knoppndvlem15 - Mathbox for Asger C. Ipsen

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Theorem knoppndvlem15 37488

Description: Lemma for knoppndv 37496. (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 37476	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1))
3	2	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ ℝ)
4	3	recnd 11344	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℂ)
5	4	abscld 15615	. . . . . . 7 ⊢ (𝜑 → (abs‘𝐶) ∈ ℝ)
6		knoppndvlem15.j	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ ℕ₀)
7	5, 6	reexpcld 14313	. . . . . 6 ⊢ (𝜑 → ((abs‘𝐶)↑𝐽) ∈ ℝ)
8		2re 12424	. . . . . . 7 ⊢ 2 ∈ ℝ
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ∈ ℝ)
10		2ne0 12456	. . . . . . 7 ⊢ 2 ≠ 0
11	10	a1i 11	. . . . . 6 ⊢ (𝜑 → 2 ≠ 0)
12	7, 9, 11	redivcld 12152	. . . . 5 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℝ)
13		1red 11316	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ)
14		knoppndvlem15.n	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ)
15	14	nnred 12357	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℝ)
16	9, 15	remulcld 11346	. . . . . . . . 9 ⊢ (𝜑 → (2 · 𝑁) ∈ ℝ)
17	16, 5	remulcld 11346	. . . . . . . 8 ⊢ (𝜑 → ((2 · 𝑁) · (abs‘𝐶)) ∈ ℝ)
18	17, 13	resubcld 11751	. . . . . . 7 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ)
19		0red 11318	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℝ)
20		0lt1 11845	. . . . . . . . . . 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 37485	. . . . . . . . . . 11 ⊢ (𝜑 → (((2 · 𝑁) · (abs‘𝐶)) ≠ 1 ∧ 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
24	23	simprd 501	. . . . . . . . . 10 ⊢ (𝜑 → 1 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
25	19, 13, 18, 21, 24	lttrd 11478	. . . . . . . . 9 ⊢ (𝜑 → 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1))
26	18, 25	jca 521	. . . . . . . 8 ⊢ (𝜑 → ((((2 · 𝑁) · (abs‘𝐶)) − 1) ∈ ℝ ∧ 0 < (((2 · 𝑁) · (abs‘𝐶)) − 1)))
27		gt0ne0 11788	. . . . . . . 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 12152	. . . . . 6 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ)
30	13, 29	resubcld 11751	. . . . 5 ⊢ (𝜑 → (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
31	12, 30	remulcld 11346	. . . 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 12725	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ℤ)
37		knoppndvlem15.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
38	14, 36, 37	knoppndvlem1 37474	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑ -𝐽) / 2) · 𝑀) ∈ ℝ)
39	35, 38	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
40	32, 33, 14, 3, 39, 6	knoppcnlem3 37457	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) ∈ ℝ)
41	40	recnd 11344	. . . . . . 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 12813	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
45	14, 36, 44	knoppndvlem1 37474	. . . . . . . . . 10 ⊢ (𝜑 → ((((2 · 𝑁)↑ -𝐽) / 2) · (𝑀 + 1)) ∈ ℝ)
46	43, 45	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ)
47	32, 33, 14, 3, 46, 6	knoppcnlem3 37457	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℝ)
48	47	recnd 11344	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)‘𝐽) ∈ ℂ)
49	41, 48	subcld 11676	. . . . . 6 ⊢ (𝜑 → (((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽)) ∈ ℂ)
50	49	abscld 15615	. . . . 5 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) ∈ ℝ)
51	32, 33, 46, 3, 14	knoppndvlem5 37478	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℝ)
52	51	recnd 11344	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) ∈ ℂ)
53	32, 33, 39, 3, 14	knoppndvlem5 37478	. . . . . . . 8 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℝ)
54	53	recnd 11344	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) ∈ ℂ)
55	52, 54	subcld 11676	. . . . . 6 ⊢ (𝜑 → (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖)) ∈ ℂ)
56	55	abscld 15615	. . . . 5 ⊢ (𝜑 → (abs‘(Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) − Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖))) ∈ ℝ)
57	50, 56	resubcld 11751	. . . 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 11676	. . . . 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 15615	. . . 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 521	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℝ))
61		remulcl 11292	. . . . . . . 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 521	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) ∈ ℝ ∧ ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ))
64		resubcl 11629	. . . . . 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 11344	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ∈ ℂ)
67		1cnd 11309	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
68	29	recnd 11344	. . . . . . 7 ⊢ (𝜑 → (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)) ∈ ℂ)
69	66, 67, 68	subdid 11779	. . . . . 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 11333	. . . . . . . 8 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · 1) = (((abs‘𝐶)↑𝐽) / 2))
