itgsinexplem1 - Mathbox for Glauco Siliprandi

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Theorem itgsinexplem1 46405

Description: Integration by parts is applied to integrate sin^(N+1). (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
itgsinexplem1.1	⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
itgsinexplem1.2	⊢ 𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
itgsinexplem1.3	⊢ 𝐻 = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
itgsinexplem1.4	⊢ 𝐼 = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
itgsinexplem1.5	⊢ 𝐿 = (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)))
itgsinexplem1.6	⊢ 𝑀 = (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
itgsinexplem1.7	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
itgsinexplem1	⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))

Distinct variable groups: 𝑥,𝑁 𝜑,𝑥 Allowed substitution hints: 𝐹(𝑥) 𝐺(𝑥) 𝐻(𝑥) 𝐼(𝑥) 𝐿(𝑥) 𝑀(𝑥)

**Proof of Theorem itgsinexplem1**
Step	Hyp	Ref	Expression
1		0m0e0 12288	. . . . 5 ⊢ (0 − 0) = 0
2	1	oveq1i 7367	. . . 4 ⊢ ((0 − 0) − ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥) = (0 − ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥)
3		0re 11138	. . . . . 6 ⊢ 0 ∈ ℝ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
5		pire 26440	. . . . . 6 ⊢ π ∈ ℝ
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → π ∈ ℝ)
7		pipos 26442	. . . . . . 7 ⊢ 0 < π
8	3, 5, 7	ltleii 11261	. . . . . 6 ⊢ 0 ≤ π
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ π)
10	3, 5	pm3.2i 471	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ ∧ π ∈ ℝ)
11		iccssre 13374	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13		ax-resscn 11087	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14	12, 13	sstri 3924	. . . . . . . . . . 11 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3911	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
17	15	sincld 16089	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
18	17	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
19		itgsinexplem1.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
20	19	nnnn0d 12490	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
22	18, 21	expcld 14100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
23		itgsinexplem1.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
24	23	fvmpt2 6948	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → (𝐹‘𝑥) = ((sin‘𝑥)↑𝑁))
25	16, 22, 24	syl2anc 590	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝐹‘𝑥) = ((sin‘𝑥)↑𝑁))
26	25	eqcomd 2745	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = (𝐹‘𝑥))
27	26	mpteq2dva 5166	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)))
28		nfmpt1 5172	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
29	23, 28	nfcxfr 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
30		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑥sin
31		sincn 26428	. . . . . . . . . 10 ⊢ sin ∈ (ℂ–cn→ℂ)
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
33	30, 32, 20	expcnfg 46044	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
34	23, 33	eqeltrid 2843	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
35	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
36	29, 34, 35	cncfmptss 46040	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
37	27, 36	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
38	15	coscld 16090	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
39	38	negcld 11484	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
40		itgsinexplem1.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
41	40	fvmpt2 6948	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → (𝐺‘𝑥) = -(cos‘𝑥))
42	15, 39, 41	syl2anc 590	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → (𝐺‘𝑥) = -(cos‘𝑥))
43	42	eqcomd 2745	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = (𝐺‘𝑥))
44	43	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) = (𝐺‘𝑥))
45	44	mpteq2dva 5166	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)))
46		nfmpt1 5172	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
47	40, 46	nfcxfr 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝐺
48		coscn 26429	. . . . . . . . 9 ⊢ cos ∈ (ℂ–cn→ℂ)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
50	40	negfcncf 24909	. . . . . . . 8 ⊢ (cos ∈ (ℂ–cn→ℂ) → 𝐺 ∈ (ℂ–cn→ℂ))
51	49, 50	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (ℂ–cn→ℂ))
52	47, 51, 35	cncfmptss 46040	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
53	45, 52	eqeltrd 2839	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
54		itgsinexplem1.3	. . . . . 6 ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
55		ssidd 3938	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
56	19	nncnd 12182	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
57	55, 56, 55	constcncfg 46323	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑁) ∈ (ℂ–cn→ℂ))
58		nnm1nn0 12470	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
59	19, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
60	30, 32, 59	expcnfg 46044	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) ∈ (ℂ–cn→ℂ))
61	57, 60	mulcncf 25432	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑁 · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
62		cosf 16084	. . . . . . . . . . 11 ⊢ cos:ℂ⟶ℂ
63	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → cos:ℂ⟶ℂ)
64	63	feqmptd 6896	. . . . . . . . 9 ⊢ (𝜑 → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
