itgsinexp - Mathbox for Glauco Siliprandi

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Theorem itgsinexp 41670

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 12062	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		zcn 11792	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		1cnd 10428	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
6	4, 5	npcand 10796	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7	6	eqcomd 2778	. . . . 5 ⊢ (𝜑 → 𝑁 = ((𝑁 − 1) + 1))
8	7	oveq1d 6985	. . . 4 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = (((𝑁 − 1) + 1) · (𝐼‘𝑁)))
9		uz2m1nn 12131	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ)
11	10	nncnd 11451	. . . . 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 6978	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
15	14	ad2antlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
16	15	itgeq2dv 24079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
17		2cnd 11512	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
18		npcan 10690	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑁 − 2) + 2) = 𝑁)
19	18	eqcomd 2778	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → 𝑁 = ((𝑁 − 2) + 2))
20	4, 17, 19	syl2anc 576	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = ((𝑁 − 2) + 2))
21		uznn0sub 12085	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ ℕ₀)
22	1, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
23		2nn0 11720	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ₀)
25	22, 24	nn0addcld 11765	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 2) + 2) ∈ ℕ₀)
26	20, 25	eqeltrd 2860	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
27		itgex 24068	. . . . . . . 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 6595	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
30		ioosscn 41200	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
31	30	sseli 3848	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
32	31	sincld 15337	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
33	32	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
34	26	adantr 473	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
35	33, 34	expcld 13319	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
36		ioossicc 12632	. . . . . . . . 9 ⊢ (0(,)π) ⊆ (0[,]π)
37	36	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
38		ioombl 23863	. . . . . . . . 9 ⊢ (0(,)π) ∈ dom vol
39	38	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
40		0re 10435	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
41		pire 24741	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
42		iccssre 12628	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
43	40, 41, 42	mp2an 679	. . . . . . . . . . . . 13 ⊢ (0[,]π) ⊆ ℝ
44		ax-resscn 10386	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
45	43, 44	sstri 3861	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℂ
46	45	sseli 3848	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
47	46	sincld 15337	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
48	47	adantl 474	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
49	26	adantr 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
50	48, 49	expcld 13319	. . . . . . . 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 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
54		eqid 2772	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
55	54	fvmpt2 6599	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
56	53, 50, 55	syl2anc 576	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
57	56	eqcomd 2778	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥))
58	57	mpteq2dva 5016	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)))
59		nfmpt1 5019	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
60		nfcv 2926	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥sin
61		sincn 24729	. . . . . . . . . . . . 13 ⊢ sin ∈ (ℂ–cn→ℂ)
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
63	60, 62, 26	expcnfg 41303	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
64	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
65	59, 63, 64	cncfmptss 41299	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
66	58, 65	eqeltrd 2860	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
67		cniccibl 24138	. . . . . . . . 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 1351	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
69	37, 39, 50, 68	iblss 24102	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
70	35, 69	itgcl 24081	. . . . . 6 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ ℂ)
71	29, 70	eqeltrd 2860	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑁) ∈ ℂ)
72	11, 71	adddirp1d 10460	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) + 1) · (𝐼‘𝑁)) = (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)))
73		eluz2b2 12129	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
74	1, 73	sylib 210	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
75	74	simpld 487	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
76		expm1t 13266	. . . . . . . . . 10 ⊢ (((sin‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
77	32, 75, 76	syl2anr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
78	77	itgeq2dv 24079	. . . . . . . 8 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 = ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥)
79		eqid 2772	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1)))
80		eqid 2772	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
81		eqid 2772	. . . . . . . . . 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 2772	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
83		eqid 2772	. . . . . . . . . 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 2772	. . . . . . . . . 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 41669	. . . . . . . . 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 10823	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
87		1p1e2 11566	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + 1) = 2)
89	88	oveq2d 6986	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − (1 + 1)) = (𝑁 − 2))
90	86, 89	eqtrd 2808	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
91	90	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 − 1) − 1) = (𝑁 − 2))
92	91	oveq2d 6986	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑((𝑁 − 1) − 1)) = ((sin‘𝑥)↑(𝑁 − 2)))
93	92	oveq2d 6986	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
94	93	itgeq2dv 24079	. . . . . . . . . 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 6986	. . . . . . . . 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 15383	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
97		1cnd 10428	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → 1 ∈ ℂ)
