dirkeritg - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dirkeritg		Structured version Visualization version GIF version

Theorem dirkeritg 43533

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 6756	. . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
2	1	cbvitgv 24846	. . 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 13038	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
5	4	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
6		halfre 12117	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (1 / 2) ∈ ℝ)
8		fzfid 13621	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → (1...𝑁) ∈ Fin)
9		elfzelz 13185	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℤ)
10	9	zred 12355	. . . . . . . . . . . 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 10936	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
14	13	recoscld 15781	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
15	8, 14	fsumrecl 15374	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
16	7, 15	readdcld 10935	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
17		pire 25520	. . . . . . . 8 ⊢ π ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ∈ ℝ)
19		pipos 25522	. . . . . . . . 9 ⊢ 0 < π
20	17, 19	gt0ne0ii 11441	. . . . . . . 8 ⊢ π ≠ 0
21	20	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ≠ 0)
22	16, 18, 21	redivcld 11733	. . . . . 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 2738	. . . . . 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 6868	. . . . 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 583	. . . 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 7262	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝑥 mod (2 · π)) = (𝑠 mod (2 · π)))
29	28	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝑥 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
30		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑛 + (1 / 2)) · 𝑥) = ((𝑛 + (1 / 2)) · 𝑠))
31	30	fveq2d 6760	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (sin‘((𝑛 + (1 / 2)) · 𝑥)) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
32		oveq1 7262	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2))
33	32	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2)))
34	33	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((2 · π) · (sin‘(𝑥 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
35	31, 34	oveq12d 7273	. . . . . . . . . . 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 4482	. . . . . . . . . 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 5183	. . . . . . . . 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 5175	. . . . . . . 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 2766	. . . . . . 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 43532	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
43	42	fveq1d 6758	. . . . 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 7263	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑘 · 𝑥) = (𝑘 · 𝑠))
47	46	fveq2d 6760	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑘 · 𝑥)) = (sin‘(𝑘 · 𝑠)))
48	47	oveq1d 7270	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((sin‘(𝑘 · 𝑥)) / 𝑘) = ((sin‘(𝑘 · 𝑠)) / 𝑘))
49	48	sumeq2sdv 15344	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘) = Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))
50	32, 49	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) = ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))
51	50	oveq1d 7270	. . . . . . . . 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 5183	. . . . . . . 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 2766	. . . . . . 7 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
54	53	oveq2i 7266	. . . . . 6 ⊢ (ℝ D 𝐺) = (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)))
55		reelprrecn 10894	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		recn 10892	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
58	57	halfcld 12148	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℂ)
59	9	zcnd 12356	. . . . . . . . . . . . . . 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 10926	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℂ)
63	62	sincld 15767	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
64		0red 10909	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 ∈ ℝ)
65		1red 10907	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℝ)
66		0lt1 11427	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 1)
68		elfzle1 13188	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ≤ 𝑘)
69	64, 65, 10, 67, 68	ltletrd 11065	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 𝑘)
70	69	gt0ne0d 11469	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
72	63, 60, 71	divcld 11681	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
73	8, 72	fsumcl 15373	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
74	58, 73	addcld 10925	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
75		picn 25521	. . . . . . . . . 10 ⊢ π ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → π ∈ ℂ)
77	74, 76, 21	divcld 11681	. . . . . . . 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 10907	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
86	56	dvmptid 25026	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
87		2cnd 11981	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
88		2ne0 12007	. . . . . . . . . . 11 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
90	56, 84, 85, 86, 87, 89	dvmptdivc 25034	. . . . . . . . 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		eqid 2738	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
94	93	tgioo2 23872	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
95		reopn 42717	. . . . . . . . . . 11 ⊢ ℝ ∈ (topGen‘ran (,))
96	95	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
97		fzfid 13621	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