65	64, 48	eqeltrrdi 2848	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (cos‘𝑥)) ∈ (ℂ–cn→ℂ))
66	61, 65	mulcncf 25432	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ (ℂ–cn→ℂ))
67	54, 66	eqeltrid 2843	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (ℂ–cn→ℂ))
68		ioosscn 13353	. . . . . . 7 ⊢ (0(,)π) ⊆ ℂ
69	68	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ⊆ ℂ)
70	56	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℂ)
71	68	sseli 3911	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
72	71	sincld 16089	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
73	72	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
74	59	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 1) ∈ ℕ₀)
75	73, 74	expcld 14100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
76	70, 75	mulcld 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
77	71	coscld 16090	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (cos‘𝑥) ∈ ℂ)
78	77	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (cos‘𝑥) ∈ ℂ)
79	76, 78	mulcld 11157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
80	54, 67, 69, 55, 79	cncfmptssg 46322	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ ((0(,)π)–cn→ℂ))
81	30, 32, 69	cncfmptss 46040	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
82		ioossicc 13378	. . . . . . 7 ⊢ (0(,)π) ⊆ (0[,]π)
83	82	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
84		ioombl 25551	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
85	84	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
86	22, 18	mulcld 11157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) ∈ ℂ)
87		itgsinexplem1.4	. . . . . . . . . . . 12 ⊢ 𝐼 = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
88	87	fvmpt2 6948	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) ∈ ℂ) → (𝐼‘𝑥) = (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
89	16, 86, 88	syl2anc 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝐼‘𝑥) = (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
90	89	eqcomd 2745	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) = (𝐼‘𝑥))
91	90	mpteq2dva 5166	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) = (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)))
92		nfmpt1 5172	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
93	87, 92	nfcxfr 2899	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐼
94		sinf 16083	. . . . . . . . . . . . . 14 ⊢ sin:ℂ⟶ℂ
95	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → sin:ℂ⟶ℂ)
96	95	feqmptd 6896	. . . . . . . . . . . 12 ⊢ (𝜑 → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
97	96, 31	eqeltrrdi 2848	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (sin‘𝑥)) ∈ (ℂ–cn→ℂ))
98	33, 97	mulcncf 25432	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ (ℂ–cn→ℂ))
99	87, 98	eqeltrid 2843	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (ℂ–cn→ℂ))
100	93, 99, 35	cncfmptss 46040	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
101	91, 100	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
102		cniccibl 25827	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ 𝐿¹)
103	4, 6, 101, 102	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ 𝐿¹)
104	83, 85, 86, 103	iblss 25791	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ 𝐿¹)
105	56	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℂ)
106	59	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 1) ∈ ℕ₀)
107	18, 106	expcld 14100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
108	105, 107	mulcld 11157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
109	38	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
110	108, 109	mulcld 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
111	39	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
112	110, 111	mulcld 11157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) ∈ ℂ)
113		itgsinexplem1.5	. . . . . . . 8 ⊢ 𝐿 = (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)))
114		eqid 2739	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
115	114	negfcncf 24909	. . . . . . . . . . 11 ⊢ (cos ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) ∈ (ℂ–cn→ℂ))
116	49, 115	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) ∈ (ℂ–cn→ℂ))
117	66, 116	mulcncf 25432	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ (ℂ–cn→ℂ))
118	113, 117	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℂ–cn→ℂ))
119	113, 118, 35, 55, 112	cncfmptssg 46322	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
120		cniccibl 25827	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
121	4, 6, 119, 120	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
122	83, 85, 112, 121	iblss 25791	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
123		reelprrecn 11122	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
124	123	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
125		recn 11120	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
126	125	sincld 16089	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
127	126	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
128	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑁 ∈ ℕ₀)
129	127, 128	expcld 14100	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
130	56	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑁 ∈ ℂ)
131	59	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑁 − 1) ∈ ℕ₀)
132	127, 131	expcld 14100	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
133	130, 132	mulcld 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
