itgsinexplem1 - Mathbox for Glauco Siliprandi

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Theorem itgsinexplem1 45959

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 12308	. . . . 5 ⊢ (0 − 0) = 0
2	1	oveq1i 7400	. . . 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 11183	. . . . . 6 ⊢ 0 ∈ ℝ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
5		pire 26373	. . . . . 6 ⊢ π ∈ ℝ
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → π ∈ ℝ)
7		pipos 26375	. . . . . . 7 ⊢ 0 < π
8	3, 5, 7	ltleii 11304	. . . . . 6 ⊢ 0 ≤ π
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ π)
10	3, 5	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ ∧ π ∈ ℝ)
11		iccssre 13397	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13		ax-resscn 11132	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14	12, 13	sstri 3959	. . . . . . . . . . 11 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3945	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
17	15	sincld 16105	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
18	17	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
19		itgsinexplem1.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
20	19	nnnn0d 12510	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
22	18, 21	expcld 14118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
23		itgsinexplem1.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
24	23	fvmpt2 6982	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → (𝐹‘𝑥) = ((sin‘𝑥)↑𝑁))
25	16, 22, 24	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝐹‘𝑥) = ((sin‘𝑥)↑𝑁))
26	25	eqcomd 2736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = (𝐹‘𝑥))
27	26	mpteq2dva 5203	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)))
28		nfmpt1 5209	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
29	23, 28	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
30		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥sin
31		sincn 26361	. . . . . . . . . 10 ⊢ sin ∈ (ℂ–cn→ℂ)
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
33	30, 32, 20	expcnfg 45596	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
34	23, 33	eqeltrid 2833	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
35	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
36	29, 34, 35	cncfmptss 45592	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
37	27, 36	eqeltrd 2829	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
38	15	coscld 16106	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
39	38	negcld 11527	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
40		itgsinexplem1.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
41	40	fvmpt2 6982	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → (𝐺‘𝑥) = -(cos‘𝑥))
42	15, 39, 41	syl2anc 584	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → (𝐺‘𝑥) = -(cos‘𝑥))
43	42	eqcomd 2736	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = (𝐺‘𝑥))
44	43	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) = (𝐺‘𝑥))
45	44	mpteq2dva 5203	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)))
46		nfmpt1 5209	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
47	40, 46	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑥𝐺
48		coscn 26362	. . . . . . . . 9 ⊢ cos ∈ (ℂ–cn→ℂ)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
50	40	negfcncf 24824	. . . . . . . 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 45592	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
53	45, 52	eqeltrd 2829	. . . . 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 3973	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
56	19	nncnd 12209	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
57	55, 56, 55	constcncfg 45877	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑁) ∈ (ℂ–cn→ℂ))
58		nnm1nn0 12490	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
59	19, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
60	30, 32, 59	expcnfg 45596	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) ∈ (ℂ–cn→ℂ))
61	57, 60	mulcncf 25353	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑁 · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
62		cosf 16100	. . . . . . . . . . 11 ⊢ cos:ℂ⟶ℂ
63	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → cos:ℂ⟶ℂ)
64	63	feqmptd 6932	. . . . . . . . 9 ⊢ (𝜑 → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
65	64, 48	eqeltrrdi 2838	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (cos‘𝑥)) ∈ (ℂ–cn→ℂ))
66	61, 65	mulcncf 25353	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ (ℂ–cn→ℂ))
67	54, 66	eqeltrid 2833	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (ℂ–cn→ℂ))
68		ioosscn 13376	. . . . . . 7 ⊢ (0(,)π) ⊆ ℂ
69	68	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ⊆ ℂ)
70	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℂ)
71	68	sseli 3945	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
72	71	sincld 16105	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
73	72	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
74	59	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 1) ∈ ℕ₀)
75	73, 74	expcld 14118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
76	70, 75	mulcld 11201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
77	71	coscld 16106	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (cos‘𝑥) ∈ ℂ)
78	77	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (cos‘𝑥) ∈ ℂ)