98	72	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
99	98	3adant1 1128	. . . . . . . . . 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 10934	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
102	101	3adant1 1128	. . . . . . . . . 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 10926	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · 𝑠) ∈ ℂ)
108	107	coscld 15768	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
109	105, 108	mulcld 10926	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
110	57, 109	sylan2 592	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
111		ax-resscn 10859	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
112		resmpt 5934	. . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))) = ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ))
115	114	oveq2d 7271	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)))
116	107	sincld 15767	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
117	116	fmpttd 6971	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ)
118	109	ralrimiva 3107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝑁) → ∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
119		dmmptg 6134	. . . . . . . . . . . . . . . . . 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 3970	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
122		dvsinax 43344	. . . . . . . . . . . . . . . . . 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 5803	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
125	121, 124	sseqtrrd 3958	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))))
126		dvcnre 43347	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)) = ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ))
128	123	reseq1d 5879	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ))
129		resmpt 5934	. . . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
132	115, 127, 131	3eqtrd 2782	. . . . . . . . . . . . 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 25034	. . . . . . . . . . . 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 11686	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘) = (cos‘(𝑘 · 𝑠)))
137	136	mpteq2dva 5170	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
138	133, 137	eqtrd 2778	. . . . . . . . . . 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	94, 93, 56, 96, 97, 99, 102, 139	dvmptfsum 25044	. . . . . . . . 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 25029	. . . . . . . 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 25034	. . . . . . 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 13095	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
148		iccntr 23890	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
149	145, 146, 148	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
150	56, 78, 79, 144, 147, 94, 93, 149	dvmptres2 25031	. . . . . 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 2791	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
152	151, 23	fvmpt2d 6870	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
153	26, 44, 152	3eqtr4d 2788	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((ℝ D 𝐺)‘𝑠))
154	153	itgeq2dv 24851	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠)
155		dirkeritg.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
156		ioosscn 13070	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
157	156	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
158		halfcn 12118	. . . . . . . 8 ⊢ (1 / 2) ∈ ℂ
159	158	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
160		ssid 3939	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
162	157, 159, 161	constcncfg 43303	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163		eqid 2738	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠)))
164		coscn 25509	. . . . . . . . . . 11 ⊢ cos ∈ (ℂ–cn→ℂ)
165	164	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → cos ∈ (ℂ–cn→ℂ))
166		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠))
167	166	mulc1cncf 23974	. . . . . . . . . . 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 23983	. . . . . . . . 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 592	. . . . . . . . 9 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
173	163, 169, 170, 171, 172	cncfmptssg 43302	. . . . . . . 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 43309	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
176	162, 175	addcncf 24513	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
177		eqid 2738	. . . . . 6 ⊢ (𝑠 ∈ ℂ ↦ π) = (𝑠 ∈ ℂ ↦ π)
178		cncfmptc 23981	. . . . . . . 8 ⊢ ((π ∈ ℂ ∧ ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
179	75, 160, 160, 178	mp3an 1459	. . . . . . 7 ⊢ (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ)
180	179	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ π) ∈ (ℂ–cn→ℂ))
181		difssd 4063	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
182		eldifsn 4717	. . . . . . . 8 ⊢ (π ∈ (ℂ ∖ {0}) ↔ (π ∈ ℂ ∧ π ≠ 0))
183	75, 20, 182	mpbir2an 707	. . . . . . 7 ⊢ π ∈ (ℂ ∖ {0})
184	183	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → π ∈ (ℂ ∖ {0}))
185	177, 180, 157, 181, 184	cncfmptssg 43302	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0})))
186	176, 185	divcncf 24516	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
187	151, 186	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
188		ioossicc 13094	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
189	188	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
190		ioombl 24634	. . . . . 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 13621	. . . . . . . 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 3917	. . . . . . . . . . 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 10936	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
