itgsinexp - Mathbox for Glauco Siliprandi

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Theorem itgsinexp 46909

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 12956	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		zcn 12679	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	1, 2, 3	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		1cnd 11283	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
6	4, 5	npcand 11654	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7	6	eqcomd 2767	. . . . 5 ⊢ (𝜑 → 𝑁 = ((𝑁 − 1) + 1))
8	7	oveq1d 7427	. . . 4 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = (((𝑁 − 1) + 1) · (𝐼‘𝑁)))
9		uz2m1nn 13031	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
10	1, 9	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ)
11	10	nncnd 12332	. . . . 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 7420	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
15	14	ad2antlr 740	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
16	15	itgeq2dv 26082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
17		2cnd 12402	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
18		npcan 11547	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑁 − 2) + 2) = 𝑁)
19	18	eqcomd 2767	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → 𝑁 = ((𝑁 − 2) + 2))
20	4, 17, 19	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = ((𝑁 − 2) + 2))
21		uznn0sub 12981	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ ℕ₀)
22	1, 21	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
23		2nn0 12604	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ₀)
25	22, 24	nn0addcld 12652	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 2) + 2) ∈ ℕ₀)
26	20, 25	eqeltrd 2861	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
27		itgex 26071	. . . . . . . 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 6993	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
30		ioosscn 13520	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
31	30	sseli 3927	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
32	31	sincld 16278	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
33	32	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
34	26	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
35	33, 34	expcld 14269	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) ∈ ℂ)
36		ioossicc 13545	. . . . . . . . 9 ⊢ (0(,)π) ⊆ (0[,]π)
37	36	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ⊆ (0[,]π))
38		ioombl 25866	. . . . . . . . 9 ⊢ (0(,)π) ∈ dom vol
39	38	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
40		0re 11291	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
41		pire 26765	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
42		iccssre 13541	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
43	40, 41, 42	mp2an 705	. . . . . . . . . . . . 13 ⊢ (0[,]π) ⊆ ℝ
44		ax-resscn 11238	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
45	43, 44	sstri 3940	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℂ
46	45	sseli 3927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
47	46	sincld 16278	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
48	47	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
49	26	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
50	48, 49	expcld 14269	. . . . . . . 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 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
54		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
55	54	fvmpt2 6997	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
56	53, 50, 55	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
57	56	eqcomd 2767	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥))
58	57	mpteq2dva 5198	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)))
59		nfmpt1 5204	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
60		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥sin
61		sincn 26753	. . . . . . . . . . . . 13 ⊢ sin ∈ (ℂ–cn→ℂ)
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
63	60, 62, 26	expcnfg 46547	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
64	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
65	59, 63, 64	cncfmptss 46543	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
66	58, 65	eqeltrd 2861	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
67		cniccibl 26141	. . . . . . . . 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 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
69	37, 39, 50, 68	iblss 26105	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
70	35, 69	itgcl 26084	. . . . . 6 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ ℂ)
71	29, 70	eqeltrd 2861	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑁) ∈ ℂ)
72	11, 71	adddirp1d 11316	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) + 1) · (𝐼‘𝑁)) = (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)))
73		eluz2b2 13029	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
74	1, 73	sylib 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
75	74	simpld 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
76		expm1t 14213	. . . . . . . . . 10 ⊢ (((sin‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
77	32, 75, 76	syl2anr 609	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
78	77	itgeq2dv 26082	. . . . . . . 8 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 = ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥)
79		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1)))
80		eqid 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
81		eqid 2761	. . . . . . . . . 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 2761	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
83		eqid 2761	. . . . . . . . . 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 2761	. . . . . . . . . 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 46908	. . . . . . . . 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 11681	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
87		1p1e2 12447	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + 1) = 2)
89	88	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − (1 + 1)) = (𝑁 − 2))
90	86, 89	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
91	90	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 − 1) − 1) = (𝑁 − 2))
92	91	oveq2d 7428	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑((𝑁 − 1) − 1)) = ((sin‘𝑥)↑(𝑁 − 2)))
93	92	oveq2d 7428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
94	93	itgeq2dv 26082	. . . . . . . . . 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 7428	. . . . . . . . 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 16324	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
97		1cnd 11283	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → 1 ∈ ℂ)
98		sincl 16274	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
99	98	sqcld 14267	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((sin‘𝑥)↑2) ∈ ℂ)
100		coscl 16275	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