79	76, 78	mulcld 11201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
80	54, 67, 69, 55, 79	cncfmptssg 45876	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ ((0(,)π)–cn→ℂ))
81	30, 32, 69	cncfmptss 45592	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
82		ioossicc 13401	. . . . . . 7 ⊢ (0(,)π) ⊆ (0[,]π)
83	82	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
84		ioombl 25473	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
85	84	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
86	22, 18	mulcld 11201	. . . . . 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 6982	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) ∈ ℂ) → (𝐼‘𝑥) = (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
89	16, 86, 88	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝐼‘𝑥) = (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
90	89	eqcomd 2736	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) = (𝐼‘𝑥))
91	90	mpteq2dva 5203	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) = (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)))
92		nfmpt1 5209	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
93	87, 92	nfcxfr 2890	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐼
94		sinf 16099	. . . . . . . . . . . . . 14 ⊢ sin:ℂ⟶ℂ
95	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → sin:ℂ⟶ℂ)
96	95	feqmptd 6932	. . . . . . . . . . . 12 ⊢ (𝜑 → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
97	96, 31	eqeltrrdi 2838	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (sin‘𝑥)) ∈ (ℂ–cn→ℂ))
98	33, 97	mulcncf 25353	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ (ℂ–cn→ℂ))
99	87, 98	eqeltrid 2833	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (ℂ–cn→ℂ))
100	93, 99, 35	cncfmptss 45592	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
101	91, 100	eqeltrd 2829	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
102		cniccibl 25749	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ 𝐿¹)
104	83, 85, 86, 103	iblss 25713	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ 𝐿¹)
105	56	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℂ)
106	59	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 1) ∈ ℕ₀)
107	18, 106	expcld 14118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
108	105, 107	mulcld 11201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
109	38	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
110	108, 109	mulcld 11201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
111	39	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
112	110, 111	mulcld 11201	. . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
115	114	negfcncf 24824	. . . . . . . . . . 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 25353	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ (ℂ–cn→ℂ))
118	113, 117	eqeltrid 2833	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℂ–cn→ℂ))
119	113, 118, 35, 55, 112	cncfmptssg 45876	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
120		cniccibl 25749	. . . . . . 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 1373	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
122	83, 85, 112, 121	iblss 25713	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
123		reelprrecn 11167	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
124	123	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
125		recn 11165	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
126	125	sincld 16105	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
127	126	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
128	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑁 ∈ ℕ₀)
129	127, 128	expcld 14118	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
130	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑁 ∈ ℂ)
131	59	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑁 − 1) ∈ ℕ₀)
132	127, 131	expcld 14118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
133	130, 132	mulcld 11201	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
134	125	coscld 16106	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
135	134	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
136	133, 135	mulcld 11201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
137		sincl 16101	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
138	137	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (sin‘𝑥) ∈ ℂ)
139	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℕ₀)
140	138, 139	expcld 14118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
141	140, 23	fmptd 7089	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
142	125	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
143		elex 3471	. . . . . . . . . . . . . . 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 3430	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V})
147	54	dmmpt 6216	. . . . . . . . . . . . 13 ⊢ dom 𝐻 = {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V}
148	146, 147	eleqtrrdi 2840	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ dom 𝐻)
149	148	ex 412	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
150	149	alrimiv 1927	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
151		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ℝ
152		nfmpt1 5209	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
153	54, 152	nfcxfr 2890	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐻
