itgsin0pilem1 - Mathbox for Glauco Siliprandi

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Theorem itgsin0pilem1 44666

Description: Calculation of the integral for sine on the (0,π) interval. (Contributed by Glauco Siliprandi, 29-Jun-2017.)

**Hypothesis**
Ref	Expression
itgsin0pilem1.1	⊢ 𝐶 = (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡))

**Assertion**
Ref	Expression
itgsin0pilem1	⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Distinct variable groups: 𝑥,𝑡 𝑥,𝐶 Allowed substitution hint: 𝐶(𝑡)

**Proof of Theorem itgsin0pilem1**
Step	Hyp	Ref	Expression
1		itgsin0pilem1.1	. . . . . . . . . . 11 ⊢ 𝐶 = (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡))
2		fveq2 6892	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (cos‘𝑡) = (cos‘𝑥))
3	2	negeqd 11454	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → -(cos‘𝑡) = -(cos‘𝑥))
4	3	cbvmptv 5262	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡)) = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
5	1, 4	eqtri 2761	. . . . . . . . . 10 ⊢ 𝐶 = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
6	5	oveq2i 7420	. . . . . . . . 9 ⊢ (ℝ D 𝐶) = (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)))
7		ax-resscn 11167	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ ℂ)
9		0re 11216	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
10		pire 25968	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
11		iccssre 13406	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	9, 10, 11	mp2an 691	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℝ)
14	12, 7	sstri 3992	. . . . . . . . . . . . . . 15 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3979	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	coscld 16074	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
17	16	adantl 483	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
18	17	negcld 11558	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
19		eqid 2733	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20	19	tgioo2 24319	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		iccntr 24337	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
22	9, 10, 21	mp2an 691	. . . . . . . . . . . 12 ⊢ ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π)
23	22	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
24	8, 13, 18, 20, 19, 23	dvmptntr 25488	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))))
25	24	mptru 1549	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥)))
26		reelprrecn 11202	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
28		recn 11200	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
29	28	coscld 16074	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
30	29	adantl 483	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
31	30	negcld 11558	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
32	28	sincld 16073	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
33	32	adantl 483	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
34	32	negcld 11558	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
35	34	adantl 483	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
36		dvcosre 44628	. . . . . . . . . . . . . 14 ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))
37	36	a1i 11	. . . . . . . . . . . . 13 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)))
38	27, 30, 35, 37	dvmptneg 25483	. . . . . . . . . . . 12 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
39	32	negnegd 11562	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
40	39	mpteq2ia 5252	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥))
41	38, 40	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
42		ioossre 13385	. . . . . . . . . . . 12 ⊢ (0(,)π) ⊆ ℝ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ⊆ ℝ)
44		iooretop 24282	. . . . . . . . . . . 12 ⊢ (0(,)π) ∈ (topGen‘ran (,))
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ∈ (topGen‘ran (,)))
46	27, 31, 33, 41, 43, 20, 19, 45	dvmptres 25480	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
47	46	mptru 1549	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
48	6, 25, 47	3eqtri 2765	. . . . . . . 8 ⊢ (ℝ D 𝐶) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
49	48	fveq1i 6893	. . . . . . 7 ⊢ ((ℝ D 𝐶)‘𝑥) = ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥)
50	42, 7	sstri 3992	. . . . . . . . . 10 ⊢ (0(,)π) ⊆ ℂ
51	50	sseli 3979	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
52	51	sincld 16073	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
53		eqid 2733	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
54	53	fvmpt2 7010	. . . . . . . 8 ⊢ ((𝑥 ∈ (0(,)π) ∧ (sin‘𝑥) ∈ ℂ) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
55	52, 54	mpdan 686	. . . . . . 7 ⊢ (𝑥 ∈ (0(,)π) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
56	49, 55	eqtrid 2785	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
57	56	adantl 483	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (0(,)π)) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
58	57	itgeq2dv 25299	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
59	58	mptru 1549	. . 3 ⊢ ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥
60	9	a1i 11	. . . . 5 ⊢ (⊤ → 0 ∈ ℝ)
61	10	a1i 11	. . . . 5 ⊢ (⊤ → π ∈ ℝ)
62		pipos 25970	. . . . . . 7 ⊢ 0 < π
63	9, 10, 62	ltleii 11337	. . . . . 6 ⊢ 0 ≤ π
64	63	a1i 11	. . . . 5 ⊢ (⊤ → 0 ≤ π)
65		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑥sin
66		sincn 25956	. . . . . . . 8 ⊢ sin ∈ (ℂ–cn→ℂ)
