itgsinexp - Mathbox for Glauco Siliprandi

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Theorem itgsinexp 44186

Description: A recursive formula for the integral of sin^N on the interval (0,π) . (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypotheses**
Ref	Expression
itgsinexp.1	⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)
itgsinexp.2	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))

**Assertion**
Ref	Expression
itgsinexp	⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Distinct variable groups: 𝑥,𝑛,𝑁 𝜑,𝑛,𝑥 Allowed substitution hints: 𝐼(𝑥,𝑛)

**Proof of Theorem itgsinexp**
Step	Hyp	Ref	Expression
1		itgsinexp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘2))
2		eluzelz 12773	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		zcn 12504	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		1cnd 11150	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
6	4, 5	npcand 11516	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7	6	eqcomd 2742	. . . . 5 ⊢ (𝜑 → 𝑁 = ((𝑁 − 1) + 1))
8	7	oveq1d 7372	. . . 4 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = (((𝑁 − 1) + 1) · (𝐼‘𝑁)))
9		uz2m1nn 12848	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ)
11	10	nncnd 12169	. . . . 5 ⊢ (𝜑 → (𝑁 − 1) ∈ ℂ)
12		itgsinexp.1	. . . . . . . 8 ⊢ 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐼 = (𝑛 ∈ ℕ₀ ↦ ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥))
14		oveq2 7365	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
15	14	ad2antlr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
16	15	itgeq2dv 25146	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
17		2cnd 12231	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
18		npcan 11410	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑁 − 2) + 2) = 𝑁)
19	18	eqcomd 2742	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → 𝑁 = ((𝑁 − 2) + 2))
20	4, 17, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = ((𝑁 − 2) + 2))
21		uznn0sub 12802	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ ℕ₀)
22	1, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
23		2nn0 12430	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ₀)
25	22, 24	nn0addcld 12477	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 2) + 2) ∈ ℕ₀)
26	20, 25	eqeltrd 2838	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
27		itgex 25135	. . . . . . . 8 ⊢ ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ V)
29	13, 16, 26, 28	fvmptd 6955	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
30		ioosscn 13326	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
31	30	sseli 3940	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
32	31	sincld 16012	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
33	32	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
34	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
35	33, 34	expcld 14051	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
36		ioossicc 13350	. . . . . . . . 9 ⊢ (0(,)π) ⊆ (0[,]π)
37	36	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
38		ioombl 24929	. . . . . . . . 9 ⊢ (0(,)π) ∈ dom vol
39	38	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
40		0re 11157	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
41		pire 25815	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
42		iccssre 13346	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
43	40, 41, 42	mp2an 690	. . . . . . . . . . . . 13 ⊢ (0[,]π) ⊆ ℝ
44		ax-resscn 11108	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
45	43, 44	sstri 3953	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℂ
46	45	sseli 3940	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
47	46	sincld 16012	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
48	47	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
49	26	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
50	48, 49	expcld 14051	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
51	40	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ ℝ)
52	41	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → π ∈ ℝ)
53	46	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
54		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
55	54	fvmpt2 6959	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
56	53, 50, 55	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
57	56	eqcomd 2742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥))
58	57	mpteq2dva 5205	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)))
59		nfmpt1 5213	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
60		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥sin
61		sincn 25803	. . . . . . . . . . . . 13 ⊢ sin ∈ (ℂ–cn→ℂ)
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
63	60, 62, 26	expcnfg 43822	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
64	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
65	59, 63, 64	cncfmptss 43818	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
66	58, 65	eqeltrd 2838	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
67		cniccibl 25205	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
68	51, 52, 66, 67	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
69	37, 39, 50, 68	iblss 25169	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
70	35, 69	itgcl 25148	. . . . . 6 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ ℂ)
71	29, 70	eqeltrd 2838	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑁) ∈ ℂ)
72	11, 71	adddirp1d 11181	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) + 1) · (𝐼‘𝑁)) = (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)))
73		eluz2b2 12846	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
74	1, 73	sylib 217	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
75	74	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
76		expm1t 13996	. . . . . . . . . 10 ⊢ (((sin‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
77	32, 75, 76	syl2anr 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
78	77	itgeq2dv 25146	. . . . . . . 8 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 = ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥)
79		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1)))
80		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
81		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((𝑁 − 1) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) · (cos‘𝑥))) = (𝑥 ∈ ℂ ↦ (((𝑁 − 1) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) · (cos‘𝑥)))
82		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
83		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ ((((𝑁 − 1) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) · (cos‘𝑥)) · -(cos‘𝑥))) = (𝑥 ∈ ℂ ↦ ((((𝑁 − 1) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) · (cos‘𝑥)) · -(cos‘𝑥)))
84		eqid 2736	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1)))) = (𝑥 ∈ ℂ ↦ (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))))
85	79, 80, 81, 82, 83, 84, 10	itgsinexplem1 44185	. . . . . . . . 9 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥 = ((𝑁 − 1) · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) d𝑥))
86	4, 5, 5	subsub4d 11543	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
87		1p1e2 12278	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + 1) = 2)
89	88	oveq2d 7373	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − (1 + 1)) = (𝑁 − 2))
90	86, 89	eqtrd 2776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
91	90	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 − 1) − 1) = (𝑁 − 2))
92	91	oveq2d 7373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑((𝑁 − 1) − 1)) = ((sin‘𝑥)↑(𝑁 − 2)))
93	92	oveq2d 7373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
94	93	itgeq2dv 25146	. . . . . . . . . 10 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) d𝑥 = ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥)
95	94	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 1) · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) d𝑥) = ((𝑁 − 1) · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥))
96		sincossq 16058	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
97		1cnd 11150	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → 1 ∈ ℂ)
98		sincl 16008	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