154	153	nfdm 5918	. . . . . . . . . . 11 ⊢ Ⅎ𝑥dom 𝐻
155	151, 154	dfssf 3940	. . . . . . . . . 10 ⊢ (ℝ ⊆ dom 𝐻 ↔ ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
156	150, 155	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ dom 𝐻)
157	19	dvsinexp 45916	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
158	23	oveq2i 7401	. . . . . . . . . . 11 ⊢ (ℂ D 𝐹) = (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)))
159	157, 158, 54	3eqtr4g 2790	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐻)
160	159	dmeqd 5872	. . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom 𝐻)
161	156, 160	sseqtrrd 3987	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
162		dvres3 25821	. . . . . . . 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 838	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝐹 ↾ ℝ)) = ((ℂ D 𝐹) ↾ ℝ))
164	23	reseq1i 5949	. . . . . . . . . 10 ⊢ (𝐹 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ↾ ℝ)
165		resmpt 6011	. . . . . . . . . . 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 2753	. . . . . . . . 9 ⊢ (𝐹 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))
168	167	oveq2i 7401	. . . . . . . 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 5952	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝐻 ↾ ℝ))
171	54	reseq1i 5949	. . . . . . . . 9 ⊢ (𝐻 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ↾ ℝ)
172		resmpt 6011	. . . . . . . . . 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 2753	. . . . . . . 8 ⊢ (𝐻 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
175	170, 174	eqtrdi 2781	. . . . . . 7 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
176	163, 169, 175	3eqtr3d 2773	. . . . . 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 24700	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
179		eqid 2730	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
180	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0 ∈ ℝ ∧ π ∈ ℝ))
181		iccntr 24717	. . . . . . 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 25873	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
184	134	negcld 11527	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → -(cos‘𝑥) ∈ ℂ)
185	184	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
186	126	negcld 11527	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
187	186	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
188		dvcosre 45917	. . . . . . . . 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 25877	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
191	126	negnegd 11531	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
192	191	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → --(sin‘𝑥) = (sin‘𝑥))
193	192	mpteq2dva 5203	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
194	190, 193	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
195	124, 185, 127, 194, 177, 178, 179, 182	dvmptres2 25873	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
196		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (sin‘𝑥) = (sin‘0))
197		sin0 16124	. . . . . . . . . . 11 ⊢ (sin‘0) = 0
198	196, 197	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (sin‘𝑥) = 0)
199	198	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
200	199	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 0) → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
201	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 0) → 𝑁 ∈ ℕ)
202	201	0expd 14111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0↑𝑁) = 0)
203	200, 202	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → ((sin‘𝑥)↑𝑁) = 0)
204	203	oveq1d 7405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
205		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 0 → 𝑥 = 0)
206		0cn 11173	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
207	205, 206	eqeltrdi 2837	. . . . . . . . 9 ⊢ (𝑥 = 0 → 𝑥 ∈ ℂ)
208		coscl 16102	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
209	208	negcld 11527	. . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → -(cos‘𝑥) ∈ ℂ)
210	207, 209	syl 17	. . . . . . . 8 ⊢ (𝑥 = 0 → -(cos‘𝑥) ∈ ℂ)
211	210	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → -(cos‘𝑥) ∈ ℂ)
212	211	mul02d 11379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0 · -(cos‘𝑥)) = 0)
213	204, 212	eqtrd 2765	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = 0)
214		fveq2 6861	. . . . . . . . . . 11 ⊢ (𝑥 = π → (sin‘𝑥) = (sin‘π))
215		sinpi 26372	. . . . . . . . . . 11 ⊢ (sin‘π) = 0
216	214, 215	eqtrdi 2781	. . . . . . . . . 10 ⊢ (𝑥 = π → (sin‘𝑥) = 0)
217	216	oveq1d 7405	. . . . . . . . 9 ⊢ (𝑥 = π → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
218	217	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = π) → ((sin‘𝑥)↑𝑁) = (0↑𝑁))
219	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = π) → 𝑁 ∈ ℕ)
220	219	0expd 14111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = π) → (0↑𝑁) = 0)
221	218, 220	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → ((sin‘𝑥)↑𝑁) = 0)
222	221	oveq1d 7405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
223		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = π → 𝑥 = π)
224		picn 26374	. . . . . . . . . . 11 ⊢ π ∈ ℂ
225	223, 224	eqeltrdi 2837	. . . . . . . . . 10 ⊢ (𝑥 = π → 𝑥 ∈ ℂ)
226	225	coscld 16106	. . . . . . . . 9 ⊢ (𝑥 = π → (cos‘𝑥) ∈ ℂ)
227	226	negcld 11527	. . . . . . . 8 ⊢ (𝑥 = π → -(cos‘𝑥) ∈ ℂ)
228	227	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → -(cos‘𝑥) ∈ ℂ)