71	70	oveq1d 7439	. . . . . . 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 11889	. . . . . . 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 5126	. . . . . 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 5126	. . . . 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 11346	. . . . . 6 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1))) ∈ ℝ)
76	12	leidd 11889	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (((abs‘𝐶)↑𝐽) / 2))
77	41, 48	abssubd 15632	. . . . . . . . 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 37483	. . . . . . . . 9 ⊢ (𝜑 → (abs‘(((𝐹‘𝐵)‘𝐽) − ((𝐹‘𝐴)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
79	77, 78	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))) = (((abs‘𝐶)↑𝐽) / 2))
80	79	eqcomd 2766	. . . . . . 7 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) = (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
81	76, 80	breqtrd 5130	. . . . . 6 ⊢ (𝜑 → (((abs‘𝐶)↑𝐽) / 2) ≤ (abs‘(((𝐹‘𝐴)‘𝐽) − ((𝐹‘𝐵)‘𝐽))))
82	32, 33, 34, 42, 1, 6, 37, 14, 22	knoppndvlem14 37487	. . . . . 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 11943	. . . . 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 11474	. . . 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 15636	. . . 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 11474	. . 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 15632	. . 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 5130	. 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 37479	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐵) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖))
91		elnn0uz 13013	. . . . . . . . 9 ⊢ (𝐽 ∈ ℕ₀ ↔ 𝐽 ∈ (ℤ_≥‘0))
92	6, 91	sylib 221	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (ℤ_≥‘0))
93	14	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑁 ∈ ℕ)
94	3	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐶 ∈ ℝ)
95	46	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐵 ∈ ℝ)
96		elfznn0 13761	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0...𝐽) → 𝑖 ∈ ℕ₀)
97	96	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝑖 ∈ ℕ₀)
98	32, 33, 93, 94, 95, 97	knoppcnlem3 37457	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℝ)
99	98	recnd 11344	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐵)‘𝑖) ∈ ℂ)
100		fveq2 6890	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐵)‘𝑖) = ((𝐹‘𝐵)‘𝐽))
101	92, 99, 100	fsumm1 15926	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐵)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
102	90, 101	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐵) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐵)‘𝑖) + ((𝐹‘𝐵)‘𝐽)))
103	32, 33, 89, 34, 1, 6, 37, 14	knoppndvlem6 37479	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐴) = Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖))
104	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → 𝐴 ∈ ℝ)
105	32, 33, 93, 94, 104, 97	knoppcnlem3 37457	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℝ)
106	105	recnd 11344	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐽)) → ((𝐹‘𝐴)‘𝑖) ∈ ℂ)
107		fveq2 6890	. . . . . . . 8 ⊢ (𝑖 = 𝐽 → ((𝐹‘𝐴)‘𝑖) = ((𝐹‘𝐴)‘𝐽))
108	92, 106, 107	fsumm1 15926	. . . . . . 7 ⊢ (𝜑 → Σ𝑖 ∈ (0...𝐽)((𝐹‘𝐴)‘𝑖) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
109	103, 108	eqtrd 2795	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐴) = (Σ𝑖 ∈ (0...(𝐽 − 1))((𝐹‘𝐴)‘𝑖) + ((𝐹‘𝐴)‘𝐽)))
110	102, 109	oveq12d 7442	. . . . 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 11724	. . . . . 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 2766	. . . . 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 2795	. . . 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 6894	. . 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 2766	. 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 5130	1 ⊢ (𝜑 → ((((abs‘𝐶)↑𝐽) / 2) · (1 − (1 / (((2 · 𝑁) · (abs‘𝐶)) − 1)))) ≤ (abs‘((𝑊‘𝐵) − (𝑊‘𝐴))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 class class class wbr 5102 ↦ cmpt 5185 ‘cfv 6543 (class class class)co 7424 ℝcr 11206 0cc0 11207 1c1 11208 + caddc 11210 · cmul 11212 < clt 11350 ≤ cle 11351 − cmin 11548 -cneg 11549 / cdiv 11980 ℕcn 12342 2c2 12404 ℕ₀cn0 12613 ℤcz 12700 ℤ_≥cuz 12972 (,)cioo 13483 ...cfz 13646 ⌊cfl 13937 ↑cexp 14211 abscabs 15410 Σcsu 15862

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8306 df-wrecs 8337 df-recs 8386 df-rdg 8425 df-1o 8483 df-er 8724 df-map 8856 df-pm 8857 df-en 8981 df-dom 8982 df-sdom 8983 df-fin 8984 df-sup 9441 df-inf 9442 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-n0 12614 df-z 12701 df-uz 12973 df-rp 13128 df-ioo 13487 df-ico 13489 df-fz 13647 df-fzo 13796 df-fl 13939 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-limsup 15647 df-clim 15664 df-rlim 15665 df-sum 15863 df-dvds 16432 df-ulm 26713

This theorem is used by: knoppndvlem17 37490

W3C validator