itgsin0pilem1 - Mathbox for Glauco Siliprandi

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Theorem itgsin0pilem1 46366

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 6829	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (cos‘𝑡) = (cos‘𝑥))
3	2	negeqd 11376	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → -(cos‘𝑡) = -(cos‘𝑥))
4	3	cbvmptv 5178	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡)) = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
5	1, 4	eqtri 2758	. . . . . . . . . 10 ⊢ 𝐶 = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
6	5	oveq2i 7367	. . . . . . . . 9 ⊢ (ℝ D 𝐶) = (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)))
7		ax-resscn 11084	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ ℂ)
9		0re 11135	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
10		pire 26409	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
11		iccssre 13371	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	9, 10, 11	mp2an 693	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℝ)
14	12, 7	sstri 3926	. . . . . . . . . . . . . . 15 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3913	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	coscld 16087	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
18	17	negcld 11481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
19		tgioo4 24758	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
20		eqid 2735	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21		iccntr 24775	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
22	9, 10, 21	mp2an 693	. . . . . . . . . . . 12 ⊢ ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π)
23	22	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
24	8, 13, 18, 19, 20, 23	dvmptntr 25926	. . . . . . . . . 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 11119	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
28		recn 11117	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
29	28	coscld 16087	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
30	29	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
31	30	negcld 11481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
32	28	sincld 16086	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
34	32	negcld 11481	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
36		dvcosre 46328	. . . . . . . . . . . . . 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 25921	. . . . . . . . . . . 12 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
39	32	negnegd 11485	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
40	39	mpteq2ia 5169	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥))
41	38, 40	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
42		ioossre 13349	. . . . . . . . . . . 12 ⊢ (0(,)π) ⊆ ℝ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ⊆ ℝ)
44		iooretop 24718	. . . . . . . . . . . 12 ⊢ (0(,)π) ∈ (topGen‘ran (,))
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ∈ (topGen‘ran (,)))
46	27, 31, 33, 41, 43, 19, 20, 45	dvmptres 25918	. . . . . . . . . 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 2762	. . . . . . . 8 ⊢ (ℝ D 𝐶) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
49	48	fveq1i 6830	. . . . . . 7 ⊢ ((ℝ D 𝐶)‘𝑥) = ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥)
50	42, 7	sstri 3926	. . . . . . . . . 10 ⊢ (0(,)π) ⊆ ℂ
51	50	sseli 3913	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
52	51	sincld 16086	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
53		eqid 2735	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
54	53	fvmpt2 6948	. . . . . . . 8 ⊢ ((𝑥 ∈ (0(,)π) ∧ (sin‘𝑥) ∈ ℂ) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
55	52, 54	mpdan 688	. . . . . . 7 ⊢ (𝑥 ∈ (0(,)π) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
56	49, 55	eqtrid 2782	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
57	56	adantl 481	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (0(,)π)) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
58	57	itgeq2dv 25737	. . . 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 26411	. . . . . . 7 ⊢ 0 < π
63	9, 10, 62	ltleii 11258	. . . . . 6 ⊢ 0 ≤ π
64	63	a1i 11	. . . . 5 ⊢ (⊤ → 0 ≤ π)
65		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥sin
66		sincn 26397	. . . . . . . 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 46005	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
70	48, 69	eqeltrid 2839	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ ((0(,)π)–cn→ℂ))
71		ioossicc 13375	. . . . . . . 8 ⊢ (0(,)π) ⊆ (0[,]π)
72	71	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ (0[,]π))
73		ioombl 25520	. . . . . . . 8 ⊢ (0(,)π) ∈ dom vol
74	73	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ∈ dom vol)
75	15	sincld 16086	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
76	75	adantl 481	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
77	14	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℂ)
78	65, 67, 77	cncfmptss 46005	. . . . . . . . . 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 25796	. . . . . . . . 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 1464	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹
82	81	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
83	72, 74, 76, 82	iblss 25760	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
84	48, 83	eqeltrid 2839	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ 𝐿¹)
85	16	negcld 11481	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
86		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
87	86	fvmpt2 6948	. . . . . . . . . . 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 2741	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
90	89	mpteq2ia 5169	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
91		nfmpt1 5173	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
92		coscn 26398	. . . . . . . . . . . 12 ⊢ cos ∈ (ℂ–cn→ℂ)
93	86	negfcncf 24878	. . . . . . . . . . . 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 46005	. . . . . . . . 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 2831	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
99	5, 98	eqeltri 2831	. . . . . 6 ⊢ 𝐶 ∈ ((0[,]π)–cn→ℂ)
100	99	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ ((0[,]π)–cn→ℂ))
101	60, 61, 64, 70, 84, 100	ftc2 25999	. . . 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 2760	. 2 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
104		0xr 11181	. . . . 5 ⊢ 0 ∈ ℝ^*
105	10	rexri 11192	. . . . 5 ⊢ π ∈ ℝ^*
106		ubicc2 13407	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → π ∈ (0[,]π))
107	104, 105, 63, 106	mp3an 1464	. . . 4 ⊢ π ∈ (0[,]π)
108		fveq2 6829	. . . . . . . 8 ⊢ (𝑡 = π → (cos‘𝑡) = (cos‘π))
109		cospi 26424	. . . . . . . 8 ⊢ (cos‘π) = -1
110	108, 109	eqtrdi 2786	. . . . . . 7 ⊢ (𝑡 = π → (cos‘𝑡) = -1)
111	110	negeqd 11376	. . . . . 6 ⊢ (𝑡 = π → -(cos‘𝑡) = --1)
112		ax-1cn 11085	. . . . . . . 8 ⊢ 1 ∈ ℂ
113	112	a1i 11	. . . . . . 7 ⊢ (𝑡 = π → 1 ∈ ℂ)
114	113	negnegd 11485	. . . . . 6 ⊢ (𝑡 = π → --1 = 1)
115	111, 114	eqtrd 2770	. . . . 5 ⊢ (𝑡 = π → -(cos‘𝑡) = 1)
116		1ex 11129	. . . . 5 ⊢ 1 ∈ V
117	115, 1, 116	fvmpt 6936	. . . 4 ⊢ (π ∈ (0[,]π) → (𝐶‘π) = 1)
118	107, 117	ax-mp 5	. . 3 ⊢ (𝐶‘π) = 1
119		lbicc2 13406	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → 0 ∈ (0[,]π))
120	104, 105, 63, 119	mp3an 1464	. . . . 5 ⊢ 0 ∈ (0[,]π)
121		fveq2 6829	. . . . . . 7 ⊢ (𝑡 = 0 → (cos‘𝑡) = (cos‘0))
122	121	negeqd 11376	. . . . . 6 ⊢ (𝑡 = 0 → -(cos‘𝑡) = -(cos‘0))
123		negex 11380	. . . . . 6 ⊢ -(cos‘0) ∈ V
124	122, 1, 123	fvmpt 6936	. . . . 5 ⊢ (0 ∈ (0[,]π) → (𝐶‘0) = -(cos‘0))
125	120, 124	ax-mp 5	. . . 4 ⊢ (𝐶‘0) = -(cos‘0)
126		cos0 16106	. . . . 5 ⊢ (cos‘0) = 1
127	126	negeqi 11375	. . . 4 ⊢ -(cos‘0) = -1
128	125, 127	eqtri 2758	. . 3 ⊢ (𝐶‘0) = -1
129	118, 128	oveq12i 7368	. 2 ⊢ ((𝐶‘π) − (𝐶‘0)) = (1 − -1)
130	112, 112	subnegi 11462	. . 3 ⊢ (1 − -1) = (1 + 1)
131		1p1e2 12290	. . 3 ⊢ (1 + 1) = 2
132	130, 131	eqtri 2758	. 2 ⊢ (1 − -1) = 2
133	103, 129, 132	3eqtri 2762	1 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ⊆ wss 3885 {cpr 4559 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 ran crn 5621 ‘cfv 6487 (class class class)co 7356 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 ℝ^*cxr 11167 ≤ cle 11169 − cmin 11366 -cneg 11367 2c2 12225 (,)cioo 13287 [,]cicc 13290 sincsin 16017 cosccos 16018 πcpi 16020 TopOpenctopn 17373 topGenctg 17389 ℂ_fldccnfld 21341 intcnt 22970 –cn→ccncf 24831 volcvol 25418 𝐿¹cibl 25572 ∫citg 25573 D cdv 25818

