itgsin0pilem1 - Mathbox for Glauco Siliprandi

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Theorem itgsin0pilem1 40165

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 6191	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (cos‘𝑡) = (cos‘𝑥))
3	2	negeqd 10275	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → -(cos‘𝑡) = -(cos‘𝑥))
4	3	cbvmptv 4750	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡)) = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
5	1, 4	eqtri 2644	. . . . . . . . . 10 ⊢ 𝐶 = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
6	5	oveq2i 6661	. . . . . . . . 9 ⊢ (ℝ D 𝐶) = (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)))
7		ax-resscn 9993	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ ℂ)
9		0re 10040	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
10		pire 24210	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
11		iccssre 12255	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	9, 10, 11	mp2an 708	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℝ)
14	12, 7	sstri 3612	. . . . . . . . . . . . . . 15 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3599	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	coscld 14861	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
17	16	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
18	17	negcld 10379	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
19		eqid 2622	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
20	19	tgioo2 22606	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		iccntr 22624	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
22	9, 10, 21	mp2an 708	. . . . . . . . . . . 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 23734	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))))
25	24	trud 1493	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥)))
26		reelprrecn 10028	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
28		recn 10026	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
29	28	coscld 14861	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
30	29	adantl 482	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
31	30	negcld 10379	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
32	28	sincld 14860	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
33	32	adantl 482	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
34	32	negcld 10379	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
35	34	adantl 482	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
36		dvcosre 40126	. . . . . . . . . . . . . 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 23729	. . . . . . . . . . . 12 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
39	32	negnegd 10383	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
40	39	mpteq2ia 4740	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥))
41	38, 40	syl6eq 2672	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
42		ioossre 12235	. . . . . . . . . . . 12 ⊢ (0(,)π) ⊆ ℝ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ⊆ ℝ)
44		iooretop 22569	. . . . . . . . . . . 12 ⊢ (0(,)π) ∈ (topGen‘ran (,))
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ∈ (topGen‘ran (,)))
46	27, 31, 33, 41, 43, 20, 19, 45	dvmptres 23726	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
47	46	trud 1493	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
48	6, 25, 47	3eqtri 2648	. . . . . . . 8 ⊢ (ℝ D 𝐶) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
49	48	fveq1i 6192	. . . . . . 7 ⊢ ((ℝ D 𝐶)‘𝑥) = ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥)
50	42, 7	sstri 3612	. . . . . . . . . 10 ⊢ (0(,)π) ⊆ ℂ
51	50	sseli 3599	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
52	51	sincld 14860	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
53		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
54	53	fvmpt2 6291	. . . . . . . 8 ⊢ ((𝑥 ∈ (0(,)π) ∧ (sin‘𝑥) ∈ ℂ) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
55	52, 54	mpdan 702	. . . . . . 7 ⊢ (𝑥 ∈ (0(,)π) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
56	49, 55	syl5eq 2668	. . . . . 6 ⊢ (𝑥 ∈ (0(,)π) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
57	56	adantl 482	. . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (0(,)π)) → ((ℝ D 𝐶)‘𝑥) = (sin‘𝑥))
58	57	itgeq2dv 23548	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
59	58	trud 1493	. . 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 24212	. . . . . . 7 ⊢ 0 < π
63	9, 10, 62	ltleii 10160	. . . . . 6 ⊢ 0 ≤ π
64	63	a1i 11	. . . . 5 ⊢ (⊤ → 0 ≤ π)
65		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥sin
