itgsinexp - Mathbox for Glauco Siliprandi

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Theorem itgsinexp 46387

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 12762	. . . . . . . 8 ⊢ (𝑁 ∈ (ℤ_≥‘2) → 𝑁 ∈ ℤ)
3		zcn 12494	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	1, 2, 3	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		1cnd 11128	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
6	4, 5	npcand 11497	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7	6	eqcomd 2743	. . . . 5 ⊢ (𝜑 → 𝑁 = ((𝑁 − 1) + 1))
8	7	oveq1d 7373	. . . 4 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = (((𝑁 − 1) + 1) · (𝐼‘𝑁)))
9		uz2m1nn 12837	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 1) ∈ ℕ)
10	1, 9	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ)
11	10	nncnd 12162	. . . . 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 7366	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
15	14	ad2antlr 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑁) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑𝑁))
16	15	itgeq2dv 25727	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
17		2cnd 12224	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℂ)
18		npcan 11390	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝑁 − 2) + 2) = 𝑁)
19	18	eqcomd 2743	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ) → 𝑁 = ((𝑁 − 2) + 2))
20	4, 17, 19	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = ((𝑁 − 2) + 2))
21		uznn0sub 12787	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘2) → (𝑁 − 2) ∈ ℕ₀)
22	1, 21	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
23		2nn0 12419	. . . . . . . . . 10 ⊢ 2 ∈ ℕ₀
24	23	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 2 ∈ ℕ₀)
25	22, 24	nn0addcld 12467	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 2) + 2) ∈ ℕ₀)
26	20, 25	eqeltrd 2837	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
27		itgex 25715	. . . . . . . 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 6947	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)
30		ioosscn 13325	. . . . . . . . . . 11 ⊢ (0(,)π) ⊆ ℂ
31	30	sseli 3918	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
32	31	sincld 16056	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (sin‘𝑥) ∈ ℂ)
34	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 𝑁 ∈ ℕ₀)
35	33, 34	expcld 14070	. . . . . . 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 25510	. . . . . . . . 9 ⊢ (0(,)π) ∈ dom vol
39	38	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (0(,)π) ∈ dom vol)
40		0re 11135	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ
41		pire 26406	. . . . . . . . . . . . . 14 ⊢ π ∈ ℝ
42		iccssre 13346	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
43	40, 41, 42	mp2an 693	. . . . . . . . . . . . 13 ⊢ (0[,]π) ⊆ ℝ
44		ax-resscn 11084	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
45	43, 44	sstri 3932	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℂ
46	45	sseli 3918	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
47	46	sincld 16056	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
48	47	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
49	26	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑁 ∈ ℕ₀)
50	48, 49	expcld 14070	. . . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → 𝑥 ∈ ℂ)
54		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
55	54	fvmpt2 6951	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑𝑁) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
56	53, 50, 55	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥) = ((sin‘𝑥)↑𝑁))
57	56	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑𝑁) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥))
58	57	mpteq2dva 5179	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)))
59		nfmpt1 5185	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))
60		nfcv 2899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥sin
61		sincn 26394	. . . . . . . . . . . . 13 ⊢ sin ∈ (ℂ–cn→ℂ)
62	61	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → sin ∈ (ℂ–cn→ℂ))
63	60, 62, 26	expcnfg 46025	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁)) ∈ (ℂ–cn→ℂ))
64	45	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (0[,]π) ⊆ ℂ)
65	59, 63, 64	cncfmptss 46021	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑𝑁))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
66	58, 65	eqeltrd 2837	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ ((0[,]π)–cn→ℂ))
67		cniccibl 25786	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
69	37, 39, 50, 68	iblss 25750	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑𝑁)) ∈ 𝐿¹)
70	35, 69	itgcl 25729	. . . . . 6 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 ∈ ℂ)
71	29, 70	eqeltrd 2837	. . . . 5 ⊢ (𝜑 → (𝐼‘𝑁) ∈ ℂ)
72	11, 71	adddirp1d 11159	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) + 1) · (𝐼‘𝑁)) = (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)))
73		eluz2b2 12835	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁))
74	1, 73	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁))
