itgsinexplem1 - Mathbox for Glauco Siliprandi

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Theorem itgsinexplem1 45945

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 12243	. . . . 5 ⊢ (0 − 0) = 0
2	1	oveq1i 7359	. . . 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 11117	. . . . . 6 ⊢ 0 ∈ ℝ
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
5		pire 26364	. . . . . 6 ⊢ π ∈ ℝ
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → π ∈ ℝ)
7		pipos 26366	. . . . . . 7 ⊢ 0 < π
8	3, 5, 7	ltleii 11239	. . . . . 6 ⊢ 0 ≤ π
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → 0 ≤ π)
10	3, 5	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ (0 ∈ ℝ ∧ π ∈ ℝ)
11		iccssre 13332	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	10, 11	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13		ax-resscn 11066	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
14	12, 13	sstri 3945	. . . . . . . . . . 11 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3931	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
17	15	sincld 16039	. . . . . . . . . . 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 12445	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
21	20	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
22	18, 21	expcld 14053	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
23		itgsinexplem1.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
24	23	fvmpt2 6941	. . . . . . . . 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 2735	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = (𝐹‘𝑥))
27	26	mpteq2dva 5185	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)))
28		nfmpt1 5191	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
29	23, 28	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
30		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑥sin
31		sincn 26352	. . . . . . . . . 10 ⊢ sin ∈ (ℂ–cn→ℂ)
32	31	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
33	30, 32, 20	expcnfg 45582	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
34	23, 33	eqeltrid 2832	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (ℂ–cn→ℂ))
35	14	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
36	29, 34, 35	cncfmptss 45578	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐹‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
37	27, 36	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
38	15	coscld 16040	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
39	38	negcld 11462	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
40		itgsinexplem1.2	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
41	40	fvmpt2 6941	. . . . . . . . . 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 2735	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = (𝐺‘𝑥))
44	43	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) = (𝐺‘𝑥))
45	44	mpteq2dva 5185	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)))
46		nfmpt1 5191	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
47	40, 46	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑥𝐺
48		coscn 26353	. . . . . . . . 9 ⊢ cos ∈ (ℂ–cn→ℂ)
49	48	a1i 11	. . . . . . . 8 ⊢ (𝜑 → cos ∈ (ℂ–cn→ℂ))
50	40	negfcncf 24815	. . . . . . . 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 45578	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐺‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
53	45, 52	eqeltrd 2828	. . . . 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 3959	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ⊆ ℂ)
56	19	nncnd 12144	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℂ)
57	55, 56, 55	constcncfg 45863	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑁) ∈ (ℂ–cn→ℂ))
58		nnm1nn0 12425	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
59	19, 58	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
60	30, 32, 59	expcnfg 45582	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) ∈ (ℂ–cn→ℂ))
61	57, 60	mulcncf 25344	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑁 · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
62		cosf 16034	. . . . . . . . . . 11 ⊢ cos:ℂ⟶ℂ
63	62	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → cos:ℂ⟶ℂ)
64	63	feqmptd 6891	. . . . . . . . 9 ⊢ (𝜑 → cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)))
65	64, 48	eqeltrrdi 2837	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (cos‘𝑥)) ∈ (ℂ–cn→ℂ))
66	61, 65	mulcncf 25344	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ (ℂ–cn→ℂ))
67	54, 66	eqeltrid 2832	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ (ℂ–cn→ℂ))
68		ioosscn 13311	. . . . . . 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 3931	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
72	71	sincld 16039	. . . . . . . . . 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 14053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
76	70, 75	mulcld 11135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
77	71	coscld 16040	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (cos‘𝑥) ∈ ℂ)
78	77	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (cos‘𝑥) ∈ ℂ)
79	76, 78	mulcld 11135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
80	54, 67, 69, 55, 79	cncfmptssg 45862	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ∈ ((0(,)π)–cn→ℂ))
81	30, 32, 69	cncfmptss 45578	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
82		ioossicc 13336	. . . . . . 7 ⊢ (0(,)π) ⊆ (0[,]π)
83	82	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
84		ioombl 25464	. . . . . . 7 ⊢ (0(,)π) ∈ dom vol
85	84	a1i 11	. . . . . 6 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
86	22, 18	mulcld 11135	. . . . . 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 6941	. . . . . . . . . . 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 2735	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((sin‘𝑥)↑𝑁) · (sin‘𝑥)) = (𝐼‘𝑥))
91	90	mpteq2dva 5185	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) = (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)))
92		nfmpt1 5191	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥)))
93	87, 92	nfcxfr 2889	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐼
94		sinf 16033	. . . . . . . . . . . . . 14 ⊢ sin:ℂ⟶ℂ
95	94	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → sin:ℂ⟶ℂ)
96	95	feqmptd 6891	. . . . . . . . . . . 12 ⊢ (𝜑 → sin = (𝑥 ∈ ℂ ↦ (sin‘𝑥)))
97	96, 31	eqeltrrdi 2837	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (sin‘𝑥)) ∈ (ℂ–cn→ℂ))