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-symdif 4183 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-disj 5042 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-isom 6496 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8632 df-map 8764 df-pm 8765 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9264 df-fi 9313 df-sup 9344 df-inf 9345 df-oi 9414 df-dju 9814 df-card 9852 df-acn 9855 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-4 12235 df-5 12236 df-6 12237 df-7 12238 df-8 12239 df-9 12240 df-n0 12427 df-z 12514 df-dec 12634 df-uz 12778 df-q 12888 df-rp 12932 df-xneg 13052 df-xadd 13053 df-xmul 13054 df-ioo 13291 df-ioc 13292 df-ico 13293 df-icc 13294 df-fz 13451 df-fzo 13598 df-fl 13740 df-mod 13818 df-seq 13953 df-exp 14013 df-fac 14225 df-bc 14254 df-hash 14282 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15422 df-clim 15439 df-rlim 15440 df-sum 15638 df-ef 16021 df-sin 16023 df-cos 16024 df-pi 16026 df-struct 17106 df-sets 17123 df-slot 17141 df-ndx 17153 df-base 17169 df-ress 17190 df-plusg 17222 df-mulr 17223 df-starv 17224 df-sca 17225 df-vsca 17226 df-ip 17227 df-tset 17228 df-ple 17229 df-ds 17231 df-unif 17232 df-hom 17233 df-cco 17234 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18741 df-mulg 19033 df-cntz 19281 df-cmn 19746 df-psmet 21333 df-xmet 21334 df-met 21335 df-bl 21336 df-mopn 21337 df-fbas 21338 df-fg 21339 df-cnfld 21342 df-top 22847 df-topon 22864 df-topsp 22886 df-bases 22899 df-cld 22972 df-ntr 22973 df-cls 22974 df-nei 23051 df-lp 23089 df-perf 23090 df-cn 23180 df-cnp 23181 df-haus 23268 df-cmp 23340 df-tx 23515 df-hmeo 23708 df-fil 23799 df-fm 23891 df-flim 23892 df-flf 23893 df-xms 24273 df-ms 24274 df-tms 24275 df-cncf 24833 df-ovol 25419 df-vol 25420 df-mbf 25574 df-itg1 25575 df-itg2 25576 df-ibl 25577 df-itg 25578 df-0p 25625 df-limc 25821 df-dv 25822

This theorem is referenced by: itgsin0pi 46368

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