134	125	coscld 16090	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
135	134	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
136	133, 135	mulcld 11157	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
137		sincl 16085	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
138	137	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
139	20	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℕ₀)
140	138, 139	expcld 14100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
141	140, 23	fmptd 7056	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
142	125	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
143		elex 3452	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V)
144	136, 143	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V)
145		rabid 3412	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V} ↔ (𝑥 ∈ ℂ ∧ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V))
146	142, 144, 145	sylanbrc 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V})
147	54	dmmpt 6192	. . . . . . . . . . . . 13 ⊢ dom 𝐻 = {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V}
148	146, 147	eleqtrrdi 2850	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ dom 𝐻)
149	148	ex 413	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
150	149	alrimiv 1934	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
151		nfcv 2901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ℝ
152		nfmpt1 5172	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
153	54, 152	nfcxfr 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐻
154	153	nfdm 5894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥dom 𝐻
155	151, 154	dfssf 3906	. . . . . . . . . 10 ⊢ (ℝ ⊆ dom 𝐻 ↔ ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
156	150, 155	sylibr 235	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ dom 𝐻)
157	19	dvsinexp 46362	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
158	23	oveq2i 7368	. . . . . . . . . . 11 ⊢ (ℂ D 𝐹) = (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)))
159	157, 158, 54	3eqtr4g 2799	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐻)
160	159	dmeqd 5848	. . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom 𝐻)
161	156, 160	sseqtrrd 3952	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
162		dvres3 25899	. . . . . . . 8 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ 𝐹:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D 𝐹))) → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
163	124, 141, 55, 161, 162	syl22anc 844	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
164	23	reseq1i 5928	. . . . . . . . . 10 ⊢ (𝐹 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ↾ ℝ)
165		resmpt 5990	. . . . . . . . . . 11 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁)))
166	13, 165	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))
167	164, 166	eqtri 2762	. . . . . . . . 9 ⊢ (𝐹 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))
168	167	oveq2i 7368	. . . . . . . 8 ⊢ (ℝ D (𝐹 ↾ ℝ)) = (ℝ D (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁)))
169	168	a1i 11	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = (ℝ D (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))))
170	159	reseq1d 5931	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝐻 ↾ ℝ))
171	54	reseq1i 5928	. . . . . . . . 9 ⊢ (𝐻 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ↾ ℝ)
172		resmpt 5990	. . . . . . . . . 10 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
173	13, 172	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
174	171, 173	eqtri 2762	. . . . . . . 8 ⊢ (𝐻 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
175	170, 174	eqtrdi 2790	. . . . . . 7 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
176	163, 169, 175	3eqtr3d 2782	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
177	12	a1i 11	. . . . . 6 ⊢ (𝜑 → (0[,]π) ⊆ ℝ)
178		tgioo4 24789	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
179		eqid 2739	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
180	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0 ∈ ℝ ∧ π ∈ ℝ))
181		iccntr 24806	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
182	180, 181	syl 17	. . . . . 6 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
183	124, 129, 136, 176, 177, 178, 179, 182	dvmptres2 25948	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
184	134	negcld 11484	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → -(cos‘𝑥) ∈ ℂ)
185	184	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
186	126	negcld 11484	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
187	186	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
188		dvcosre 46363	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))
189	188	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)))
190	124, 135, 187, 189	dvmptneg 25952	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
191	126	negnegd 11488	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
192	191	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → --(sin‘𝑥) = (sin‘𝑥))
193	192	mpteq2dva 5166	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
194	190, 193	eqtrd 2774	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
195	124, 185, 127, 194, 177, 178, 179, 182	dvmptres2 25948	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
196		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (sin‘𝑥) = (sin‘0))
197		sin0 16108	. . . . . . . . . . 11 ⊢ (sin‘0) = 0
198	196, 197	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (sin‘𝑥) = 0)