67	66	a1i 11	. . . . . . 7 ⊢ (⊤ → sin ∈ (ℂ–cn→ℂ))
68	50	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ ℂ)
69	65, 67, 68	cncfmptss 44303	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
70	48, 69	eqeltrid 2838	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ ((0(,)π)–cn→ℂ))
71		ioossicc 13410	. . . . . . . 8 ⊢ (0(,)π) ⊆ (0[,]π)
72	71	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ (0[,]π))
73		ioombl 25082	. . . . . . . 8 ⊢ (0(,)π) ∈ dom vol
74	73	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ∈ dom vol)
75	15	sincld 16073	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
76	75	adantl 483	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
77	14	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℂ)
78	65, 67, 77	cncfmptss 44303	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
79	78	mptru 1549	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
80		cniccibl 25358	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ ∧ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ)) → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
81	9, 10, 79, 80	mp3an 1462	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹
82	81	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
83	72, 74, 76, 82	iblss 25322	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
84	48, 83	eqeltrid 2838	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ 𝐿¹)
85	16	negcld 11558	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
86		eqid 2733	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
87	86	fvmpt2 7010	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
88	15, 85, 87	syl2anc 585	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
89	88	eqcomd 2739	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
90	89	mpteq2ia 5252	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
91		nfmpt1 5257	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
92		coscn 25957	. . . . . . . . . . . 12 ⊢ cos ∈ (ℂ–cn→ℂ)
93	86	negfcncf 24439	. . . . . . . . . . . 12 ⊢ (cos ∈ (ℂ–cn→ℂ) → (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) ∈ (ℂ–cn→ℂ))
94	92, 93	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) ∈ (ℂ–cn→ℂ)
95	94	a1i 11	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) ∈ (ℂ–cn→ℂ))
96	91, 95, 77	cncfmptss 44303	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
97	96	mptru 1549	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
98	90, 97	eqeltri 2830	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
99	5, 98	eqeltri 2830	. . . . . 6 ⊢ 𝐶 ∈ ((0[,]π)–cn→ℂ)
100	99	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ ((0[,]π)–cn→ℂ))
101	60, 61, 64, 70, 84, 100	ftc2 25561	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0)))
102	101	mptru 1549	. . 3 ⊢ ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
103	59, 102	eqtr3i 2763	. 2 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
104		0xr 11261	. . . . 5 ⊢ 0 ∈ ℝ^*
105	10	rexri 11272	. . . . 5 ⊢ π ∈ ℝ^*
106		ubicc2 13442	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → π ∈ (0[,]π))
107	104, 105, 63, 106	mp3an 1462	. . . 4 ⊢ π ∈ (0[,]π)
108		fveq2 6892	. . . . . . . 8 ⊢ (𝑡 = π → (cos‘𝑡) = (cos‘π))
109		cospi 25982	. . . . . . . 8 ⊢ (cos‘π) = -1
110	108, 109	eqtrdi 2789	. . . . . . 7 ⊢ (𝑡 = π → (cos‘𝑡) = -1)
111	110	negeqd 11454	. . . . . 6 ⊢ (𝑡 = π → -(cos‘𝑡) = --1)
112		ax-1cn 11168	. . . . . . . 8 ⊢ 1 ∈ ℂ
113	112	a1i 11	. . . . . . 7 ⊢ (𝑡 = π → 1 ∈ ℂ)
114	113	negnegd 11562	. . . . . 6 ⊢ (𝑡 = π → --1 = 1)
115	111, 114	eqtrd 2773	. . . . 5 ⊢ (𝑡 = π → -(cos‘𝑡) = 1)
116		1ex 11210	. . . . 5 ⊢ 1 ∈ V
117	115, 1, 116	fvmpt 6999	. . . 4 ⊢ (π ∈ (0[,]π) → (𝐶‘π) = 1)
118	107, 117	ax-mp 5	. . 3 ⊢ (𝐶‘π) = 1
119		lbicc2 13441	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → 0 ∈ (0[,]π))
120	104, 105, 63, 119	mp3an 1462	. . . . 5 ⊢ 0 ∈ (0[,]π)
121		fveq2 6892	. . . . . . 7 ⊢ (𝑡 = 0 → (cos‘𝑡) = (cos‘0))
122	121	negeqd 11454	. . . . . 6 ⊢ (𝑡 = 0 → -(cos‘𝑡) = -(cos‘0))
123		negex 11458	. . . . . 6 ⊢ -(cos‘0) ∈ V
124	122, 1, 123	fvmpt 6999	. . . . 5 ⊢ (0 ∈ (0[,]π) → (𝐶‘0) = -(cos‘0))
125	120, 124	ax-mp 5	. . . 4 ⊢ (𝐶‘0) = -(cos‘0)
126		cos0 16093	. . . . 5 ⊢ (cos‘0) = 1
127	126	negeqi 11453	. . . 4 ⊢ -(cos‘0) = -1
128	125, 127	eqtri 2761	. . 3 ⊢ (𝐶‘0) = -1
129	118, 128	oveq12i 7421	. 2 ⊢ ((𝐶‘π) − (𝐶‘0)) = (1 − -1)
130	112, 112	subnegi 11539	. . 3 ⊢ (1 − -1) = (1 + 1)
131		1p1e2 12337	. . 3 ⊢ (1 + 1) = 2
132	130, 131	eqtri 2761	. 2 ⊢ (1 − -1) = 2
133	103, 129, 132	3eqtri 2765	1 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 ⊆ wss 3949 {cpr 4631 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ran crn 5678 ‘cfv 6544 (class class class)co 7409 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 ℝ^*cxr 11247 ≤ cle 11249 − cmin 11444 -cneg 11445 2c2 12267 (,)cioo 13324 [,]cicc 13327 sincsin 16007 cosccos 16008 πcpi 16010 TopOpenctopn 17367 topGenctg 17383 ℂ_fldccnfld 20944 intcnt 22521 –cn→ccncf 24392 volcvol 24980 𝐿¹cibl 25134 ∫citg 25135 D cdv 25380

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cc 10430 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-symdif 4243 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-ofr 7671 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-mulg 18951 df-cntz 19181 df-cmn 19650 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-ovol 24981 df-vol 24982 df-mbf 25136 df-itg1 25137 df-itg2 25138 df-ibl 25139 df-itg 25140 df-0p 25187 df-limc 25383 df-dv 25384

This theorem is referenced by: itgsin0pi 44668

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