99	98	sqcld 14049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((sin‘𝑥)↑2) ∈ ℂ)
100		coscl 16009	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
101	100	sqcld 14049	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) ∈ ℂ)
102	97, 99, 101	subaddd 11530	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2) ↔ (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1))
103	96, 102	mpbird 256	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2))
104	103	eqcomd 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) = (1 − ((sin‘𝑥)↑2)))
105	31, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0(,)π) → ((cos‘𝑥)↑2) = (1 − ((sin‘𝑥)↑2)))
106	105	oveq1d 7372	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0(,)π) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
107	106	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
108	107	itgeq2dv 25146	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥 = ∫(0(,)π)((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥)
109		1cnd 11150	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 1 ∈ ℂ)
110	32	sqcld 14049	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0(,)π) → ((sin‘𝑥)↑2) ∈ ℂ)
111	110	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑2) ∈ ℂ)
112	90	eqcomd 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 2) = ((𝑁 − 1) − 1))
113		nnm1nn0 12454	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 − 1) − 1) ∈ ℕ₀)
114	10, 113	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) − 1) ∈ ℕ₀)
115	112, 114	eqeltrd 2838	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
116	115	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 2) ∈ ℕ₀)
117	33, 116	expcld 14051	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
118	109, 111, 117	subdird 11612	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 · ((sin‘𝑥)↑(𝑁 − 2))) − (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2)))))
119	117	mulid2d 11173	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (1 · ((sin‘𝑥)↑(𝑁 − 2))) = ((sin‘𝑥)↑(𝑁 − 2)))
120	23	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 2 ∈ ℕ₀)
121	33, 116, 120	expaddd 14053	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
122	17, 4	pncan3d 11515	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 + (𝑁 − 2)) = 𝑁)
123	122	oveq2d 7373	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
124	123	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
125	121, 124	eqtr3d 2778	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
126	119, 125	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 · ((sin‘𝑥)↑(𝑁 − 2))) − (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2)))) = (((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)))
127	118, 126	eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) = (((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)))
128	127	itgeq2dv 25146	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(0(,)π)((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥 = ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)) d𝑥)
129	115	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 2) ∈ ℕ₀)
130	48, 129	expcld 14051	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
131		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
132	131	fvmpt2 6959	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
133	53, 130, 132	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
134	133	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥))
135	134	mpteq2dva 5205	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)))
136		nfmpt1 5213	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
137	60, 62, 115	expcnfg 43822	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ (ℂ–cn→ℂ))
138	136, 137, 64	cncfmptss 43818	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
139	135, 138	eqeltrd 2838	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ ((0[,]π)–cn→ℂ))
140		cniccibl 25205	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
141	51, 52, 139, 140	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
142	37, 39, 130, 141	iblss 25169	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
143	117, 142, 35, 69	itgsub 25190	. . . . . . . . . . 11 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)) d𝑥 = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
144	108, 128, 143	3eqtrd 2780	. . . . . . . . . 10 ⊢ (𝜑 → ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥 = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
145	144	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑁 − 1) · ∫(0(,)π)(((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) d𝑥) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
146	85, 95, 145	3eqtrd 2780	. . . . . . . 8 ⊢ (𝜑 → ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥 = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
147	29, 78, 146	3eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → (𝐼‘𝑁) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
148		oveq2 7365	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑁 − 2) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
149	148	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑛 = (𝑁 − 2) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
150	149	itgeq2dv 25146	. . . . . . . . . 10 ⊢ (𝑛 = (𝑁 − 2) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
151		itgex 25135	. . . . . . . . . . 11 ⊢ ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ V
152	151	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ V)
153	12, 150, 115, 152	fvmptd3 6971	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
154	153, 29	oveq12d 7375	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁)) = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
155	154	oveq2d 7373	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) · ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁))) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
156	117, 142	itgcl 25148	. . . . . . . . 9 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ ℂ)
157	153, 156	eqeltrd 2838	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) ∈ ℂ)
158	11, 157, 71	subdid 11611	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) · ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁))) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
159	147, 155, 158	3eqtr2d 2782	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
160	159	eqcomd 2742	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))) = (𝐼‘𝑁))
161	11, 157	mulcld 11175	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘(𝑁 − 2))) ∈ ℂ)
162	11, 71	mulcld 11175	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘𝑁)) ∈ ℂ)
163	161, 162, 71	subaddd 11530	. . . . 5 ⊢ (𝜑 → ((((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))) = (𝐼‘𝑁) ↔ (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2)))))
164	160, 163	mpbid 231	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
165	8, 72, 164	3eqtrd 2780	. . 3 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
166	165	oveq1d 7372	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁))
167	75	nnne0d 12203	. . 3 ⊢ (𝜑 → 𝑁 ≠ 0)
168	71, 4, 167	divcan3d 11936	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (𝐼‘𝑁))
169	11, 157, 4, 167	div23d 11968	. 2 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
170	166, 168, 169	3eqtr3d 2784	1 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 < clt 11189 − cmin 11385 -cneg 11386 / cdiv 11812 ℕcn 12153 2c2 12208 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 (,)cioo 13264 [,]cicc 13267 ↑cexp 13967 sincsin 15946 cosccos 15947 πcpi 15949 –cn→ccncf 24239 volcvol 24827 𝐿¹cibl 24981 ∫citg 24982

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-symdif 4202 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-disj 5071 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-ofr 7618 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-omul 8417 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-dju 9837 df-card 9875 df-acn 9878 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-cmp 22738 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-ovol 24828 df-vol 24829 df-mbf 24983 df-itg1 24984 df-itg2 24985 df-ibl 24986 df-itg 24987 df-0p 25034 df-limc 25230 df-dv 25231

This theorem is referenced by: wallispilem2 44297 wallispilem4 44299 wallispilem5 44300

W3C validator