itgsin0pilem1 - Mathbox for Glauco Siliprandi

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Theorem itgsin0pilem1 46529

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 6869	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (cos‘𝑡) = (cos‘𝑥))
3	2	negeqd 11426	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → -(cos‘𝑡) = -(cos‘𝑥))
4	3	cbvmptv 5206	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡)) = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
5	1, 4	eqtri 2787	. . . . . . . . . 10 ⊢ 𝐶 = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
6	5	oveq2i 7409	. . . . . . . . 9 ⊢ (ℝ D 𝐶) = (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)))
7		ax-resscn 11132	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ ℂ)
9		0re 11185	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
10		pire 26521	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
11		iccssre 13435	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	9, 10, 11	mp2an 702	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℝ)
14	12, 7	sstri 3947	. . . . . . . . . . . . . . 15 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3934	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	coscld 16165	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
17	16	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
18	17	negcld 11531	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
19		tgioo4 24867	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
20		eqid 2764	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21		iccntr 24884	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
22	9, 10, 21	mp2an 702	. . . . . . . . . . . 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 26035	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))))
25	24	mptru 1569	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥)))
26		reelprrecn 11167	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
28		recn 11165	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
29	28	coscld 16165	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
30	29	adantl 485	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
31	30	negcld 11531	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
32	28	sincld 16164	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
33	32	adantl 485	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
34	32	negcld 11531	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
35	34	adantl 485	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
36		dvcosre 46491	. . . . . . . . . . . . . 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 26030	. . . . . . . . . . . 12 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
39	32	negnegd 11535	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
40	39	mpteq2ia 5197	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥))
41	38, 40	eqtrdi 2815	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
42		ioossre 13413	. . . . . . . . . . . 12 ⊢ (0(,)π) ⊆ ℝ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ⊆ ℝ)
44		iooretop 24827	. . . . . . . . . . . 12 ⊢ (0(,)π) ∈ (topGen‘ran (,))
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ∈ (topGen‘ran (,)))
46	27, 31, 33, 41, 43, 19, 20, 45	dvmptres 26027	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
47	46	mptru 1569	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
48	6, 25, 47	3eqtri 2791	. . . . . . . 8 ⊢ (ℝ D 𝐶) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
49	48	fveq1i 6870	. . . . . . 7 ⊢ ((ℝ D 𝐶)‘𝑥) = ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥)
50	42, 7	sstri 3947	. . . . . . . . . 10 ⊢ (0(,)π) ⊆ ℂ
51	50	sseli 3934	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
52	51	sincld 16164	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
53		eqid 2764	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
54	53	fvmpt2 6989	. . . . . . . 8 ⊢ ((𝑥 ∈ (0(,)π) ∧ (sin‘𝑥) ∈ ℂ) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
55	52, 54	mpdan 697	. . . . . . 7 ⊢ (𝑥 ∈ (0(,)π) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
56	49, 55	eqtrid 2811	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
57	56	adantl 485	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (0(,)π)) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
58	57	itgeq2dv 25846	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
59	58	mptru 1569	. . 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 26525	. . . . . . 7 ⊢ 0 < π
63	9, 10, 62	ltleii 11308	. . . . . 6 ⊢ 0 ≤ π
64	63	a1i 11	. . . . 5 ⊢ (⊤ → 0 ≤ π)
65		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑥sin
66		sincn 26509	. . . . . . . 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 46168	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
70	48, 69	eqeltrid 2868	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ ((0(,)π)–cn→ℂ))
71		ioossicc 13439	. . . . . . . 8 ⊢ (0(,)π) ⊆ (0[,]π)
72	71	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ (0[,]π))
73		ioombl 25629	. . . . . . . 8 ⊢ (0(,)π) ∈ dom vol
74	73	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ∈ dom vol)
75	15	sincld 16164	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
