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 46117

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 6906	. . . 4 ⊢ (𝑥 = 𝑠 → (𝐹‘𝑥) = (𝐹‘𝑠))
2	1	cbvitgv 25812	. . 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 13417	. . . . . 6 ⊢ (𝑠 ∈ (𝐴(,)𝐵) → 𝑠 ∈ ℝ)
5	4	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → 𝑠 ∈ ℝ)
6		halfre 12480	. . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ
7	6	a1i 11	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → (1 / 2) ∈ ℝ)
8		fzfid 14014	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → (1...𝑁) ∈ Fin)
9		elfzelz 13564	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℤ)
10	9	zred 12722	. . . . . . . . . . . 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 11291	. . . . . . . . . 10 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
14	13	recoscld 16180	. . . . . . . . 9 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
15	8, 14	fsumrecl 15770	. . . . . . . 8 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
16	7, 15	readdcld 11290	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ℝ)
17		pire 26500	. . . . . . . 8 ⊢ π ∈ ℝ
18	17	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ∈ ℝ)
19		pipos 26502	. . . . . . . . 9 ⊢ 0 < π
20	17, 19	gt0ne0ii 11799	. . . . . . . 8 ⊢ π ≠ 0
21	20	a1i 11	. . . . . . 7 ⊢ (𝑠 ∈ ℝ → π ≠ 0)
22	16, 18, 21	redivcld 12095	. . . . . 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 2737	. . . . . 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 7027	. . . . 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 7438	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (𝑥 mod (2 · π)) = (𝑠 mod (2 · π)))
29	28	eqeq1d 2739	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → ((𝑥 mod (2 · π)) = 0 ↔ (𝑠 mod (2 · π)) = 0))
30		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → ((𝑛 + (1 / 2)) · 𝑥) = ((𝑛 + (1 / 2)) · 𝑠))
31	30	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → (sin‘((𝑛 + (1 / 2)) · 𝑥)) = (sin‘((𝑛 + (1 / 2)) · 𝑠)))
32		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥 / 2) = (𝑠 / 2))
33	32	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑥 / 2)) = (sin‘(𝑠 / 2)))
34	33	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((2 · π) · (sin‘(𝑥 / 2))) = ((2 · π) · (sin‘(𝑠 / 2))))
35	31, 34	oveq12d 7449	. . . . . . . . . . 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 4552	. . . . . . . . . 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 5255	. . . . . . . . 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 5247	. . . . . . . 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 2765	. . . . . . 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 46116	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑠 ∈ ℝ ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
43	42	fveq1d 6908	. . . . 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 7439	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑘 · 𝑥) = (𝑘 · 𝑠))
47	46	fveq2d 6910	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (sin‘(𝑘 · 𝑥)) = (sin‘(𝑘 · 𝑠)))
48	47	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑠 → ((sin‘(𝑘 · 𝑥)) / 𝑘) = ((sin‘(𝑘 · 𝑠)) / 𝑘))
49	48	sumeq2sdv 15739	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑠 → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘) = Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))
50	32, 49	oveq12d 7449	. . . . . . . . . 10 ⊢ (𝑥 = 𝑠 → ((𝑥 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑥)) / 𝑘)) = ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)))
51	50	oveq1d 7446	. . . . . . . . 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 5255	. . . . . . . 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 2765	. . . . . . 7 ⊢ 𝐺 = (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π))
54	53	oveq2i 7442	. . . . . 6 ⊢ (ℝ D 𝐺) = (ℝ D (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)))
55		reelprrecn 11247	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
57		recn 11245	. . . . . . . . . . 11 ⊢ (𝑠 ∈ ℝ → 𝑠 ∈ ℂ)
58	57	halfcld 12511	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → (𝑠 / 2) ∈ ℂ)
59	9	zcnd 12723	. . . . . . . . . . . . . . 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 11281	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℂ)
63	62	sincld 16166	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
64		0red 11264	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 ∈ ℝ)
65		1red 11262	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ∈ ℝ)
66		0lt1 11785	. . . . . . . . . . . . . . . 16 ⊢ 0 < 1
67	66	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 1)
68		elfzle1 13567	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → 1 ≤ 𝑘)
69	64, 65, 10, 67, 68	ltletrd 11421	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → 0 < 𝑘)
70	69	gt0ne0d 11827	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ≠ 0)
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → 𝑘 ≠ 0)
72	63, 60, 71	divcld 12043	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ℝ ∧ 𝑘 ∈ (1...𝑁)) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
73	8, 72	fsumcl 15769	. . . . . . . . . 10 ⊢ (𝑠 ∈ ℝ → Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
74	58, 73	addcld 11280	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ℂ)
75		picn 26501	. . . . . . . . . 10 ⊢ π ∈ ℂ
76	75	a1i 11	. . . . . . . . 9 ⊢ (𝑠 ∈ ℝ → π ∈ ℂ)
77	74, 76, 21	divcld 12043	. . . . . . . 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 11262	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈ ℝ)
86	56	dvmptid 25995	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D (𝑠 ∈ ℝ ↦ 𝑠)) = (𝑠 ∈ ℝ ↦ 1))
87		2cnd 12344	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℂ)
88		2ne0 12370	. . . . . . . . . . 11 ⊢ 2 ≠ 0
