dirkeritg - Mathbox for Glauco Siliprandi

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Theorem dirkeritg 42744

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 6645	. . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
2	1	cbvitgv 24380	. . 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 12756	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
5	4	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
6		halfre 11839	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (1 / 2) ∈ ℝ)
8		fzfid 13336	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → (1...𝑁) ∈ Fin)
9		elfzelz 12902	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℤ)
10	9	zred 12075	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℝ)
11	10	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
12		simpl 486	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
13	11, 12	remulcld 10660	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
14	13	recoscld 15489	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
15	8, 14	fsumrecl 15083	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
16	7, 15	readdcld 10659	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
17		pire 25051	. . . . . . . 8 ⊢ π ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ∈ ℝ)
19		pipos 25053	. . . . . . . . 9 ⊢ 0 < π
20	17, 19	gt0ne0ii 11165	. . . . . . . 8 ⊢ π ≠ 0
21	20	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ≠ 0)
22	16, 18, 21	redivcld 11457	. . . . . 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 2798	. . . . . 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 6756	. . . . 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 587	. . . 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 7142	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝑥 mod (2 · π)) = (𝑠 mod (2 · π)))
29	28	eqeq1d 2800	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝑥 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
30		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑛 + (1 / 2)) · 𝑥) = ((𝑛 + (1 / 2)) · 𝑠))
31	30	fveq2d 6649	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (sin‘((𝑛 + (1 / 2)) · 𝑥)) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
32		oveq1 7142	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2))
33	32	fveq2d 6649	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2)))
34	33	oveq2d 7151	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((2 · π) · (sin‘(𝑥 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
35	31, 34	oveq12d 7153	. . . . . . . . . . 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 4450	. . . . . . . . . 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 5133	. . . . . . . . 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 5122	. . . . . . . 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 2821	. . . . . . 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 42743	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
43	42	fveq1d 6647	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑠) = ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠))
44	43	adantr 484	. . . 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 7143	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑘 · 𝑥) = (𝑘 · 𝑠))
47	46	fveq2d 6649	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑘 · 𝑥)) = (sin‘(𝑘 · 𝑠)))
48	47	oveq1d 7150	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((sin‘(𝑘 · 𝑥)) / 𝑘) = ((sin‘(𝑘 · 𝑠)) / 𝑘))
49	48	sumeq2sdv 15053	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘) = Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))
50	32, 49	oveq12d 7153	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) = ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))
51	50	oveq1d 7150	. . . . . . . . 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 5133	. . . . . . . 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 2821	. . . . . . 7 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
54	53	oveq2i 7146	. . . . . 6 ⊢ (ℝ D 𝐺) = (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)))
55		reelprrecn 10618	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		recn 10616	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
58	57	halfcld 11870	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℂ)
59	9	zcnd 12076	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℂ)
60	59	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
61	57	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℂ)
62	60, 61	mulcld 10650	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℂ)
63	62	sincld 15475	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
64		0red 10633	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 ∈ ℝ)
65		1red 10631	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℝ)
66		0lt1 11151	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 1)
68		elfzle1 12905	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ≤ 𝑘)
69	64, 65, 10, 67, 68	ltletrd 10789	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 𝑘)
70	69	gt0ne0d 11193	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
71	70	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
72	63, 60, 71	divcld 11405	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
73	8, 72	fsumcl 15082	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
74	58, 73	addcld 10649	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
75		picn 25052	. . . . . . . . . 10 ⊢ π ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → π ∈ ℂ)
77	74, 76, 21	divcld 11405	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
78	77	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
79	22	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
80	74	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
81	16	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
82	58	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑠 / 2) ∈ ℂ)
83	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (1 / 2) ∈ ℝ)
84	57	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
85		1red 10631	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
86	56	dvmptid 24560	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
87		2cnd 11703	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
