dirkeritg - Mathbox for Glauco Siliprandi

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Theorem dirkeritg 46084

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 6826	. . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
2	1	cbvitgv 25694	. . 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 13296	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
5	4	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
6		halfre 12355	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (1 / 2) ∈ ℝ)
8		fzfid 13898	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → (1...𝑁) ∈ Fin)
9		elfzelz 13445	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℤ)
10	9	zred 12598	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℝ)
11	10	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
12		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
13	11, 12	remulcld 11164	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
14	13	recoscld 16071	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
15	8, 14	fsumrecl 15659	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
16	7, 15	readdcld 11163	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
17		pire 26382	. . . . . . . 8 ⊢ π ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ∈ ℝ)
19		pipos 26384	. . . . . . . . 9 ⊢ 0 < π
20	17, 19	gt0ne0ii 11674	. . . . . . . 8 ⊢ π ≠ 0
21	20	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ≠ 0)
22	16, 18, 21	redivcld 11970	. . . . . 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 2729	. . . . . 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 6945	. . . . 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 584	. . . 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 7360	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝑥 mod (2 · π)) = (𝑠 mod (2 · π)))
29	28	eqeq1d 2731	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝑥 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
30		oveq2 7361	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑛 + (1 / 2)) · 𝑥) = ((𝑛 + (1 / 2)) · 𝑠))
31	30	fveq2d 6830	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (sin‘((𝑛 + (1 / 2)) · 𝑥)) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
32		oveq1 7360	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2))
33	32	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2)))
34	33	oveq2d 7369	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((2 · π) · (sin‘(𝑥 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
35	31, 34	oveq12d 7371	. . . . . . . . . . 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 4505	. . . . . . . . . 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 5199	. . . . . . . . 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 5191	. . . . . . . 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 2752	. . . . . . 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 46083	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
43	42	fveq1d 6828	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑠) = ((𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))‘𝑠))
44	43	adantr 480	. . . 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 7361	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑘 · 𝑥) = (𝑘 · 𝑠))
47	46	fveq2d 6830	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑘 · 𝑥)) = (sin‘(𝑘 · 𝑠)))
48	47	oveq1d 7368	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((sin‘(𝑘 · 𝑥)) / 𝑘) = ((sin‘(𝑘 · 𝑠)) / 𝑘))
49	48	sumeq2sdv 15628	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘) = Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))
50	32, 49	oveq12d 7371	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) = ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))
51	50	oveq1d 7368	. . . . . . . . 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 5199	. . . . . . . 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 2752	. . . . . . 7 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
54	53	oveq2i 7364	. . . . . 6 ⊢ (ℝ D 𝐺) = (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)))
55		reelprrecn 11120	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		recn 11118	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
58	57	halfcld 12387	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℂ)
59	9	zcnd 12599	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℂ)
60	59	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℂ)
61	57	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℂ)
62	60, 61	mulcld 11154	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℂ)
63	62	sincld 16057	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
64		0red 11137	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 ∈ ℝ)
65		1red 11135	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℝ)
66		0lt1 11660	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 1)
68		elfzle1 13448	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ≤ 𝑘)
69	64, 65, 10, 67, 68	ltletrd 11294	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 𝑘)
70	69	gt0ne0d 11702	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
72	63, 60, 71	divcld 11918	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
73	8, 72	fsumcl 15658	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
74	58, 73	addcld 11153	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
75		picn 26383	. . . . . . . . . 10 ⊢ π ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → π ∈ ℂ)
77	74, 76, 21	divcld 11918	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
78	77	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π) ∈ ℂ)
79	22	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
80	74	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
81	16	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
82	58	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (𝑠 / 2) ∈ ℂ)
83	6	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (1 / 2) ∈ ℝ)
84	57	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℂ)
85		1red 11135	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
86	56	dvmptid 25877	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
87		2cnd 12224	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
88		2ne0 12250	. . . . . . . . . . 11 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
90	56, 84, 85, 86, 87, 89	dvmptdivc 25885	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ (𝑠 / 2))) = (𝑠 ∈ ℝ ↦ (1 / 2)))