66		sincn 24198	. . . . . . . 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 39819	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
70	48, 69	syl5eqel 2705	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ ((0(,)π)–cn→ℂ))
71		ioossicc 12259	. . . . . . . 8 ⊢ (0(,)π) ⊆ (0[,]π)
72	71	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ (0[,]π))
73		ioombl 23333	. . . . . . . 8 ⊢ (0(,)π) ∈ dom vol
74	73	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ∈ dom vol)
75	15	sincld 14860	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) → (sin‘𝑥) ∈ ℂ)
76	75	adantl 482	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (sin‘𝑥) ∈ ℂ)
77	14	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℂ)
78	65, 67, 77	cncfmptss 39819	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
79	78	trud 1493	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
80		cniccibl 23607	. . . . . . . . 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 1424	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹
82	81	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
83	72, 74, 76, 82	iblss 23571	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
84	48, 83	syl5eqel 2705	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ 𝐿¹)
85	16	negcld 10379	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
86		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
87	86	fvmpt2 6291	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
88	15, 85, 87	syl2anc 693	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
89	88	eqcomd 2628	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
90	89	mpteq2ia 4740	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
91		nfmpt1 4747	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
92		coscn 24199	. . . . . . . . . . . 12 ⊢ cos ∈ (ℂ–cn→ℂ)
93	86	negfcncf 22722	. . . . . . . . . . . 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 39819	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
97	96	trud 1493	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
98	90, 97	eqeltri 2697	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
99	5, 98	eqeltri 2697	. . . . . 6 ⊢ 𝐶 ∈ ((0[,]π)–cn→ℂ)
100	99	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ ((0[,]π)–cn→ℂ))
101	60, 61, 64, 70, 84, 100	ftc2 23807	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0)))
102	101	trud 1493	. . 3 ⊢ ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
103	59, 102	eqtr3i 2646	. 2 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
104		0xr 10086	. . . . 5 ⊢ 0 ∈ ℝ^*
105	10	rexri 10097	. . . . 5 ⊢ π ∈ ℝ^*
106		ubicc2 12289	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → π ∈ (0[,]π))
107	104, 105, 63, 106	mp3an 1424	. . . 4 ⊢ π ∈ (0[,]π)
108		fveq2 6191	. . . . . . . 8 ⊢ (𝑡 = π → (cos‘𝑡) = (cos‘π))
109		cospi 24224	. . . . . . . 8 ⊢ (cos‘π) = -1
110	108, 109	syl6eq 2672	. . . . . . 7 ⊢ (𝑡 = π → (cos‘𝑡) = -1)
111	110	negeqd 10275	. . . . . 6 ⊢ (𝑡 = π → -(cos‘𝑡) = --1)
112		ax-1cn 9994	. . . . . . . 8 ⊢ 1 ∈ ℂ
113	112	a1i 11	. . . . . . 7 ⊢ (𝑡 = π → 1 ∈ ℂ)
114	113	negnegd 10383	. . . . . 6 ⊢ (𝑡 = π → --1 = 1)
115	111, 114	eqtrd 2656	. . . . 5 ⊢ (𝑡 = π → -(cos‘𝑡) = 1)
116		1ex 10035	. . . . 5 ⊢ 1 ∈ V
117	115, 1, 116	fvmpt 6282	. . . 4 ⊢ (π ∈ (0[,]π) → (𝐶‘π) = 1)
118	107, 117	ax-mp 5	. . 3 ⊢ (𝐶‘π) = 1
119		lbicc2 12288	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → 0 ∈ (0[,]π))
120	104, 105, 63, 119	mp3an 1424	. . . . 5 ⊢ 0 ∈ (0[,]π)
121		fveq2 6191	. . . . . . 7 ⊢ (𝑡 = 0 → (cos‘𝑡) = (cos‘0))
122	121	negeqd 10275	. . . . . 6 ⊢ (𝑡 = 0 → -(cos‘𝑡) = -(cos‘0))
123		negex 10279	. . . . . 6 ⊢ -(cos‘0) ∈ V
124	122, 1, 123	fvmpt 6282	. . . . 5 ⊢ (0 ∈ (0[,]π) → (𝐶‘0) = -(cos‘0))
125	120, 124	ax-mp 5	. . . 4 ⊢ (𝐶‘0) = -(cos‘0)
126		cos0 14880	. . . . 5 ⊢ (cos‘0) = 1
127	126	negeqi 10274	. . . 4 ⊢ -(cos‘0) = -1
128	125, 127	eqtri 2644	. . 3 ⊢ (𝐶‘0) = -1
129	118, 128	oveq12i 6662	. 2 ⊢ ((𝐶‘π) − (𝐶‘0)) = (1 − -1)
130	112, 112	subnegi 10360	. . 3 ⊢ (1 − -1) = (1 + 1)
131		1p1e2 11134	. . 3 ⊢ (1 + 1) = 2
132	130, 131	eqtri 2644	. 2 ⊢ (1 − -1) = 2
133	103, 129, 132	3eqtri 2648	1 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 ⊆ wss 3574 {cpr 4179 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ^*cxr 10073 ≤ cle 10075 − cmin 10266 -cneg 10267 2c2 11070 (,)cioo 12175 [,]cicc 12178 sincsin 14794 cosccos 14795 πcpi 14797 TopOpenctopn 16082 topGenctg 16098 ℂ_fldccnfld 19746 intcnt 20821 –cn→ccncf 22679 volcvol 23232 𝐿¹cibl 23386 ∫citg 23387 D cdv 23627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-limc 23630 df-dv 23631

This theorem is referenced by: itgsin0pi 40167

W3C validator