dirkeritg - Mathbox for Glauco Siliprandi

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Theorem dirkeritg 41246

Description: The definite integral of the Dirichlet Kernel. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
dirkeritg.d	⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2)))))))
dirkeritg.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
dirkeritg.f	⊢ 𝐹 = (𝐷‘𝑁)
dirkeritg.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
dirkeritg.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
dirkeritg.aleb	⊢ (𝜑 → 𝐴 ≤ 𝐵)
dirkeritg.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π))

**Assertion**
Ref	Expression
dirkeritg	⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))

Distinct variable groups: 𝐴,𝑘,𝑥 𝐵,𝑘,𝑥 𝑥,𝐹 𝑘,𝑁,𝑥 𝜑,𝑘 𝑥,𝑛 Allowed substitution hints: 𝜑(𝑥,𝑛) 𝐴(𝑛) 𝐵(𝑛) 𝐷(𝑥,𝑘,𝑛) 𝐹(𝑘,𝑛) 𝐺(𝑥,𝑘,𝑛) 𝑁(𝑛)

Proof of Theorem dirkeritg Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6446	. . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
2	1	cbvitgv 23980	. . 3 ⊢ ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠
3	2	a1i 11	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠)
4		elioore 12517	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
5	4	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
6		halfre 11596	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (1 / 2) ∈ ℝ)
8		fzfid 13091	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → (1...𝑁) ∈ Fin)
9		elfzelz 12659	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℤ)
10	9	zred 11834	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℝ)
11	10	adantl 475	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
12		simpl 476	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
13	11, 12	remulcld 10407	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
14	13	recoscld 15276	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
15	8, 14	fsumrecl 14872	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
16	7, 15	readdcld 10406	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
17		pire 24648	. . . . . . . 8 ⊢ π ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ∈ ℝ)
19		pipos 24650	. . . . . . . . 9 ⊢ 0 < π
20	17, 19	gt0ne0ii 10911	. . . . . . . 8 ⊢ π ≠ 0
21	20	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ≠ 0)
22	16, 18, 21	redivcld 11203	. . . . . 6 ⊢ (𝑠 ∈ ℝ → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
23	5, 22	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
24		eqid 2778	. . . . . 6 ⊢ (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
25	24	fvmpt2 6552	. . . . 5 ⊢ ((𝑠 ∈ ℝ ∧ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ) → ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
26	5, 23, 25	syl2anc 579	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
27		dirkeritg.d	. . . . . . . 8 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2)))))))
28		oveq1 6929	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝑥 mod (2 · π)) = (𝑠 mod (2 · π)))
29	28	eqeq1d 2780	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝑥 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
30		oveq2 6930	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑛 + (1 / 2)) · 𝑥) = ((𝑛 + (1 / 2)) · 𝑠))
31	30	fveq2d 6450	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (sin‘((𝑛 + (1 / 2)) · 𝑥)) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
32		oveq1 6929	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2))
33	32	fveq2d 6450	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2)))
34	33	oveq2d 6938	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((2 · π) · (sin‘(𝑥 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
35	31, 34	oveq12d 6940	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2)))) = ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))
36	29, 35	ifbieq2d 4332	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2))))) = if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
37	36	cbvmptv 4985	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ ↦ if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2)))))) = (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2))))))
38	37	mpteq2i 4976	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝑥 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑥)) / ((2 · π) · (sin‘(𝑥 / 2))))))) = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
39	27, 38	eqtri 2802	. . . . . . 7 ⊢ 𝐷 = (𝑛 ∈ ℕ ↦ (𝑠 ∈ ℝ ↦ if((𝑠 mod (2 · π)) = 0, (((2 · 𝑛) + 1) / (2 · π)), ((sin‘((𝑛 + (1 / 2)) · 𝑠)) / ((2 · π) · (sin‘(𝑠 / 2)))))))
40		dirkeritg.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
41		dirkeritg.f	. . . . . . 7 ⊢ 𝐹 = (𝐷‘𝑁)
42	39, 40, 41, 24	dirkertrigeq 41245	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
43	42	fveq1d 6448	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑠) = ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠))
44	43	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠))
45		dirkeritg.g	. . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π))
46		oveq2 6930	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑘 · 𝑥) = (𝑘 · 𝑠))
47	46	fveq2d 6450	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑘 · 𝑥)) = (sin‘(𝑘 · 𝑠)))
48	47	oveq1d 6937	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((sin‘(𝑘 · 𝑥)) / 𝑘) = ((sin‘(𝑘 · 𝑠)) / 𝑘))
49	48	sumeq2sdv 14842	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘) = Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))
50	32, 49	oveq12d 6940	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) = ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))
51	50	oveq1d 6937	. . . . . . . . 9 ⊢ (𝑥 = 𝑠 → (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π) = (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
52	51	cbvmptv 4985	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) / π)) = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