88		2ne0 11729	. . . . . . . . . . 11 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
90	56, 84, 85, 86, 87, 89	dvmptdivc 24568	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ (𝑠 / 2))) = (𝑠 ∈ ℝ ↦ (1 / 2)))
91	73	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
92	15	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
93		eqid 2798	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
94	93	tgioo2 23408	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
95		reopn 41920	. . . . . . . . . . 11 ⊢ ℝ ∈ (topGen‘ran (,))
96	95	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
97		fzfid 13336	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
98	72	ancoms 462	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
99	98	3adant1 1127	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
100	14	ancoms 462	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
101	100	recnd 10658	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
102	101	3adant1 1127	. . . . . . . . . 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 462	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
105	59	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑘 ∈ ℂ)
106		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
107	105, 106	mulcld 10650	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · 𝑠) ∈ ℂ)
108	107	coscld 15476	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
109	105, 108	mulcld 10650	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
110	57, 109	sylan2 595	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
111		ax-resscn 10583	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
112		resmpt 5872	. . . . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))) = ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ))
115	114	oveq2d 7151	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)))
116	107	sincld 15475	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
117	116	fmpttd 6856	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ)
118	109	ralrimiva 3149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝑁) → ∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
119		dmmptg 6063	. . . . . . . . . . . . . . . . . 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	sseqtrrid 3968	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
122		dvsinax 42555	. . . . . . . . . . . . . . . . . 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 5738	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
125	121, 124	sseqtrrd 3956	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))))
126		dvcnre 42558	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)) = ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ))
128	123	reseq1d 5817	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ))
129		resmpt 5872	. . . . . . . . . . . . . . . 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	eqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
132	115, 127, 131	3eqtrd 2837	. . . . . . . . . . . . 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 24568	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)))
134	59	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ∈ ℂ)
135	70	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ≠ 0)
136	101, 134, 135	divcan3d 11410	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘) = (cos‘(𝑘 · 𝑠)))
137	136	mpteq2dva 5125	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
138	133, 137	eqtrd 2833	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
139	138	adantl 485	. . . . . . . . . 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 24578	. . . . . . . . 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 24563	. . . . . . . 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 24568	. . . . . . 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 12812	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
148		iccntr 23426	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
149	145, 146, 148	syl2anc 587	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
150	56, 78, 79, 144, 147, 94, 93, 149	dvmptres2 24565	. . . . . 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 2845	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
152	151, 23	fvmpt2d 6758	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
153	26, 44, 152	3eqtr4d 2843	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((ℝ D 𝐺)‘𝑠))
154	153	itgeq2dv 24385	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠)
155		dirkeritg.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
156		ioosscn 12787	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
157	156	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
158		halfcn 11840	. . . . . . . 8 ⊢ (1 / 2) ∈ ℂ
159	158	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
160		ssid 3937	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
162	157, 159, 161	constcncfg 42514	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163		eqid 2798	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠)))
164		coscn 25040	. . . . . . . . . . 11 ⊢ cos ∈ (ℂ–cn→ℂ)
165	164	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → cos ∈ (ℂ–cn→ℂ))
166		eqid 2798	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠))
167	166	mulc1cncf 23510	. . . . . . . . . . 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 23519	. . . . . . . . 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 595	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
173	163, 169, 170, 171, 172	cncfmptssg 42513	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
174	173	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
175	157, 97, 174	fsumcncf 42520	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
176	162, 175	addcncf 24048	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
177		eqid 2798	. . . . . 6 ⊢ (𝑠 ∈ ℂ ↦ π) = (𝑠 ∈ ℂ ↦ π)
178		cncfmptc 23517	. . . . . . . 8 ⊢ ((π ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
179	75, 160, 160, 178	mp3an 1458	. . . . . . 7 ⊢ (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ)
180	179	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
181		difssd 4060	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
182		eldifsn 4680	. . . . . . . 8 ⊢ (π ∈ (ℂ ∖ {0}) ↔ (π ∈ ℂ ∧ π ≠ 0))