91	73	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
92	15	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
93		tgioo4 24709	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
94		eqid 2729	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
95		reopn 45271	. . . . . . . . . . 11 ⊢ ℝ ∈ (topGen‘ran (,))
96	95	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
97		fzfid 13898	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
98	72	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
99	98	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
100	14	ancoms 458	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
101	100	recnd 11162	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
102	101	3adant1 1130	. . . . . . . . . 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 458	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
105	59	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑘 ∈ ℂ)
106		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → 𝑠 ∈ ℂ)
107	105, 106	mulcld 11154	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · 𝑠) ∈ ℂ)
108	107	coscld 16058	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
109	105, 108	mulcld 11154	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
110	57, 109	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
111		ax-resscn 11085	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
112		resmpt 5992	. . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))) = ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ))
115	114	oveq2d 7369	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)))
116	107	sincld 16057	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
117	116	fmpttd 7053	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ)
118	109	ralrimiva 3121	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝑁) → ∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
119		dmmptg 6195	. . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
122		dvsinax 45895	. . . . . . . . . . . . . . . . . 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 5852	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
125	121, 124	sseqtrrd 3975	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))))
126		dvcnre 45898	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)) = ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ))
128	123	reseq1d 5933	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ))
129		resmpt 5992	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
132	115, 127, 131	3eqtrd 2768	. . . . . . . . . . . . 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 25885	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)))
134	59	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ∈ ℂ)
135	70	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → 𝑘 ≠ 0)
136	101, 134, 135	divcan3d 11923	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘) = (cos‘(𝑘 · 𝑠)))
137	136	mpteq2dva 5188	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
138	133, 137	eqtrd 2764	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
139	138	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℝ D (𝑠 ∈ ℝ ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘))) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
140	93, 94, 56, 96, 97, 99, 102, 139	dvmptfsum 25895	. . . . . . . . 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 25880	. . . . . . . 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 25885	. . . . . . 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 13355	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
148		iccntr 24726	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
149	145, 146, 148	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
150	56, 78, 79, 144, 147, 93, 94, 149	dvmptres2 25882	. . . . . 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	eqtrid 2776	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
152	151, 23	fvmpt2d 6947	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
153	26, 44, 152	3eqtr4d 2774	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((ℝ D 𝐺)‘𝑠))
154	153	itgeq2dv 25699	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠)
155		dirkeritg.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
156		ioosscn 13329	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
157	156	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
158		halfcn 12356	. . . . . . . 8 ⊢ (1 / 2) ∈ ℂ
159	158	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
160		ssid 3960	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
162	157, 159, 161	constcncfg 45854	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163		eqid 2729	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠)))
164		coscn 26371	. . . . . . . . . . 11 ⊢ cos ∈ (ℂ–cn→ℂ)
165	164	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → cos ∈ (ℂ–cn→ℂ))
166		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠))
167	166	mulc1cncf 24814	. . . . . . . . . . 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 24823	. . . . . . . . 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 593	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
173	163, 169, 170, 171, 172	cncfmptssg 45853	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
174	173	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴(,)𝐵) ↦ (cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
175	157, 97, 174	fsumcncf 45860	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
176	162, 175	addcncf 25360	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
177		eqid 2729	. . . . . 6 ⊢ (𝑠 ∈ ℂ ↦ π) = (𝑠 ∈ ℂ ↦ π)
178		cncfmptc 24821	. . . . . . . 8 ⊢ ((π ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
179	75, 160, 160, 178	mp3an 1463	. . . . . . 7 ⊢ (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ)
180	179	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
181		difssd 4090	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
182		eldifsn 4740	. . . . . . . 8 ⊢ (π ∈ (ℂ ∖ {0}) ↔ (π ∈ ℂ ∧ π ≠ 0))
183	75, 20, 182	mpbir2an 711	. . . . . . 7 ⊢ π ∈ (ℂ ∖ {0})
184	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → π ∈ (ℂ ∖ {0}))
185	177, 180, 157, 181, 184	cncfmptssg 45853	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0})))
186	176, 185	divcncf 25364	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
