knoppndvlem15 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem15 36492

Description: Lemma for knoppndv 36500. (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 36480	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
3	2	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11318	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5	4	abscld 15485	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
6		knoppndvlem15.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	5, 6	reexpcld 14213	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶)↑𝐽) ∈ ℝ)
8		2re 12367	. . . . . . 7 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
10		2ne0 12397	. . . . . . 7 ⊢ 2 ≠ 0
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
12	7, 9, 11	redivcld 12122	. . . . 5 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℝ)
13		1red 11291	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
14		knoppndvlem15.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
15	14	nnred 12308	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
16	9, 15	remulcld 11320	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
17	16, 5	remulcld 11320	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ)
18	17, 13	resubcld 11718	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ)
19		0red 11293	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
20		0lt1 11812	. . . . . . . . . . 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 36489	. . . . . . . . . . 11 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
24	23	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
25	19, 13, 18, 21, 24	lttrd 11451	. . . . . . . . 9 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
26	18, 25	jca 511	. . . . . . . 8 ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
27		gt0ne0 11755	. . . . . . . 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 12122	. . . . . 6 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ)
30	13, 29	resubcld 11718	. . . . 5 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
31	12, 30	remulcld 11320	. . . 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 12665	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
37		knoppndvlem15.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38	14, 36, 37	knoppndvlem1 36478	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) ∈ ℝ)
39	35, 38	eqeltrd 2844	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
40	32, 33, 14, 3, 39, 6	knoppcnlem3 36461	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
41	40	recnd 11318	. . . . . . 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 12750	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
45	14, 36, 44	knoppndvlem1 36478	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑-𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
46	43, 45	eqeltrd 2844	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
47	32, 33, 14, 3, 46, 6	knoppcnlem3 36461	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
48	47	recnd 11318	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
49	41, 48	subcld 11647	. . . . . 6 ⊢ (𝜑 → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) ∈ ℂ)
50	49	abscld 15485	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) ∈ ℝ)
51	32, 33, 46, 3, 14	knoppndvlem5 36482	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℝ)
52	51	recnd 11318	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℂ)
53	32, 33, 39, 3, 14	knoppndvlem5 36482	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℝ)
54	53	recnd 11318	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℂ)
55	52, 54	subcld 11647	. . . . . 6 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) ∈ ℂ)
56	55	abscld 15485	. . . . 5 ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ∈ ℝ)
57	50, 56	resubcld 11718	. . . 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 11647	. . . . 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 15485	. . . 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 511	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ))
61		remulcl 11269	. . . . . . . 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 511	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ))
64		resubcl 11600	. . . . . 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 11318	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℂ)
67		1cnd 11285	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
68	29	recnd 11318	. . . . . . 7 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℂ)
69	66, 67, 68	subdid 11746	. . . . . 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 11307	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · 1) = (((abs‘𝐶)↑𝐽) / 2))
71	70	oveq1d 7463	. . . . . . 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 11856	. . . . . . 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 5188	. . . . . 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 5188	. . . . 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 11320	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
76	12	leidd 11856	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (((abs‘𝐶)↑𝐽) / 2))
77	41, 48	abssubd 15502	. . . . . . . . 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 36487	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
79	77, 78	eqtrd 2780	. . . . . . . 8 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
80	79	eqcomd 2746	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
81	76, 80	breqtrd 5192	. . . . . 6 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
82	32, 33, 34, 42, 1, 6, 37, 14, 22	knoppndvlem14 36491	. . . . . 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 11910	. . . . 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 11447	. . . 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 15506	. . . 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 11447	. . 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 15502	. . 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 5192	. 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 36483	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐵) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖))
91		elnn0uz 12948	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ₀ ↔ 𝐽 ∈ (ℤ_≥‘0))
92	6, 91	sylib 218	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘0))
93	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑁 ∈ ℕ)
94	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐶 ∈ ℝ)
95	46	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐵 ∈ ℝ)
96		elfznn0 13677	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝐽) → 𝑖 ∈ ℕ₀)
97	96	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑖 ∈ ℕ₀)
98	32, 33, 93, 94, 95, 97	knoppcnlem3 36461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℝ)
99	98	recnd 11318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℂ)
100		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐵)‘𝑖) = ((𝐹‘𝐵)‘𝐽))
101	92, 99, 100	fsumm1 15799	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
102	90, 101	eqtrd 2780	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐵) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
103	32, 33, 89, 34, 1, 6, 37, 14	knoppndvlem6 36483	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
104	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐴 ∈ ℝ)
105	32, 33, 93, 94, 104, 97	knoppcnlem3 36461	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ)
106	105	recnd 11318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℂ)
107		fveq2 6920	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐴)‘𝑖) = ((𝐹‘𝐴)‘𝐽))
108	92, 106, 107	fsumm1 15799	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
109	103, 108	eqtrd 2780	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐴) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
110	102, 109	oveq12d 7466	. . . . 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 11695	. . . . . 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 2746	. . . . 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 2780	. . . 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 6924	. . 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 2746	. 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 5192	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 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 class class class wbr 5166 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 · cmul 11189 < clt 11324 ≤ cle 11325 − cmin 11520 -cneg 11521 / cdiv 11947 ℕcn 12293 2c2 12348 ℕ₀cn0 12553 ℤcz 12639 ℤ_≥cuz 12903 (,)cioo 13407 ...cfz 13567 ⌊cfl 13841 ↑cexp 14112 abscabs 15283 Σcsu 15734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-ioo 13411 df-ico 13413 df-fz 13568 df-fzo 13712 df-fl 13843 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-dvds 16303 df-ulm 26438

This theorem is referenced by: knoppndvlem17 36494

W3C validator