98		sincl 15333	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
99	98	sqcld 13317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((sin‘𝑥)↑2) ∈ ℂ)
100		coscl 15334	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
101	100	sqcld 13317	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) ∈ ℂ)
102	97, 99, 101	subaddd 10810	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2) ↔ (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1))
103	96, 102	mpbird 249	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2))
104	103	eqcomd 2778	. . . . . . . . . . . . . . 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 6985	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0(,)π) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
107	106	adantl 474	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
108	107	itgeq2dv 24079	. . . . . . . . . . 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 10428	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 1 ∈ ℂ)
110	32	sqcld 13317	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0(,)π) → ((sin‘𝑥)↑2) ∈ ℂ)
111	110	adantl 474	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑2) ∈ ℂ)
112	90	eqcomd 2778	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 2) = ((𝑁 − 1) − 1))
113		nnm1nn0 11744	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 − 1) − 1) ∈ ℕ₀)
114	10, 113	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) − 1) ∈ ℕ₀)
115	112, 114	eqeltrd 2860	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
116	115	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 2) ∈ ℕ₀)
117	33, 116	expcld 13319	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
118	109, 111, 117	subdird 10892	. . . . . . . . . . . . 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 10452	. . . . . . . . . . . . . 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 13321	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
122	17, 4	pncan3d 10795	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 + (𝑁 − 2)) = 𝑁)
123	122	oveq2d 6986	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
124	123	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
125	121, 124	eqtr3d 2810	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
126	119, 125	oveq12d 6988	. . . . . . . . . . . . 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 2808	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) = (((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)))
128	127	itgeq2dv 24079	. . . . . . . . . . 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 473	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 2) ∈ ℕ₀)
130	48, 129	expcld 13319	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
131		eqid 2772	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
132	131	fvmpt2 6599	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
133	53, 130, 132	syl2anc 576	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
134	133	eqcomd 2778	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥))
135	134	mpteq2dva 5016	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)))
136		nfmpt1 5019	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
137	60, 62, 115	expcnfg 41303	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ (ℂ–cn→ℂ))
138	136, 137, 64	cncfmptss 41299	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
139	135, 138	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ ((0[,]π)–cn→ℂ))
140		cniccibl 24138	. . . . . . . . . . . . . 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 1351	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
142	37, 39, 130, 141	iblss 24102	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
143	117, 142, 35, 69	itgsub 24123	. . . . . . . . . . 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 2812	. . . . . . . . . 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 6986	. . . . . . . . 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 2812	. . . . . . . 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 2812	. . . . . . 7 ⊢ (𝜑 → (𝐼‘𝑁) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
148		oveq2 6978	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑁 − 2) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
149	148	adantr 473	. . . . . . . . . . 11 ⊢ ((𝑛 = (𝑁 − 2) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
150	149	itgeq2dv 24079	. . . . . . . . . 10 ⊢ (𝑛 = (𝑁 − 2) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
151		itgex 24068	. . . . . . . . . . 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 6611	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
154	153, 29	oveq12d 6988	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁)) = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
155	154	oveq2d 6986	. . . . . . 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 24081	. . . . . . . . 9 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ ℂ)
157	153, 156	eqeltrd 2860	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) ∈ ℂ)
158	11, 157, 71	subdid 10891	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) · ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁))) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
159	147, 155, 158	3eqtr2d 2814	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
160	159	eqcomd 2778	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))) = (𝐼‘𝑁))
161	11, 157	mulcld 10454	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘(𝑁 − 2))) ∈ ℂ)
162	11, 71	mulcld 10454	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘𝑁)) ∈ ℂ)
163	161, 162, 71	subaddd 10810	. . . . 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 224	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
165	8, 72, 164	3eqtrd 2812	. . 3 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
166	165	oveq1d 6985	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁))
167	75	nnne0d 11484	. . 3 ⊢ (𝜑 → 𝑁 ≠ 0)
168	71, 4, 167	divcan3d 11216	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (𝐼‘𝑁))
169	11, 157, 4, 167	div23d 11248	. 2 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
170	166, 168, 169	3eqtr3d 2816	1 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3409 ⊆ wss 3823 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5401 ‘cfv 6182 (class class class)co 6970 ℂcc 10327 ℝcr 10328 0cc0 10329 1c1 10330 + caddc 10332 · cmul 10334 < clt 10468 − cmin 10664 -cneg 10665 / cdiv 11092 ℕcn 11433 2c2 11489 ℕ₀cn0 11701 ℤcz 11787 ℤ_≥cuz 12052 (,)cioo 12548 [,]cicc 12551 ↑cexp 13238 sincsin 15271 cosccos 15272 πcpi 15274 –cn→ccncf 23181 volcvol 23761 𝐿¹cibl 23915 ∫citg 23916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8892 ax-cc 9649 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 ax-addf 10408 ax-mulf 10409