187	151, 186	eqeltrd 2828	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
188		ioossicc 13354	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
189	188	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
190		ioombl 25482	. . . . . 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 13898	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (1...𝑁) ∈ Fin)
194	10	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ ℝ)
195	147	sselda 3937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℝ)
196	195	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → 𝑠 ∈ ℝ)
197	194, 196	remulcld 11164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
198	197	recoscld 16071	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
199	193, 198	fsumrecl 15659	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
200	192, 199	readdcld 11163	. . . . . 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 11970	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
204	147, 111	sstrdi 3950	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
205	204, 159, 161	constcncfg 45854	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (1 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
206		eqid 2729	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))
207	169	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
208	161, 97, 207	fsumcncf 45860	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
209	199	recnd 11162	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℂ)
210	206, 208, 204, 161, 209	cncfmptssg 45853	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
211	205, 210	addcncf 25360	. . . . . . 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 45854	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ π) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
214	211, 213	divcncf 25364	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
215		cniccibl 25758	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
217	189, 191, 203, 216	iblss 25722	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
218	151, 217	eqeltrd 2828	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
219	204, 161	idcncfg 45855	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑠) ∈ ((𝐴[,]𝐵)–cn→ℂ))
220		2cn 12221	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
221		eldifsn 4740	. . . . . . . . . 10 ⊢ (2 ∈ (ℂ ∖ {0}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0))
222	220, 88, 221	mpbir2an 711	. . . . . . . . 9 ⊢ 2 ∈ (ℂ ∖ {0})
223	222	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0}))
224	204, 223, 181	constcncfg 45854	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
225	219, 224	divcncf 25364	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (𝑠 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
226		eqid 2729	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))
227		sincn 26370	. . . . . . . . . . . 12 ⊢ sin ∈ (ℂ–cn→ℂ)
228	227	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → sin ∈ (ℂ–cn→ℂ))
229	228, 168	cncfmpt1f 24823	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
230	229	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
231	204	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝐴[,]𝐵) ⊆ ℂ)
232	160	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → ℂ ⊆ ℂ)
233	59	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑘 ∈ ℂ)
234	195	recnd 11162	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
235	234	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
236	233, 235	mulcld 11154	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑘 · 𝑠) ∈ ℂ)
237	236	sincld 16057	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
238	226, 230, 231, 232, 237	cncfmptssg 45853	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
239		eldifsn 4740	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℂ ∖ {0}) ↔ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
240	59, 70, 239	sylanbrc 583	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ (ℂ ∖ {0}))
241	240	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (ℂ ∖ {0}))
242		difssd 4090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℂ ∖ {0}) ⊆ ℂ)
243	231, 241, 242	constcncfg 45854	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑘) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
244	238, 243	divcncf 25364	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
245	204, 97, 244	fsumcncf 45860	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
246	225, 245	addcncf 25360	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
247	246, 213	divcncf 25364	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
248	53, 247	eqeltrid 2832	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
249	145, 146, 155, 187, 218, 248	ftc2 25967	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
250	3, 154, 249	3eqtrd 2768	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∖ cdif 3902 ⊆ wss 3905 ifcif 4478 {csn 4579 {cpr 4581 class class class wbr 5095 ↦ cmpt 5176 dom cdm 5623 ran crn 5624 ↾ cres 5625 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ℝcr 11027 0cc0 11028 1c1 11029 + caddc 11031 · cmul 11033 < clt 11168 ≤ cle 11169 − cmin 11365 / cdiv 11795 ℕcn 12146 2c2 12201 (,)cioo 13266 [,]cicc 13269 ...cfz 13428 mod cmo 13791 Σcsu 15611 sincsin 15988 cosccos 15989 πcpi 15991 TopOpenctopn 17343 topGenctg 17359 ℂ_fldccnfld 21279 intcnt 22920 –cn→ccncf 24785 volcvol 25380 𝐿¹cibl 25534 ∫citg 25535 D cdv 25780

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cc 10348 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-symdif 4206 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-ofr 7618 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ioc 13271 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-fac 14199 df-bc 14228 df-hash 14256 df-shft 14992 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-ef 15992 df-sin 15994 df-cos 15995 df-pi 15997 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-lp 23039 df-perf 23040 df-cn 23130 df-cnp 23131 df-haus 23218 df-cmp 23290 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cncf 24787 df-ovol 25381 df-vol 25382 df-mbf 25536 df-itg1 25537 df-itg2 25538 df-ibl 25539 df-itg 25540 df-0p 25587 df-limc 25783 df-dv 25784

This theorem is referenced by: fourierdlem103 46191 fourierdlem104 46192

W3C validator