98	33, 97	mulcncf 25344	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ (ℂ–cn→ℂ))
99	87, 98	eqeltrid 2832	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (ℂ–cn→ℂ))
100	93, 99, 35	cncfmptss 45578	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝐼‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
101	91, 100	eqeltrd 2828	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((sin‘𝑥)↑𝑁) · (sin‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
102		cniccibl 25740	. . . . . . 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 25704	. . . . 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 14053	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
108	105, 107	mulcld 11135	. . . . . . . 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 11135	. . . . . . 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 11135	. . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
115	114	negfcncf 24815	. . . . . . . . . . 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 25344	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ (ℂ–cn→ℂ))
118	113, 117	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ (ℂ–cn→ℂ))
119	113, 118, 35, 55, 112	cncfmptssg 45862	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ ((0[,]π)–cn→ℂ))
120		cniccibl 25740	. . . . . . 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 25704	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) ∈ 𝐿¹)
123		reelprrecn 11101	. . . . . . 7 ⊢ ℝ ∈ {ℝ, ℂ}
124	123	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
125		recn 11099	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
126	125	sincld 16039	. . . . . . . 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 14053	. . . . . 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 14053	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((sin‘𝑥)↑(𝑁 − 1)) ∈ ℂ)
133	130, 132	mulcld 11135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
134	125	coscld 16040	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
135	134	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
136	133, 135	mulcld 11135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ ℂ)
137		sincl 16035	. . . . . . . . . . 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 14053	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
141	140, 23	fmptd 7048	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ)
142	125	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
143		elex 3457	. . . . . . . . . . . . . . 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 3416	. . . . . . . . . . . . . 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 6189	. . . . . . . . . . . . 13 ⊢ dom 𝐻 = {𝑥 ∈ ℂ ∣ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)) ∈ V}
148	146, 147	eleqtrrdi 2839	. . . . . . . . . . . 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 2891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥ℝ
152		nfmpt1 5191	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
153	54, 152	nfcxfr 2889	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝐻
154	153	nfdm 5893	. . . . . . . . . . 11 ⊢ Ⅎ𝑥dom 𝐻
155	151, 154	dfssf 3926	. . . . . . . . . 10 ⊢ (ℝ ⊆ dom 𝐻 ↔ ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ dom 𝐻))
156	150, 155	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ℝ ⊆ dom 𝐻)
157	19	dvsinexp 45902	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
158	23	oveq2i 7360	. . . . . . . . . . 11 ⊢ (ℂ D 𝐹) = (ℂ D (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)))
159	157, 158, 54	3eqtr4g 2789	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D 𝐹) = 𝐻)
160	159	dmeqd 5848	. . . . . . . . 9 ⊢ (𝜑 → dom (ℂ D 𝐹) = dom 𝐻)
161	156, 160	sseqtrrd 3973	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ dom (ℂ D 𝐹))
162		dvres3 25812	. . . . . . . 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 5926	. . . . . . . . . 10 ⊢ (𝐹 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ↾ ℝ)
165		resmpt 5988	. . . . . . . . . . 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 2752	. . . . . . . . 9 ⊢ (𝐹 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((sin‘𝑥)↑𝑁))
168	167	oveq2i 7360	. . . . . . . 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 5929	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝐻 ↾ ℝ))
171	54	reseq1i 5926	. . . . . . . . 9 ⊢ (𝐻 ↾ ℝ) = ((𝑥 ∈ ℂ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))) ↾ ℝ)
172		resmpt 5988	. . . . . . . . . 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 2752	. . . . . . . 8 ⊢ (𝐻 ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥)))
175	170, 174	eqtrdi 2780	. . . . . . 7 ⊢ (𝜑 → ((ℂ D 𝐹) ↾ ℝ) = (𝑥 ∈ ℝ ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
176	163, 169, 175	3eqtr3d 2772	. . . . . 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 24691	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
179		eqid 2729	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
180	10	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0 ∈ ℝ ∧ π ∈ ℝ))
181		iccntr 24708	. . . . . . 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 25864	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁))) = (𝑥 ∈ (0(,)π) ↦ ((𝑁 · ((sin‘𝑥)↑(𝑁 − 1))) · (cos‘𝑥))))
184	134	negcld 11462	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → -(cos‘𝑥) ∈ ℂ)
185	184	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
186	126	negcld 11462	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
187	186	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
188		dvcosre 45903	. . . . . . . . 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 25868	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
191	126	negnegd 11466	. . . . . . . . 9 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
192	191	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → --(sin‘𝑥) = (sin‘𝑥))
193	192	mpteq2dva 5185	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
194	190, 193	eqtrd 2764	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
195	124, 185, 127, 194, 177, 178, 179, 182	dvmptres2 25864	. . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
196		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑥 = 0 → (sin‘𝑥) = (sin‘0))
197		sin0 16058	. . . . . . . . . . 11 ⊢ (sin‘0) = 0
198	196, 197	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑥 = 0 → (sin‘𝑥) = 0)
199	198	oveq1d 7364	. . . . . . . . 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 14046	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0↑𝑁) = 0)