89	88	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2 ≠ 0)
90	56, 84, 85, 86, 87, 89	dvmptdivc 26003	. . . . . . . . 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 24826	. . . . . . . . . 10 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
94		eqid 2737	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
95		reopn 45301	. . . . . . . . . . 11 ⊢ ℝ ∈ (topGen‘ran (,))
96	95	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (topGen‘ran (,)))
97		fzfid 14014	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
98	72	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((sin‘(𝑘 · 𝑠)) / 𝑘) ∈ ℂ)
99	98	3adant1 1131	. . . . . . . . . 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 11289	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
102	101	3adant1 1131	. . . . . . . . . 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 11281	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (𝑘 · 𝑠) ∈ ℂ)
108	107	coscld 16167	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (cos‘(𝑘 · 𝑠)) ∈ ℂ)
109	105, 108	mulcld 11281	. . . . . . . . . . . . . 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 11212	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
112		resmpt 6055	. . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠))) = ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ))
115	114	oveq2d 7447	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → (ℝ D (𝑠 ∈ ℝ ↦ (sin‘(𝑘 · 𝑠)))) = (ℝ D ((𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) ↾ ℝ)))
116	107	sincld 16166	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℂ) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
117	116	fmpttd 7135	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))):ℂ⟶ℂ)
118	109	ralrimiva 3146	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1...𝑁) → ∀𝑠 ∈ ℂ (𝑘 · (cos‘(𝑘 · 𝑠))) ∈ ℂ)
119		dmmptg 6262	. . . . . . . . . . . . . . . . . 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 4027	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
122		dvsinax 45928	. . . . . . . . . . . . . . . . . 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 5916	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (1...𝑁) → dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) = dom (𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
125	121, 124	sseqtrrd 4021	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ℝ ⊆ dom (ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))))
126		dvcnre 45931	. . . . . . . . . . . . . . 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 5996	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = ((𝑠 ∈ ℂ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))) ↾ ℝ))
129		resmpt 6055	. . . . . . . . . . . . . . . 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 2793	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ (1...𝑁) → ((ℂ D (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))) ↾ ℝ) = (𝑠 ∈ ℝ ↦ (𝑘 · (cos‘(𝑘 · 𝑠)))))
132	115, 127, 131	3eqtrd 2781	. . . . . . . . . . . . 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 26003	. . . . . . . . . . . 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 12048	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ (1...𝑁) ∧ 𝑠 ∈ ℝ) → ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘) = (cos‘(𝑘 · 𝑠)))
137	136	mpteq2dva 5242	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (1...𝑁) → (𝑠 ∈ ℝ ↦ ((𝑘 · (cos‘(𝑘 · 𝑠))) / 𝑘)) = (𝑠 ∈ ℝ ↦ (cos‘(𝑘 · 𝑠))))
138	133, 137	eqtrd 2777	. . . . . . . . . . 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 26013	. . . . . . . . 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 25998	. . . . . . . 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 26003	. . . . . . 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 13474	. . . . . . 7 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
148		iccntr 24843	. . . . . . . 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 26000	. . . . . 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 2789	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)))
152	151, 23	fvmpt2d 7029	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑠) = (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π))
153	26, 44, 152	3eqtr4d 2787	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑠) = ((ℝ D 𝐺)‘𝑠))
154	153	itgeq2dv 25817	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐹‘𝑠) d𝑠 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠)
155		dirkeritg.aleb	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
156		ioosscn 13449	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℂ
157	156	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
158		halfcn 12481	. . . . . . . 8 ⊢ (1 / 2) ∈ ℂ
159	158	a1i 11	. . . . . . 7 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
160		ssid 4006	. . . . . . . 8 ⊢ ℂ ⊆ ℂ
161	160	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ⊆ ℂ)
162	157, 159, 161	constcncfg 45887	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (1 / 2)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
163		eqid 2737	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (cos‘(𝑘 · 𝑠)))
164		coscn 26489	. . . . . . . . . . 11 ⊢ cos ∈ (ℂ–cn→ℂ)
165	164	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → cos ∈ (ℂ–cn→ℂ))
166		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠)) = (𝑠 ∈ ℂ ↦ (𝑘 · 𝑠))
167	166	mulc1cncf 24931	. . . . . . . . . . 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 24940	. . . . . . . . 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 45886	. . . . . . . 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 45893	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