199	198	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
200	199	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 0) → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
201	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 0) → 𝑁 ∈ ℕ)
202	201	0expd 14093	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0↑𝑁) = 0)
203	200, 202	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → ((sin‘𝑥)↑𝑁) = 0)
204	203	oveq1d 7372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
205		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 0 → 𝑥 = 0)
206		0cn 11128	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
207	205, 206	eqeltrdi 2847	. . . . . . . . 9 ⊢ (𝑥 = 0 → 𝑥 ∈ ℂ)
208		coscl 16086	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
209	208	negcld 11484	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → -(cos‘𝑥) ∈ ℂ)
210	207, 209	syl 17	. . . . . . . 8 ⊢ (𝑥 = 0 → -(cos‘𝑥) ∈ ℂ)
211	210	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → -(cos‘𝑥) ∈ ℂ)
212	211	mul02d 11336	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0 · -(cos‘𝑥)) = 0)
213	204, 212	eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = 0)
214		fveq2 6828	. . . . . . . . . . 11 ⊢ (𝑥 = π → (sin‘𝑥) = (sin‘π))
215		sinpi 26439	. . . . . . . . . . 11 ⊢ (sin‘π) = 0
216	214, 215	eqtrdi 2790	. . . . . . . . . 10 ⊢ (𝑥 = π → (sin‘𝑥) = 0)
217	216	oveq1d 7372	. . . . . . . . 9 ⊢ (𝑥 = π → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
218	217	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = π) → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
219	19	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = π) → 𝑁 ∈ ℕ)
220	219	0expd 14093	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = π) → (0↑𝑁) = 0)
221	218, 220	eqtrd 2774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → ((sin‘𝑥)↑𝑁) = 0)
222	221	oveq1d 7372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
223		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = π → 𝑥 = π)
224		picn 26441	. . . . . . . . . . 11 ⊢ π ∈ ℂ
225	223, 224	eqeltrdi 2847	. . . . . . . . . 10 ⊢ (𝑥 = π → 𝑥 ∈ ℂ)
226	225	coscld 16090	. . . . . . . . 9 ⊢ (𝑥 = π → (cos‘𝑥) ∈ ℂ)
227	226	negcld 11484	. . . . . . . 8 ⊢ (𝑥 = π → -(cos‘𝑥) ∈ ℂ)
228	227	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → -(cos‘𝑥) ∈ ℂ)
229	228	mul02d 11336	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (0 · -(cos‘𝑥)) = 0)
230	222, 229	eqtrd 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = π) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = 0)
231	4, 6, 9, 37, 53, 80, 81, 104, 122, 183, 195, 213, 230	itgparts 26033	. . . 4 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = ((0 − 0) − ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥))
232		df-neg 11372	. . . . 5 ⊢ -∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥 = (0 − ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥)
233	232	a1i 11	. . . 4 ⊢ (𝜑 → -∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥 = (0 − ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥))
234	2, 231, 233	3eqtr4a 2800	. . 3 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = -∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥)
235	76, 78, 78	mulassd 11160	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · (cos‘𝑥)) = ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · ((cos‘𝑥) · (cos‘𝑥))))
236		sqval 14068	. . . . . . . . . . . . . 14 ⊢ ((cos‘𝑥) ∈ ℂ → ((cos‘𝑥)↑2) = ((cos‘𝑥) · (cos‘𝑥)))
237	236	eqcomd 2745	. . . . . . . . . . . . 13 ⊢ ((cos‘𝑥) ∈ ℂ → ((cos‘𝑥) · (cos‘𝑥)) = ((cos‘𝑥)↑2))
238	77, 237	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,)π) → ((cos‘𝑥) · (cos‘𝑥)) = ((cos‘𝑥)↑2))
239	238	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((cos‘𝑥) · (cos‘𝑥)) = ((cos‘𝑥)↑2))
240	239	oveq2d 7373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · ((cos‘𝑥) · (cos‘𝑥))) = ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · ((cos‘𝑥)↑2)))
241	77	sqcld 14098	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,)π) → ((cos‘𝑥)↑2) ∈ ℂ)
242	241	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((cos‘𝑥)↑2) ∈ ℂ)
243	70, 75, 242	mulassd 11160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · ((cos‘𝑥)↑2)) = (𝑁 · (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2))))
244	240, 243	eqtrd 2774	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · ((cos‘𝑥) · (cos‘𝑥))) = (𝑁 · (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2))))
245	75, 242	mulcomd 11158	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2)) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
246	245	oveq2d 7373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 · (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2))) = (𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))))
247	235, 244, 246	3eqtrd 2778	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · (cos‘𝑥)) = (𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))))
248	247	negeqd 11379	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → -(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · (cos‘𝑥)) = -(𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))))
249	79, 78	mulneg2d 11596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) = -(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · (cos‘𝑥)))
250	242, 75	mulcld 11157	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
251	70, 250	mulneg1d 11595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (-𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) = -(𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))))