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-symdif 4100 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-disj 4892 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-se 5361 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7495 df-2nd 7496 df-supp 7628 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-2o 7900 df-oadd 7903 df-omul 7904 df-er 8083 df-map 8202 df-pm 8203 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-fsupp 8623 df-fi 8664 df-sup 8695 df-inf 8696 df-oi 8763 df-dju 9118 df-card 9156 df-acn 9159 df-cda 9382 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-ioo 12552 df-ioc 12553 df-ico 12554 df-icc 12555 df-fz 12703 df-fzo 12844 df-fl 12971 df-mod 13047 df-seq 13179 df-exp 13239 df-fac 13443 df-bc 13472 df-hash 13500 df-shft 14281 df-cj 14313 df-re 14314 df-im 14315 df-sqrt 14449 df-abs 14450 df-limsup 14683 df-clim 14700 df-rlim 14701 df-sum 14898 df-ef 15275 df-sin 15277 df-cos 15278 df-pi 15280 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-sets 16340 df-ress 16341 df-plusg 16428 df-mulr 16429 df-starv 16430 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-unif 16438 df-hom 16439 df-cco 16440 df-rest 16546 df-topn 16547 df-0g 16565 df-gsum 16566 df-topgen 16567 df-pt 16568 df-prds 16571 df-xrs 16625 df-qtop 16630 df-imas 16631 df-xps 16633 df-mre 16709 df-mrc 16710 df-acs 16712 df-mgm 17704 df-sgrp 17746 df-mnd 17757 df-submnd 17798 df-mulg 18006 df-cntz 18212 df-cmn 18662 df-psmet 20233 df-xmet 20234 df-met 20235 df-bl 20236 df-mopn 20237 df-fbas 20238 df-fg 20239 df-cnfld 20242 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-cld 21325 df-ntr 21326 df-cls 21327 df-nei 21404 df-lp 21442 df-perf 21443 df-cn 21533 df-cnp 21534 df-haus 21621 df-cmp 21693 df-tx 21868 df-hmeo 22061 df-fil 22152 df-fm 22244 df-flim 22245 df-flf 22246 df-xms 22627 df-ms 22628 df-tms 22629 df-cncf 23183 df-ovol 23762 df-vol 23763 df-mbf 23917 df-itg1 23918 df-itg2 23919 df-ibl 23920 df-itg 23921 df-0p 23968 df-limc 24161 df-dv 24162

This theorem is referenced by: wallispilem2 41782 wallispilem4 41784 wallispilem5 41785

W3C validator