53	45, 52	eqtri 2802	. . . . . . 7 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
54	53	oveq2i 6933	. . . . . 6 ⊢ (ℝ D 𝐺) = (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)))
55		reelprrecn 10364	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		recn 10362	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
58	57	halfcld 11627	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℂ)
59	9	zcnd 11835	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℂ)
60	59	adantl 475	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
61	57	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℂ)
62	60, 61	mulcld 10397	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℂ)
63	62	sincld 15262	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
64		0red 10380	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 ∈ ℝ)
65		1red 10377	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℝ)
66		0lt1 10897	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 1)
68		elfzle1 12661	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ≤ 𝑘)
69	64, 65, 10, 67, 68	ltletrd 10536	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 𝑘)
70	69	gt0ne0d 10939	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
71	70	adantl 475	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
72	63, 60, 71	divcld 11151	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
73	8, 72	fsumcl 14871	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
74	58, 73	addcld 10396	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
75		picn 24649	. . . . . . . . . 10 ⊢ π ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → π ∈ ℂ)
77	74, 76, 21	divcld 11151	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
78	77	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
79	22	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
80	74	adantl 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
81	16	adantl 475	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
82	58	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑠 / 2) ∈ ℂ)
83	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (1 / 2) ∈ ℝ)
84	57	adantl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
85		1red 10377	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
86	56	dvmptid 24157	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
87		2cnd 11453	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
88		2ne0 11486	. . . . . . . . . . 11 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
90	56, 84, 85, 86, 87, 89	dvmptdivc 24165	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ (𝑠 / 2))) = (𝑠 ∈ ℝ ↦ (1 / 2)))
91	73	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
92	15	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
93		eqid 2778	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
94	93	tgioo2 23014	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
95		reopn 40411	. . . . . . . . . . 11 ⊢ ℝ ∈ (topGen‘ran (,))
96	95	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
97		fzfid 13091	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
98	72	ancoms 452	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
99	98	3adant1 1121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
100	14	ancoms 452	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
101	100	recnd 10405	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
102	101	3adant1 1121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
103	55	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ∈ {ℝ, ℂ})
104	63	ancoms 452	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
105	59	adantr 474	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑘 ∈ ℂ)
106		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
107	105, 106	mulcld 10397	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · 𝑠) ∈ ℂ)
108	107	coscld 15263	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
109	105, 108	mulcld 10397	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
110	57, 109	sylan2 586	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
111		ax-resscn 10329	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
112		resmpt 5699	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ ⊆ ℂ → ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))))
113	111, 112	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))))
114	113	eqcomd 2784	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))) = ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ))
115	114	oveq2d 6938	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)))
116	107	sincld 15262	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
117	116	fmpttd 6649	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ)
118	109	ralrimiva 3148	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝑁) → ∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
119		dmmptg 5886	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ → dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) = ℂ)
120	118, 119	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑁) → dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) = ℂ)
121	111, 120	syl5sseqr 3873	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
122		dvsinax 41055	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ ℂ → (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
123	59, 122	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1...𝑁) → (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
124	123	dmeqd 5571	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
125	121, 124	sseqtr4d 3861	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))))
126		dvcnre 41058	. . . . . . . . . . . . . . 15 ⊢ (((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ ∧ ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))))) → (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)) = ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ))