252	248, 249, 251	3eqtr4d 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) = (-𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))))
253	252	itgeq2dv 25768	. . . . 5 ⊢ (𝜑 → ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥 = ∫(0(,)π)(-𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) d𝑥)
254	56	negcld 11484	. . . . . 6 ⊢ (𝜑 → -𝑁 ∈ ℂ)
255	38	sqcld 14098	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → ((cos‘𝑥)↑2) ∈ ℂ)
256	255	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((cos‘𝑥)↑2) ∈ ℂ)
257	256, 107	mulcld 11157	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
258		itgsinexplem1.6	. . . . . . . . . . . . 13 ⊢ 𝑀 = (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
259	258	fvmpt2 6948	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ℂ ∧ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ) → (𝑀‘𝑥) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
260	16, 257, 259	syl2anc 590	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑀‘𝑥) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
261	260	eqcomd 2745	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) = (𝑀‘𝑥))
262	261	mpteq2dva 5166	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) = (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)))
263		nfmpt1 5172	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
264	258, 263	nfcxfr 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑀
265		nfcv 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥cos
266		2nn0 12446	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
267	266	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℕ₀)
268	265, 49, 267	expcnfg 46044	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((cos‘𝑥)↑2)) ∈ (ℂ–cn→ℂ))
269	268, 60	mulcncf 25432	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
270	258, 269	eqeltrid 2843	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℂ–cn→ℂ))
271	264, 270, 35	cncfmptss 46040	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
272	262, 271	eqeltrd 2839	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ ((0[,]π)–cn→ℂ))
273		cniccibl 25827	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
274	4, 6, 272, 273	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
275	83, 85, 257, 274	iblss 25791	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
276	254, 250, 275	itgmulc2 25820	. . . . 5 ⊢ (𝜑 → (-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) = ∫(0(,)π)(-𝑁 · (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) d𝑥)
277	253, 276	eqtr4d 2777	. . . 4 ⊢ (𝜑 → ∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥 = (-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
278	277	negeqd 11379	. . 3 ⊢ (𝜑 → -∫(0(,)π)(((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥)) d𝑥 = -(-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
279	234, 278	eqtrd 2774	. 2 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = -(-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
280	250, 275	itgcl 25770	. . . 4 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥 ∈ ℂ)
281	56, 280	mulneg1d 11595	. . 3 ⊢ (𝜑 → (-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) = -(𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
282	281	negeqd 11379	. 2 ⊢ (𝜑 → -(-𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) = --(𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
283	56, 280	mulcld 11157	. . 3 ⊢ (𝜑 → (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) ∈ ℂ)
284	283	negnegd 11488	. 2 ⊢ (𝜑 → --(𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))
285	279, 282, 284	3eqtrd 2778	1 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑𝑁) · (sin‘𝑥)) d𝑥 = (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∀wal 1545 = wceq 1547 ∈ wcel 2119 {crab 3391 Vcvv 3431 ⊆ wss 3883 {cpr 4558 class class class wbr 5073 ↦ cmpt 5154 dom cdm 5619 ran crn 5620 ↾ cres 5621 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 · cmul 11035 ≤ cle 11172 − cmin 11369 -cneg 11370 ℕcn 12166 2c2 12228 ℕ₀cn0 12429 (,)cioo 13290 [,]cicc 13293 ↑cexp 14015 sincsin 16020 cosccos 16021 πcpi 16023 TopOpenctopn 17376 topGenctg 17392 ℂ_fldccnfld 21348 intcnt 23001 –cn→ccncf 24862 volcvol 25449 𝐿¹cibl 25603 ∫citg 25604 D cdv 25849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-inf2 9554 ax-cc 10349 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-symdif 4182 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-disj 5041 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 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 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-ofr 7622 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9817 df-card 9855 df-acn 9858 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-ioo 13294 df-ioc 13295 df-ico 13296 df-icc 13297 df-fz 13454 df-fzo 13601 df-fl 13743 df-mod 13821 df-seq 13956 df-exp 14016 df-fac 14228 df-bc 14257 df-hash 14285 df-shft 15021 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-limsup 15425 df-clim 15442 df-rlim 15443 df-sum 15641 df-ef 16024 df-sin 16026 df-cos 16027 df-pi 16029 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-cmp 23371 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-ovol 25450 df-vol 25451 df-mbf 25605 df-itg1 25606 df-itg2 25607 df-ibl 25608 df-itg 25609 df-0p 25656 df-limc 25852 df-dv 25853

This theorem is referenced by: itgsinexp 46406

W3C validator