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 5100	. 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 34697	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐵) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖))
91		elnn0uz 12623	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ₀ ↔ 𝐽 ∈ (ℤ_≥‘0))
92	6, 91	sylib 217	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘0))
93	14	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑁 ∈ ℕ)
94	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐶 ∈ ℝ)
95	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐵 ∈ ℝ)
96		elfznn0 13349	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝐽) → 𝑖 ∈ ℕ₀)
97	96	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑖 ∈ ℕ₀)
98	32, 33, 93, 94, 95, 97	knoppcnlem3 34675	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℝ)
99	98	recnd 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℂ)
100		fveq2 6774	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐵)‘𝑖) = ((𝐹‘𝐵)‘𝐽))
101	92, 99, 100	fsumm1 15463	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
102	90, 101	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐵) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
103	32, 33, 89, 34, 1, 6, 37, 14	knoppndvlem6 34697	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
104	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐴 ∈ ℝ)
105	32, 33, 93, 94, 104, 97	knoppcnlem3 34675	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ)
106	105	recnd 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℂ)
107		fveq2 6774	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐴)‘𝑖) = ((𝐹‘𝐴)‘𝐽))
108	92, 106, 107	fsumm1 15463	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
109	103, 108	eqtrd 2778	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐴) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
110	102, 109	oveq12d 7293	. . . . 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 11380	. . . . . 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 2744	. . . . 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 2778	. . . 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 6778	. . 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 2744	. 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 5100	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 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ↦ cmpt 5157 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 / cdiv 11632 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 (,)cioo 13079 ...cfz 13239 ⌊cfl 13510 ↑cexp 13782 abscabs 14945 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ioo 13083 df-ico 13085 df-fz 13240 df-fzo 13383 df-fl 13512 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-dvds 15964 df-ulm 25536

This theorem is referenced by: knoppndvlem17 34708

W3C validator