198	197	recoscld 15781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
199	193, 198	fsumrecl 15374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
200	192, 199	readdcld 10935	. . . . . 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 11733	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
204	147, 111	sstrdi 3929	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
205	204, 159, 161	constcncfg 43303	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (1 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
206		eqid 2738	. . . . . . . . 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 43309	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
209	199	recnd 10934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℂ)
210	206, 208, 204, 161, 209	cncfmptssg 43302	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
211	205, 210	addcncf 24513	. . . . . . 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 43303	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ π) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
214	211, 213	divcncf 24516	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
215		cniccibl 24910	. . . . . 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 1369	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
217	189, 191, 203, 216	iblss 24874	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
218	151, 217	eqeltrd 2839	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
219	204, 161	idcncfg 43304	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑠) ∈ ((𝐴[,]𝐵)–cn→ℂ))
220		2cn 11978	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
221		eldifsn 4717	. . . . . . . . . 10 ⊢ (2 ∈ (ℂ ∖ {0}) ↔ (2 ∈ ℂ ∧ 2 ≠ 0))
222	220, 88, 221	mpbir2an 707	. . . . . . . . 9 ⊢ 2 ∈ (ℂ ∖ {0})
223	222	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ (ℂ ∖ {0}))
224	204, 223, 181	constcncfg 43303	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
225	219, 224	divcncf 24516	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (𝑠 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
226		eqid 2738	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))
227		sincn 25508	. . . . . . . . . . . 12 ⊢ sin ∈ (ℂ–cn→ℂ)
228	227	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → sin ∈ (ℂ–cn→ℂ))
229	228, 168	cncfmpt1f 23983	. . . . . . . . . 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 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑘 ∈ ℂ)
234	195	recnd 10934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
235	234	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → 𝑠 ∈ ℂ)
236	233, 235	mulcld 10926	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑘 · 𝑠) ∈ ℂ)
237	236	sincld 15767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
238	226, 230, 231, 232, 237	cncfmptssg 43302	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
239		eldifsn 4717	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℂ ∖ {0}) ↔ (𝑘 ∈ ℂ ∧ 𝑘 ≠ 0))
240	59, 70, 239	sylanbrc 582	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ (ℂ ∖ {0}))
241	240	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ∈ (ℂ ∖ {0}))
242		difssd 4063	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℂ ∖ {0}) ⊆ ℂ)
243	231, 241, 242	constcncfg 43303	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑘) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
244	238, 243	divcncf 24516	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
245	204, 97, 244	fsumcncf 43309	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
246	225, 245	addcncf 24513	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
247	246, 213	divcncf 24516	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
248	53, 247	eqeltrid 2843	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
249	145, 146, 155, 187, 218, 248	ftc2 25113	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
250	3, 154, 249	3eqtrd 2782	1 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑥) d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∖ cdif 3880 ⊆ wss 3883 ifcif 4456 {csn 4558 {cpr 4560 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 < clt 10940 ≤ cle 10941 − cmin 11135 / cdiv 11562 ℕcn 11903 2c2 11958 (,)cioo 13008 [,]cicc 13011 ...cfz 13168 mod cmo 13517 Σcsu 15325 sincsin 15701 cosccos 15702 πcpi 15704 TopOpenctopn 17049 topGenctg 17065 ℂ_fldccnfld 20510 intcnt 22076 –cn→ccncf 23945 volcvol 24532 𝐿¹cibl 24686 ∫citg 24687 D cdv 24932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cc 10122 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-addf 10881 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-symdif 4173 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-disj 5036 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-ofr 7512 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-oadd 8271 df-omul 8272 df-er 8456 df-map 8575 df-pm 8576 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-dju 9590 df-card 9628 df-acn 9631 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-ioo 13012 df-ioc 13013 df-ico 13014 df-icc 13015 df-fz 13169 df-fzo 13312 df-fl 13440 df-mod 13518 df-seq 13650 df-exp 13711 df-fac 13916 df-bc 13945 df-hash 13973 df-shft 14706 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-limsup 15108 df-clim 15125 df-rlim 15126 df-sum 15326 df-ef 15705 df-sin 15707 df-cos 15708 df-pi 15710 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-fbas 20507 df-fg 20508 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-perf 22196 df-cn 22286 df-cnp 22287 df-haus 22374 df-cmp 22446 df-tx 22621 df-hmeo 22814 df-fil 22905 df-fm 22997 df-flim 22998 df-flf 22999 df-xms 23381 df-ms 23382 df-tms 23383 df-cncf 23947 df-ovol 24533 df-vol 24534 df-mbf 24688 df-itg1 24689 df-itg2 24690 df-ibl 24691 df-itg 24692 df-0p 24739 df-limc 24935 df-dv 24936

This theorem is referenced by: fourierdlem103 43640 fourierdlem104 43641

W3C validator