76	75	adantl 485	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
77	14	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℂ)
78	65, 67, 77	cncfmptss 46168	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
79	78	mptru 1569	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
80		cniccibl 25905	. . . . . . . . 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 1484	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹
82	81	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
83	72, 74, 76, 82	iblss 25869	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
84	48, 83	eqeltrid 2868	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ 𝐿¹)
85	16	negcld 11531	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
86		eqid 2764	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
87	86	fvmpt2 6989	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
88	15, 85, 87	syl2anc 593	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
89	88	eqcomd 2770	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
90	89	mpteq2ia 5197	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
91		nfmpt1 5201	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
92		coscn 26510	. . . . . . . . . . . 12 ⊢ cos ∈ (ℂ–cn→ℂ)
93	86	negfcncf 24987	. . . . . . . . . . . 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 46168	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
97	96	mptru 1569	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
98	90, 97	eqeltri 2860	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
99	5, 98	eqeltri 2860	. . . . . 6 ⊢ 𝐶 ∈ ((0[,]π)–cn→ℂ)
100	99	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ ((0[,]π)–cn→ℂ))
101	60, 61, 64, 70, 84, 100	ftc2 26108	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0)))
102	101	mptru 1569	. . 3 ⊢ ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
103	59, 102	eqtr3i 2789	. 2 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
104		0xr 11231	. . . . 5 ⊢ 0 ∈ ℝ^*
105	10	rexri 11242	. . . . 5 ⊢ π ∈ ℝ^*
106		ubicc2 13471	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → π ∈ (0[,]π))
107	104, 105, 63, 106	mp3an 1484	. . . 4 ⊢ π ∈ (0[,]π)
108		fveq2 6869	. . . . . . . 8 ⊢ (𝑡 = π → (cos‘𝑡) = (cos‘π))
109		cospi 26539	. . . . . . . 8 ⊢ (cos‘π) = -1
110	108, 109	eqtrdi 2815	. . . . . . 7 ⊢ (𝑡 = π → (cos‘𝑡) = -1)
111	110	negeqd 11426	. . . . . 6 ⊢ (𝑡 = π → -(cos‘𝑡) = --1)
112		ax-1cn 11133	. . . . . . . 8 ⊢ 1 ∈ ℂ
113	112	a1i 11	. . . . . . 7 ⊢ (𝑡 = π → 1 ∈ ℂ)
114	113	negnegd 11535	. . . . . 6 ⊢ (𝑡 = π → --1 = 1)
115	111, 114	eqtrd 2799	. . . . 5 ⊢ (𝑡 = π → -(cos‘𝑡) = 1)
116		1ex 11178	. . . . 5 ⊢ 1 ∈ V
117	115, 1, 116	fvmpt 6977	. . . 4 ⊢ (π ∈ (0[,]π) → (𝐶‘π) = 1)
118	107, 117	ax-mp 5	. . 3 ⊢ (𝐶‘π) = 1
119		lbicc2 13470	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → 0 ∈ (0[,]π))
120	104, 105, 63, 119	mp3an 1484	. . . . 5 ⊢ 0 ∈ (0[,]π)
121		fveq2 6869	. . . . . . 7 ⊢ (𝑡 = 0 → (cos‘𝑡) = (cos‘0))
122	121	negeqd 11426	. . . . . 6 ⊢ (𝑡 = 0 → -(cos‘𝑡) = -(cos‘0))
123		negex 11430	. . . . . 6 ⊢ -(cos‘0) ∈ V
124	122, 1, 123	fvmpt 6977	. . . . 5 ⊢ (0 ∈ (0[,]π) → (𝐶‘0) = -(cos‘0))
125	120, 124	ax-mp 5	. . . 4 ⊢ (𝐶‘0) = -(cos‘0)
126		cos0 16184	. . . . 5 ⊢ (cos‘0) = 1
127	126	negeqi 11425	. . . 4 ⊢ -(cos‘0) = -1
128	125, 127	eqtri 2787	. . 3 ⊢ (𝐶‘0) = -1
129	118, 128	oveq12i 7410	. 2 ⊢ ((𝐶‘π) − (𝐶‘0)) = (1 − -1)
130	112, 112	subnegi 11512	. . 3 ⊢ (1 − -1) = (1 + 1)
131		1p1e2 12343	. . 3 ⊢ (1 + 1) = 2
132	130, 131	eqtri 2787	. 2 ⊢ (1 − -1) = 2
133	103, 129, 132	3eqtri 2791	1 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 = wceq 1562 ⊤wtru 1563 ∈ wcel 2144 ⊆ wss 3906 {cpr 4586 class class class wbr 5102 ↦ cmpt 5183 dom cdm 5649 ran crn 5650 ‘cfv 6523 (class class class)co 7398 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℝ^*cxr 11217 ≤ cle 11219 − cmin 11416 -cneg 11417 2c2 12274 (,)cioo 13351 [,]cicc 13354 sincsin 16095 cosccos 16096 πcpi 16098 TopOpenctopn 17452 topGenctg 17468 ℂ_fldccnfld 21426 intcnt 23079 –cn→ccncf 24940 volcvol 25527 𝐿¹cibl 25681 ∫citg 25682 D cdv 25927

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cc 10394 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-symdif 4207 df-nul 4288 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 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-ofr 7663 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-oadd 8443 df-omul 8444 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-dju 9861 df-card 9899 df-acn 9902 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13515 df-fzo 13662 df-fl 13804 df-mod 13882 df-seq 14017 df-exp 14077 df-fac 14289 df-bc 14318 df-hash 14346 df-shft 15082 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-limsup 15500 df-clim 15517 df-rlim 15518 df-sum 15716 df-ef 16099 df-sin 16101 df-cos 16102 df-pi 16104 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cn 23289 df-cnp 23290 df-haus 23377 df-cmp 23449 df-tx 23624 df-hmeo 23817 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942 df-ovol 25528 df-vol 25529 df-mbf 25683 df-itg1 25684 df-itg2 25685 df-ibl 25686 df-itg 25687 df-0p 25734 df-limc 25930 df-dv 25931

This theorem is referenced by: itgsin0pi 46531

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