127	117, 125, 126	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)) = ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ))
128	123	reseq1d 5641	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ))
129		resmpt 5699	. . . . . . . . . . . . . . . 16 ⊢ (ℝ ⊆ ℂ → ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
130	111, 129	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠))))
131	128, 130	syl6eq 2830	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
132	115, 127, 131	3eqtrd 2818	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
133	103, 104, 110, 132, 59, 70	dvmptdivc 24165	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)))
134	59	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ∈ ℂ)
135	70	adantr 474	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ≠ 0)
136	101, 134, 135	divcan3d 11156	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘) = (cos‘(𝑘 · 𝑠)))
137	136	mpteq2dva 4979	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
138	133, 137	eqtrd 2814	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
139	138	adantl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
140	94, 93, 56, 96, 97, 99, 102, 139	dvmptfsum 24175	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))))
141	56, 82, 83, 90, 91, 92, 140	dvmptadd 24160	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))) = (𝑠 ∈ ℝ ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))))
142	75	a1i 11	. . . . . . . 8 ⊢ (𝜑 → π ∈ ℂ)
143	20	a1i 11	. . . . . . . 8 ⊢ (𝜑 → π ≠ 0)
144	56, 80, 81, 141, 142, 143	dvmptdivc 24165	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))) = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
145		dirkeritg.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ)
146		dirkeritg.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ ℝ)
147	145, 146	iccssred 40639	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
148		iccntr 23032	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
149	145, 146, 148	syl2anc 579	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
150	56, 78, 79, 144, 147, 94, 93, 149	dvmptres2 24162	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
151	54, 150	syl5eq 2826	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
152	151, 23	fvmpt2d 6554	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
153	26, 44, 152	3eqtr4d 2824	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((ℝ D 𝐺)‘𝑠))
154	153	itgeq2dv 23985	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠)
155		dirkeritg.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
156		ioosscn 40628	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
157	156	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
158		halfcn 11597	. . . . . . . 8 ⊢ (1 / 2) ∈ ℂ
159	158	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
160		ssid 3842	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
162	157, 159, 161	constcncfg 41012	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163		eqid 2778	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠)))
164		coscn 24636	. . . . . . . . . . 11 ⊢ cos ∈ (ℂ–cn→ℂ)
165	164	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → cos ∈ (ℂ–cn→ℂ))
166		eqid 2778	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠))
167	166	mulc1cncf 23116	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℂ → (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) ∈ (ℂ–cn→ℂ))
168	59, 167	syl 17	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) ∈ (ℂ–cn→ℂ))
169	165, 168	cncfmpt1f 23124	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
170	156	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → (𝐴(,)𝐵) ⊆ ℂ)
171	160	a1i 11	. . . . . . . . 9 ⊢ (𝑘 ∈ (1...𝑁) → ℂ ⊆ ℂ)
172	4, 101	sylan2 586	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
173	163, 169, 170, 171, 172	cncfmptssg 41011	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
174	173	adantl 475	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
175	157, 97, 174	fsumcncf 41019	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
176	162, 175	addcncf 41014	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
177		eqid 2778	. . . . . 6 ⊢ (𝑠 ∈ ℂ ↦ π) = (𝑠 ∈ ℂ ↦ π)
178		cncfmptc 23122	. . . . . . . 8 ⊢ ((π ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
179	75, 160, 160, 178	mp3an 1534	. . . . . . 7 ⊢ (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ)
180	179	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
181		difssd 3961	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
182		eldifsn 4550	. . . . . . . 8 ⊢ (π ∈ (ℂ ∖ {0}) ↔ (π ∈ ℂ ∧ π ≠ 0))
183	75, 20, 182	mpbir2an 701	. . . . . . 7 ⊢ π ∈ (ℂ ∖ {0})
184	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → π ∈ (ℂ ∖ {0}))
185	177, 180, 157, 181, 184	cncfmptssg 41011	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0})))
186	176, 185	divcncf 23651	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
187	151, 186	eqeltrd 2859	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
188		ioossicc 12571	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
189	188	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
190		ioombl 23769	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ dom vol
191	190	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ dom vol)
192	6	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (1 / 2) ∈ ℝ)
193		fzfid 13091	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (1...𝑁) ∈ Fin)
194	10	adantl 475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
195	147	sselda 3821	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
196	195	adantr 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
197	194, 196	remulcld 10407	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
198	197	recoscld 15276	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
199	193, 198	fsumrecl 14872	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