75	74	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ)
76		expm1t 14014	. . . . . . . . . 10 ⊢ (((sin‘𝑥) ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
77	32, 75, 76	syl2anr 598	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑁) = (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
78	77	itgeq2dv 25727	. . . . . . . 8 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥 = ∫(0(,)π)(((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)) d𝑥)
79		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 1)))
80		eqid 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
81		eqid 2737	. . . . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥))) = (𝑥 ∈ ℂ ↦ (((sin‘𝑥)↑(𝑁 − 1)) · (sin‘𝑥)))
83		eqid 2737	. . . . . . . . . 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 2737	. . . . . . . . . 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 46386	. . . . . . . . 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 11524	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − (1 + 1)))
87		1p1e2 12266	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) = 2
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 + 1) = 2)
89	88	oveq2d 7374	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − (1 + 1)) = (𝑁 − 2))
90	86, 89	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) − 1) = (𝑁 − 2))
91	90	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((𝑁 − 1) − 1) = (𝑁 − 2))
92	91	oveq2d 7374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑((𝑁 − 1) − 1)) = ((sin‘𝑥)↑(𝑁 − 2)))
93	92	oveq2d 7374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑((𝑁 − 1) − 1))) = (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
94	93	itgeq2dv 25727	. . . . . . . . . 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 7374	. . . . . . . . 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 16102	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1)
97		1cnd 11128	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → 1 ∈ ℂ)
98		sincl 16052	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (sin‘𝑥) ∈ ℂ)
99	98	sqcld 14068	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((sin‘𝑥)↑2) ∈ ℂ)
100		coscl 16053	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ → (cos‘𝑥) ∈ ℂ)
101	100	sqcld 14068	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℂ → ((cos‘𝑥)↑2) ∈ ℂ)
102	97, 99, 101	subaddd 11511	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℂ → ((1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2) ↔ (((sin‘𝑥)↑2) + ((cos‘𝑥)↑2)) = 1))
103	96, 102	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℂ → (1 − ((sin‘𝑥)↑2)) = ((cos‘𝑥)↑2))
104	103	eqcomd 2743	. . . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0(,)π) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
107	106	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((cos‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))))
108	107	itgeq2dv 25727	. . . . . . . . . . 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 11128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → 1 ∈ ℂ)
110	32	sqcld 14068	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (0(,)π) → ((sin‘𝑥)↑2) ∈ ℂ)
111	110	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑2) ∈ ℂ)
112	90	eqcomd 2743	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 2) = ((𝑁 − 1) − 1))
113		nnm1nn0 12443	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ ℕ → ((𝑁 − 1) − 1) ∈ ℕ₀)
114	10, 113	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) − 1) ∈ ℕ₀)
115	112, 114	eqeltrd 2837	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 2) ∈ ℕ₀)
116	115	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (𝑁 − 2) ∈ ℕ₀)
117	33, 116	expcld 14070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
118	109, 111, 117	subdird 11595	. . . . . . . . . . . . 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 11151	. . . . . . . . . . . . . 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 14072	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))))
122	17, 4	pncan3d 11496	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2 + (𝑁 − 2)) = 𝑁)
123	122	oveq2d 7374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
124	123	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑(2 + (𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
125	121, 124	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → (((sin‘𝑥)↑2) · ((sin‘𝑥)↑(𝑁 − 2))) = ((sin‘𝑥)↑𝑁))
126	119, 125	oveq12d 7376	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0(,)π)) → ((1 − ((sin‘𝑥)↑2)) · ((sin‘𝑥)↑(𝑁 − 2))) = (((sin‘𝑥)↑(𝑁 − 2)) − ((sin‘𝑥)↑𝑁)))
128	127	itgeq2dv 25727	. . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → (𝑁 − 2) ∈ ℕ₀)
130	48, 129	expcld 14070	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ)
131		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
132	131	fvmpt2 6951	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ ℂ ∧ ((sin‘𝑥)↑(𝑁 − 2)) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
133	53, 130, 132	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥) = ((sin‘𝑥)↑(𝑁 − 2)))