183	75, 20, 182	mpbir2an 710	. . . . . . 7 ⊢ π ∈ (ℂ ∖ {0})
184	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → π ∈ (ℂ ∖ {0}))
185	177, 180, 157, 181, 184	cncfmptssg 42513	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0})))
186	176, 185	divcncf 24051	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
187	151, 186	eqeltrd 2890	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
188		ioossicc 12811	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
189	188	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
190		ioombl 24169	. . . . . 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 13336	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (1...𝑁) ∈ Fin)
194	10	adantl 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
195	147	sselda 3915	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
196	195	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
197	194, 196	remulcld 10660	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
198	197	recoscld 15489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
199	193, 198	fsumrecl 15083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
200	192, 199	readdcld 10659	. . . . . 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 11457	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
204	147, 111	sstrdi 3927	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
205	204, 159, 161	constcncfg 42514	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (1 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
206		eqid 2798	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))
207	169	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
208	161, 97, 207	fsumcncf 42520	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
209	199	recnd 10658	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℂ)
210	206, 208, 204, 161, 209	cncfmptssg 42513	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
211	205, 210	addcncf 24048	. . . . . . 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 42514	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ π) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
214	211, 213	divcncf 24051	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
215		cniccibl 24444	. . . . . 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 1368	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
217	189, 191, 203, 216	iblss 24408	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
218	151, 217	eqeltrd 2890	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
219	204, 161	idcncfg 42515	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑠) ∈ ((𝐴[,]𝐵)–cn→ℂ))
220		2cn 11700	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
221		eldifsn 4680	. . . . . . . . . 10 ⊢ (2 ∈ (ℂ ∖ {0}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0))
222	220, 88, 221	mpbir2an 710	. . . . . . . . 9 ⊢ 2 ∈ (ℂ ∖ {0})
223	222	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0}))
224	204, 223, 181	constcncfg 42514	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
225	219, 224	divcncf 24051	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (𝑠 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
226		eqid 2798	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))
227		sincn 25039	. . . . . . . . . . . 12 ⊢ sin ∈ (ℂ–cn→ℂ)
228	227	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → sin ∈ (ℂ–cn→ℂ))
229	228, 168	cncfmpt1f 23519	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
230	229	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
231	204	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝐴[,]𝐵) ⊆ ℂ)
232	160	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → ℂ ⊆ ℂ)
233	59	ad2antlr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑘 ∈ ℂ)
234	195	recnd 10658	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
235	234	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
236	233, 235	mulcld 10650	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑘 · 𝑠) ∈ ℂ)
237	236	sincld 15475	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
238	226, 230, 231, 232, 237	cncfmptssg 42513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
239		eldifsn 4680	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℂ ∖ {0}) ↔ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
240	59, 70, 239	sylanbrc 586	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ (ℂ ∖ {0}))
241	240	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (ℂ ∖ {0}))
242		difssd 4060	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℂ ∖ {0}) ⊆ ℂ)
243	231, 241, 242	constcncfg 42514	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑘) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
244	238, 243	divcncf 24051	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
245	204, 97, 244	fsumcncf 42520	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
246	225, 245	addcncf 24048	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
247	246, 213	divcncf 24051	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
248	53, 247	eqeltrid 2894	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
249	145, 146, 155, 187, 218, 248	ftc2 24647	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
250	3, 154, 249	3eqtrd 2837	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∖ cdif 3878 ⊆ wss 3881 ifcif 4425 {csn 4525 {cpr 4527 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 (,)cioo 12726 [,]cicc 12729 ...cfz 12885 mod cmo 13232 Σcsu 15034 sincsin 15409 cosccos 15410 πcpi 15412 TopOpenctopn 16687 topGenctg 16703 ℂ_fldccnfld 20091 intcnt 21622 –cn→ccncf 23481 volcvol 24067 𝐿¹cibl 24221 ∫citg 24222 D cdv 24466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-disj 4996 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-ofr 7390 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-omul 8090 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-dju 9314 df-card 9352 df-acn 9355 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-pi 15418 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-ovol 24068 df-vol 24069 df-mbf 24223 df-itg1 24224 df-itg2 24225 df-ibl 24226 df-itg 24227 df-0p 24274 df-limc 24469 df-dv 24470

This theorem is referenced by: fourierdlem103 42851 fourierdlem104 42852

W3C validator