203	200, 202	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 0) → ((sin‘𝑥)↑𝑁) = 0)
204	203	oveq1d 7364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
205		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 0 → 𝑥 = 0)
206		0cn 11107	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
207	205, 206	eqeltrdi 2836	. . . . . . . . 9 ⊢ (𝑥 = 0 → 𝑥 ∈ ℂ)
208		coscl 16036	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
209	208	negcld 11462	. . . . . . . . 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 11314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 0) → (0 · -(cos‘𝑥)) = 0)
213	204, 212	eqtrd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 0) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = 0)
214		fveq2 6822	. . . . . . . . . . 11 ⊢ (𝑥 = π → (sin‘𝑥) = (sin‘π))
215		sinpi 26363	. . . . . . . . . . 11 ⊢ (sin‘π) = 0
216	214, 215	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑥 = π → (sin‘𝑥) = 0)
217	216	oveq1d 7364	. . . . . . . . 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 14046	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = π) → (0↑𝑁) = 0)
221	218, 220	eqtrd 2764	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → ((sin‘𝑥)↑𝑁) = 0)
222	221	oveq1d 7364	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (((sin‘𝑥)↑𝑁) · -(cos‘𝑥)) = (0 · -(cos‘𝑥)))
223		id 22	. . . . . . . . . . 11 ⊢ (𝑥 = π → 𝑥 = π)
224		picn 26365	. . . . . . . . . . 11 ⊢ π ∈ ℂ
225	223, 224	eqeltrdi 2836	. . . . . . . . . 10 ⊢ (𝑥 = π → 𝑥 ∈ ℂ)
226	225	coscld 16040	. . . . . . . . 9 ⊢ (𝑥 = π → (cos‘𝑥) ∈ ℂ)
227	226	negcld 11462	. . . . . . . 8 ⊢ (𝑥 = π → -(cos‘𝑥) ∈ ℂ)
228	227	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = π) → -(cos‘𝑥) ∈ ℂ)
229	228	mul02d 11314	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = π) → (0 · -(cos‘𝑥)) = 0)
230	222, 229	eqtrd 2764	. . . . 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 25952	. . . 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 11350	. . . . 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 2790	. . 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 11138	. . . . . . . . 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 14021	. . . . . . . . . . . . . 14 ⊢ ((cos‘𝑥) ∈ ℂ → ((cos‘𝑥)↑2) = ((cos‘𝑥) · (cos‘𝑥)))
237	236	eqcomd 2735	. . . . . . . . . . . . 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 7365	. . . . . . . . . 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 14051	. . . . . . . . . . . 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 11138	. . . . . . . . . 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 2764	. . . . . . . . 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 11136	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑(𝑁 − 1)) · ((cos‘𝑥)↑2)) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
246	245	oveq2d 7365	. . . . . . . . 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 2768	. . . . . . . 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 11357	. . . . . . 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 11574	. . . . . . 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 11135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) ∈ ℂ)
251	70, 250	mulneg1d 11573	. . . . . . 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 2774	. . . . . 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 25681	. . . . 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 11462	. . . . . 6 ⊢ (𝜑 → -𝑁 ∈ ℂ)
255	38	sqcld 14051	. . . . . . . . 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 11135	. . . . . . 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 6941	. . . . . . . . . . . 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 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) = (𝑀‘𝑥))
262	261	mpteq2dva 5185	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) = (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)))
263		nfmpt1 5191	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))))
264	258, 263	nfcxfr 2889	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑀
265		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥cos
266		2nn0 12401	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℕ₀
267	266	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℕ₀)
268	265, 49, 267	expcnfg 45582	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((cos‘𝑥)↑2)) ∈ (ℂ–cn→ℂ))
269	268, 60	mulcncf 25344	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ (ℂ–cn→ℂ))
270	258, 269	eqeltrid 2832	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (ℂ–cn→ℂ))
271	264, 270, 35	cncfmptss 45578	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (𝑀‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
272	262, 271	eqeltrd 2828	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ ((0[,]π)–cn→ℂ))
273		cniccibl 25740	. . . . . . . 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 25704	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1)))) ∈ 𝐿¹)
276	254, 250, 275	itgmulc2 25733	. . . . 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 2767	. . . 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 11357	. . 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 2764	. 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 25683	. . . 4 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥 ∈ ℂ)
281	56, 280	mulneg1d 11573	. . 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 11357	. 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 11135	. . 3 ⊢ (𝜑 → (𝑁 · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 1))) d𝑥) ∈ ℂ)
284	283	negnegd 11466	. 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 2768	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 3394 Vcvv 3436 ⊆ wss 3903 {cpr 4579 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ↾ cres 5621 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 ≤ cle 11150 − cmin 11347 -cneg 11348 ℕcn 12128 2c2 12183 ℕ₀cn0 12384 (,)cioo 13248 [,]cicc 13251 ↑cexp 13968 sincsin 15970 cosccos 15971 πcpi 15973 TopOpenctopn 17325 topGenctg 17341 ℂ_fldccnfld 21261 intcnt 22902 –cn→ccncf 24767 volcvol 25362 𝐿¹cibl 25516 ∫citg 25517 D cdv 25762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4204 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 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 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-limc 25765 df-dv 25766

This theorem is referenced by: itgsinexp 45946

W3C validator