200	192, 199	readdcld 10406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
201	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → π ∈ ℝ)
202	20	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → π ≠ 0)
203	200, 201, 202	redivcld 11203	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
204	147, 111	syl6ss 3833	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
205	204, 159, 161	constcncfg 41012	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (1 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
206		eqid 2778	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))
207	169	adantl 475	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
208	161, 97, 207	fsumcncf 41019	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
209	199	recnd 10405	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℂ)
210	206, 208, 204, 161, 209	cncfmptssg 41011	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
211	205, 210	addcncf 41014	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
212	183	a1i 11	. . . . . . . 8 ⊢ (𝜑 → π ∈ (ℂ ∖ {0}))
213	204, 212, 181	constcncfg 41012	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ π) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
214	211, 213	divcncf 23651	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
215		cniccibl 24044	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
216	145, 146, 214, 215	syl3anc 1439	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
217	189, 191, 203, 216	iblss 24008	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
218	151, 217	eqeltrd 2859	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
219	204, 161	idcncfg 41013	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑠) ∈ ((𝐴[,]𝐵)–cn→ℂ))
220		2cn 11450	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
221		eldifsn 4550	. . . . . . . . . 10 ⊢ (2 ∈ (ℂ ∖ {0}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0))
222	220, 88, 221	mpbir2an 701	. . . . . . . . 9 ⊢ 2 ∈ (ℂ ∖ {0})
223	222	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0}))
224	204, 223, 181	constcncfg 41012	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
225	219, 224	divcncf 23651	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (𝑠 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
226		eqid 2778	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))
227		sincn 24635	. . . . . . . . . . . 12 ⊢ sin ∈ (ℂ–cn→ℂ)
228	227	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → sin ∈ (ℂ–cn→ℂ))
229	228, 168	cncfmpt1f 23124	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
230	229	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
231	204	adantr 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝐴[,]𝐵) ⊆ ℂ)
232	160	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → ℂ ⊆ ℂ)
233	59	ad2antlr 717	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑘 ∈ ℂ)
234	195	recnd 10405	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
235	234	adantlr 705	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
236	233, 235	mulcld 10397	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑘 · 𝑠) ∈ ℂ)
237	236	sincld 15262	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
238	226, 230, 231, 232, 237	cncfmptssg 41011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
239		eldifsn 4550	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℂ ∖ {0}) ↔ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
240	59, 70, 239	sylanbrc 578	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ (ℂ ∖ {0}))
241	240	adantl 475	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (ℂ ∖ {0}))
242		difssd 3961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℂ ∖ {0}) ⊆ ℂ)
243	231, 241, 242	constcncfg 41012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑘) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
244	238, 243	divcncf 23651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
245	204, 97, 244	fsumcncf 41019	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
246	225, 245	addcncf 41014	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
247	246, 213	divcncf 23651	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
248	53, 247	syl5eqel 2863	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
249	145, 146, 155, 187, 218, 248	ftc2 24244	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
250	3, 154, 249	3eqtrd 2818	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∀wral 3090 ∖ cdif 3789 ⊆ wss 3792 ifcif 4307 {csn 4398 {cpr 4400 class class class wbr 4886 ↦ cmpt 4965 dom cdm 5355 ran crn 5356 ↾ cres 5357 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ℂcc 10270 ℝcr 10271 0cc0 10272 1c1 10273 + caddc 10275 · cmul 10277 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 ℕcn 11374 2c2 11430 (,)cioo 12487 [,]cicc 12490 ...cfz 12643 mod cmo 12987 Σcsu 14824 sincsin 15196 cosccos 15197 πcpi 15199 TopOpenctopn 16468 topGenctg 16484 ℂ_fldccnfld 20142 intcnt 21229 –cn→ccncf 23087 volcvol 23667 𝐿¹cibl 23821 ∫citg 23822 D cdv 24064

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-symdif 4067 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-disj 4855 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-ofr 7175 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ioc 12492 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-mod 12988 df-seq 13120 df-exp 13179 df-fac 13379 df-bc 13408 df-hash 13436 df-shft 14214 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-limsup 14610 df-clim 14627 df-rlim 14628 df-sum 14825 df-ef 15200 df-sin 15202 df-cos 15203 df-pi 15205 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cn 21439 df-cnp 21440 df-haus 21527 df-cmp 21599 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cncf 23089 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 df-itg2 23825 df-ibl 23826 df-itg 23827 df-0p 23874 df-limc 24067 df-dv 24068

This theorem is referenced by: fourierdlem103 41353 fourierdlem104 41354

W3C validator