229	228	mul02d 11379	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (0 · -(cos‘𝑥)) = 0)
230	222, 229	eqtrd 2765	. . . . 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 25961	. . . 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 11415	. . . . 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 2791	. . 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 11204	. . . . . . . . 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 14086	. . . . . . . . . . . . . 14 ⊢ ((cos‘𝑥) ∈ ℂ → ((cos‘𝑥)↑2) = ((cos‘𝑥) · (cos‘𝑥)))
237	236	eqcomd 2736	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((cos‘𝑥) · (cos‘𝑥)) = ((cos‘𝑥)↑2))
240	239	oveq2d 7406	. . . . . . . . . 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 14116	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0(,)π) → ((cos‘𝑥)↑2) ∈ ℂ)
242	241	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((cos‘𝑥)↑2) ∈ ℂ)
243	70, 75, 242	mulassd 11204	. . . . . . . . . 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 2765	. . . . . . . . 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 11202	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2)) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
246	245	oveq2d 7406	. . . . . . . . 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 2769	. . . . . . . 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 11422	. . . . . . 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 11639	. . . . . . 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 11201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
251	70, 250	mulneg1d 11638	. . . . . . 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 2775	. . . . . 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 25690	. . . . 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 11527	. . . . . 6 ⊢ (𝜑 → -𝑁 ∈ ℂ)
255	38	sqcld 14116	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → ((cos‘𝑥)↑2) ∈ ℂ)
256	255	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((cos‘𝑥)↑2) ∈ ℂ)
257	256, 107	mulcld 11201	. . . . . . 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 6982	. . . . . . . . . . . 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 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑀‘𝑥) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
261	260	eqcomd 2736	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) = (𝑀‘𝑥))
262	261	mpteq2dva 5203	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) = (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)))
263		nfmpt1 5209	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
264	258, 263	nfcxfr 2890	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑀
265		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥cos
266		2nn0 12466	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
267	266	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℕ₀)
268	265, 49, 267	expcnfg 45596	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((cos‘𝑥)↑2)) ∈ (ℂ–cn→ℂ))
269	268, 60	mulcncf 25353	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
270	258, 269	eqeltrid 2833	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℂ–cn→ℂ))
271	264, 270, 35	cncfmptss 45592	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
272	262, 271	eqeltrd 2829	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ ((0[,]π)–cn→ℂ))
273		cniccibl 25749	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
275	83, 85, 257, 274	iblss 25713	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
276	254, 250, 275	itgmulc2 25742	. . . . 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 2768	. . . 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 11422	. . 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 2765	. 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 25692	. . . 4 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥 ∈ ℂ)
281	56, 280	mulneg1d 11638	. . 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 11422	. 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 11201	. . 3 ⊢ (𝜑 → (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) ∈ ℂ)
284	283	negnegd 11531	. 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 2769	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 395 ∀wal 1538 = wceq 1540 ∈ wcel 2109 {crab 3408 Vcvv 3450 ⊆ wss 3917 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ran crn 5642 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 · cmul 11080 ≤ cle 11216 − cmin 11412 -cneg 11413 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 (,)cioo 13313 [,]cicc 13316 ↑cexp 14033 sincsin 16036 cosccos 16037 πcpi 16039 TopOpenctopn 17391 topGenctg 17407 ℂ_fldccnfld 21271 intcnt 22911 –cn→ccncf 24776 volcvol 25371 𝐿¹cibl 25525 ∫citg 25526 D cdv 25771

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cc 10395 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-symdif 4219 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-ofr 7657 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-omul 8442 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-dju 9861 df-card 9899 df-acn 9902 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-cmp 23281 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-ovol 25372 df-vol 25373 df-mbf 25527 df-itg1 25528 df-itg2 25529 df-ibl 25530 df-itg 25531 df-0p 25578 df-limc 25774 df-dv 25775

This theorem is referenced by: itgsinexp 45960

W3C validator