101	100	sqcld 14267	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) ∈ ℂ)
102	97, 99, 101	subaddd 11668	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2) ↔ (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1))
103	96, 102	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2))
104	103	eqcomd 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) = (1 − ((sin‘𝑥)↑2)))
105	31, 104	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0(,)π) → ((cos‘𝑥)↑2) = (1 − ((sin‘𝑥)↑2)))
106	105	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0(,)π) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
107	106	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
108	107	itgeq2dv 26082	. . . . . . . . . . 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 11283	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 1 ∈ ℂ)
110	32	sqcld 14267	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0(,)π) → ((sin‘𝑥)↑2) ∈ ℂ)
111	110	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑2) ∈ ℂ)
112	90	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 2) = ((𝑁 − 1) − 1))
113		nnm1nn0 12628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 − 1) − 1) ∈ ℕ₀)
114	10, 113	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) − 1) ∈ ℕ₀)
115	112, 114	eqeltrd 2861	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
116	115	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 2) ∈ ℕ₀)
117	33, 116	expcld 14269	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
118	109, 111, 117	subdird 11754	. . . . . . . . . . . . 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	mullidd 11308	. . . . . . . . . . . . . 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 14271	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
122	17, 4	pncan3d 11653	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 + (𝑁 − 2)) = 𝑁)
123	122	oveq2d 7428	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
124	123	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
125	121, 124	eqtr3d 2798	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
126	119, 125	oveq12d 7430	. . . . . . . . . . . . 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 2796	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) = (((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)))
128	127	itgeq2dv 26082	. . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 2) ∈ ℕ₀)
130	48, 129	expcld 14269	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
131		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
132	131	fvmpt2 6997	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
133	53, 130, 132	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
134	133	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥))
135	134	mpteq2dva 5198	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)))
136		nfmpt1 5204	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
137	60, 62, 115	expcnfg 46547	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ (ℂ–cn→ℂ))
138	136, 137, 64	cncfmptss 46543	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
139	135, 138	eqeltrd 2861	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ ((0[,]π)–cn→ℂ))
140		cniccibl 26141	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
142	37, 39, 130, 141	iblss 26105	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
143	117, 142, 35, 69	itgsub 26126	. . . . . . . . . . 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 2800	. . . . . . . . . 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 7428	. . . . . . . . 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 2800	. . . . . . . 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 2800	. . . . . . 7 ⊢ (𝜑 → (𝐼‘𝑁) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
148		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑁 − 2) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
149	148	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑛 = (𝑁 − 2) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
150	149	itgeq2dv 26082	. . . . . . . . . 10 ⊢ (𝑛 = (𝑁 − 2) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
151		itgex 26071	. . . . . . . . . . 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 7009	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
154	153, 29	oveq12d 7430	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁)) = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
155	154	oveq2d 7428	. . . . . . 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 26084	. . . . . . . . 9 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ ℂ)
157	153, 156	eqeltrd 2861	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) ∈ ℂ)
158	11, 157, 71	subdid 11753	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) · ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁))) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
159	147, 155, 158	3eqtr2d 2802	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
160	159	eqcomd 2767	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))) = (𝐼‘𝑁))
161	11, 157	mulcld 11310	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘(𝑁 − 2))) ∈ ℂ)
162	11, 71	mulcld 11310	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘𝑁)) ∈ ℂ)
163	161, 162, 71	subaddd 11668	. . . . 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 235	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
165	8, 72, 164	3eqtrd 2800	. . 3 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
166	165	oveq1d 7427	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁))
167	75	nnne0d 12369	. . 3 ⊢ (𝜑 → 𝑁 ≠ 0)
168	71, 4, 167	divcan3d 12079	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (𝐼‘𝑁))
169	11, 157, 4, 167	div23d 12111	. 2 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
170	166, 168, 169	3eqtr3d 2804	1 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 ℝcr 11180 0cc0 11181 1c1 11182 + caddc 11184 · cmul 11186 < clt 11324 − cmin 11522 -cneg 11523 / cdiv 11954 ℕcn 12316 2c2 12378 ℕ₀cn0 12587 ℤcz 12674 ℤ_≥cuz 12946 (,)cioo 13457 [,]cicc 13460 ↑cexp 14184 sincsin 16209 cosccos 16210 πcpi 16212 –cn→ccncf 25177 volcvol 25764 𝐿¹cibl 25918 ∫citg 25919

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-inf2 9626 ax-cc 10494 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-ofr 7683 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-omul 8465 df-er 8701 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-dju 9963 df-card 10001 df-acn 10004 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-shft 15200 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-xrs 17654 df-qtop 17659 df-imas 17660 df-xps 17662 df-mre 17736 df-mrc 17737 df-acs 17739 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-mulg 19258 df-cntz 19511 df-cmn 19976 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-cmp 23685 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-ovol 25765 df-vol 25766 df-mbf 25920 df-itg1 25921 df-itg2 25922 df-ibl 25923 df-itg 25924 df-0p 25971 df-limc 26166 df-dv 26167

This theorem is used by: wallispilem2 47020 wallispilem4 47022 wallispilem5 47023

W3C validator