176	162, 175	addcncf 25478	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ ((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)))) ∈ ((𝐴(,)𝐵)–cn→ℂ))
177		eqid 2737	. . . . . 6 ⊢ (𝑠 ∈ ℂ ↦ π) = (𝑠 ∈ ℂ ↦ π)
178		cncfmptc 24938	. . . . . . . 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 4137	. . . . . 6 ⊢ (𝜑 → (ℂ ∖ {0}) ⊆ ℂ)
182		eldifsn 4786	. . . . . . . 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 45886	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ π) ∈ ((𝐴(,)𝐵)–cn→(ℂ ∖ {0})))
186	176, 185	divcncf 25482	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
187	151, 186	eqeltrd 2841	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
188		ioossicc 13473	. . . . . 6 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
189	188	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
190		ioombl 25600	. . . . . 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 14014	. . . . . . . 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 3983	. . . . . . . . . . 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 11291	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (𝑘 · 𝑠) ∈ ℝ)
198	197	recoscld 16180	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) ∧ 𝑘 ∈ (1...𝑁)) → (cos‘(𝑘 · 𝑠)) ∈ ℝ)
199	193, 198	fsumrecl 15770	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℝ)
200	192, 199	readdcld 11290	. . . . . 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 12095	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π) ∈ ℝ)
204	147, 111	sstrdi 3996	. . . . . . . . 9 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℂ)
205	204, 159, 161	constcncfg 45887	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (1 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
206		eqid 2737	. . . . . . . . 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 45893	. . . . . . . . 9 ⊢ (𝜑 → (𝑠 ∈ ℂ ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ (ℂ–cn→ℂ))
209	199	recnd 11289	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ (𝐴[,]𝐵)) → Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠)) ∈ ℂ)
210	206, 208, 204, 161, 209	cncfmptssg 45886	. . . . . . . 8 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
211	205, 210	addcncf 25478	. . . . . . 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 45887	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ π) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
214	211, 213	divcncf 25482	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
215		cniccibl 25876	. . . . . 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 25840	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴(,)𝐵) ↦ (((1 / 2) + Σ𝑘 ∈ (1...𝑁)(cos‘(𝑘 · 𝑠))) / π)) ∈ 𝐿¹)
218	151, 217	eqeltrd 2841	. . 3 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
219	204, 161	idcncfg 45888	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑠) ∈ ((𝐴[,]𝐵)–cn→ℂ))
220		2cn 12341	. . . . . . . . . 10 ⊢ 2 ∈ ℂ
221		eldifsn 4786	. . . . . . . . . 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 45887	. . . . . . 7 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ 2) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
225	219, 224	divcncf 25482	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (𝑠 / 2)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
226		eqid 2737	. . . . . . . . 9 ⊢ (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠))) = (𝑠 ∈ ℂ ↦ (sin‘(𝑘 · 𝑠)))
227		sincn 26488	. . . . . . . . . . . 12 ⊢ sin ∈ (ℂ–cn→ℂ)
228	227	a1i 11	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...𝑁) → sin ∈ (ℂ–cn→ℂ))
229	228, 168	cncfmpt1f 24940	. . . . . . . . . 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 11289	. . . . . . . . . . . 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 11281	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (𝑘 · 𝑠) ∈ ℂ)
237	236	sincld 16166	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (1...𝑁)) ∧ 𝑠 ∈ (𝐴[,]𝐵)) → (sin‘(𝑘 · 𝑠)) ∈ ℂ)
238	226, 230, 231, 232, 237	cncfmptssg 45886	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ (sin‘(𝑘 · 𝑠))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
239		eldifsn 4786	. . . . . . . . . . 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 4137	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (ℂ ∖ {0}) ⊆ ℂ)
243	231, 241, 242	constcncfg 45887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ 𝑘) ∈ ((𝐴[,]𝐵)–cn→(ℂ ∖ {0})))
244	238, 243	divcncf 25482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
245	204, 97, 244	fsumcncf 45893	. . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
246	225, 245	addcncf 25478	. . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ ((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘))) ∈ ((𝐴[,]𝐵)–cn→ℂ))
247	246, 213	divcncf 25482	. . . 4 ⊢ (𝜑 → (𝑠 ∈ (𝐴[,]𝐵) ↦ (((𝑠 / 2) + Σ𝑘 ∈ (1...𝑁)((sin‘(𝑘 · 𝑠)) / 𝑘)) / π)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
248	53, 247	eqeltrid 2845	. . 3 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
249	145, 146, 155, 187, 218, 248	ftc2 26085	. 2 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑠) d𝑠 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
250	3, 154, 249	3eqtrd 2781	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 2108 ≠ wne 2940 ∀wral 3061 ∖ cdif 3948 ⊆ wss 3951 ifcif 4525 {csn 4626 {cpr 4628 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 ↾ cres 5687 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 (,)cioo 13387 [,]cicc 13390 ...cfz 13547 mod cmo 13909 Σcsu 15722 sincsin 16099 cosccos 16100 πcpi 16102 TopOpenctopn 17466 topGenctg 17482 ℂ_fldccnfld 21364 intcnt 23025 –cn→ccncf 24902 volcvol 25498 𝐿¹cibl 25652 ∫citg 25653 D cdv 25898

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902

This theorem is referenced by: fourierdlem103 46224 fourierdlem104 46225

W3C validator