134	133	eqcomd 2743	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]π)) → ((sin‘𝑥)↑(𝑁 − 2)) = ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥))
135	134	mpteq2dva 5179	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)))
136		nfmpt1 5185	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))
137	60, 62, 115	expcnfg 46025	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ (ℂ–cn→ℂ))
138	136, 137, 64	cncfmptss 46021	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ ((sin‘𝑥)↑(𝑁 − 2)))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
139	135, 138	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ ((0[,]π)–cn→ℂ))
140		cniccibl 25786	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (0[,]π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
142	37, 39, 130, 141	iblss 25750	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ (0(,)π) ↦ ((sin‘𝑥)↑(𝑁 − 2))) ∈ 𝐿¹)
143	117, 142, 35, 69	itgsub 25771	. . . . . . . . . . 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 2776	. . . . . . . . . 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 7374	. . . . . . . . 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 2776	. . . . . . . 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 2776	. . . . . . 7 ⊢ (𝜑 → (𝐼‘𝑁) = ((𝑁 − 1) · (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥)))
148		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑛 = (𝑁 − 2) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
149	148	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 = (𝑁 − 2) ∧ 𝑥 ∈ (0(,)π)) → ((sin‘𝑥)↑𝑛) = ((sin‘𝑥)↑(𝑁 − 2)))
150	149	itgeq2dv 25727	. . . . . . . . . 10 ⊢ (𝑛 = (𝑁 − 2) → ∫(0(,)π)((sin‘𝑥)↑𝑛) d𝑥 = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
151		itgex 25715	. . . . . . . . . . 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 6963	. . . . . . . . 9 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) = ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥)
154	153, 29	oveq12d 7376	. . . . . . . 8 ⊢ (𝜑 → ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁)) = (∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 − ∫(0(,)π)((sin‘𝑥)↑𝑁) d𝑥))
155	154	oveq2d 7374	. . . . . . 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 25729	. . . . . . . . 9 ⊢ (𝜑 → ∫(0(,)π)((sin‘𝑥)↑(𝑁 − 2)) d𝑥 ∈ ℂ)
157	153, 156	eqeltrd 2837	. . . . . . . 8 ⊢ (𝜑 → (𝐼‘(𝑁 − 2)) ∈ ℂ)
158	11, 157, 71	subdid 11594	. . . . . . 7 ⊢ (𝜑 → ((𝑁 − 1) · ((𝐼‘(𝑁 − 2)) − (𝐼‘𝑁))) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
159	147, 155, 158	3eqtr2d 2778	. . . . . 6 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))))
160	159	eqcomd 2743	. . . . 5 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) − ((𝑁 − 1) · (𝐼‘𝑁))) = (𝐼‘𝑁))
161	11, 157	mulcld 11153	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘(𝑁 − 2))) ∈ ℂ)
162	11, 71	mulcld 11153	. . . . . 6 ⊢ (𝜑 → ((𝑁 − 1) · (𝐼‘𝑁)) ∈ ℂ)
163	161, 162, 71	subaddd 11511	. . . . 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 232	. . . 4 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘𝑁)) + (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
165	8, 72, 164	3eqtrd 2776	. . 3 ⊢ (𝜑 → (𝑁 · (𝐼‘𝑁)) = ((𝑁 − 1) · (𝐼‘(𝑁 − 2))))
166	165	oveq1d 7373	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁))
167	75	nnne0d 12196	. . 3 ⊢ (𝜑 → 𝑁 ≠ 0)
168	71, 4, 167	divcan3d 11923	. 2 ⊢ (𝜑 → ((𝑁 · (𝐼‘𝑁)) / 𝑁) = (𝐼‘𝑁))
169	11, 157, 4, 167	div23d 11955	. 2 ⊢ (𝜑 → (((𝑁 − 1) · (𝐼‘(𝑁 − 2))) / 𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))
170	166, 168, 169	3eqtr3d 2780	1 ⊢ (𝜑 → (𝐼‘𝑁) = (((𝑁 − 1) / 𝑁) · (𝐼‘(𝑁 − 2))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5622 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 · cmul 11032 < clt 11167 − cmin 11365 -cneg 11366 / cdiv 11795 ℕcn 12146 2c2 12201 ℕ₀cn0 12402 ℤcz 12489 ℤ_≥cuz 12752 (,)cioo 13262 [,]cicc 13265 ↑cexp 13985 sincsin 15987 cosccos 15988 πcpi 15990 –cn→ccncf 24821 volcvol 25408 𝐿¹cibl 25562 ∫citg 25563

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cc 10346 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-symdif 4194 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-disj 5054 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-oadd 8400 df-omul 8401 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-q 12863 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-ioo 13266 df-ioc 13267 df-ico 13268 df-icc 13269 df-fz 13425 df-fzo 13572 df-fl 13713 df-mod 13791 df-seq 13926 df-exp 13986 df-fac 14198 df-bc 14227 df-hash 14255 df-shft 14991 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-limsup 15395 df-clim 15412 df-rlim 15413 df-sum 15611 df-ef 15991 df-sin 15993 df-cos 15994 df-pi 15996 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-rest 17343 df-topn 17344 df-0g 17362 df-gsum 17363 df-topgen 17364 df-pt 17365 df-prds 17368 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24263 df-ms 24264 df-tms 24265 df-cncf 24823 df-ovol 25409 df-vol 25410 df-mbf 25564 df-itg1 25565 df-itg2 25566 df-ibl 25567 df-itg 25568 df-0p 25615 df-limc 25811 df-dv 25812

This theorem is referenced by: wallispilem2 46498 wallispilem4 46500 wallispilem5 46501

W3C validator