itgsin0pilem1 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > itgsin0pilem1		Structured version Visualization version GIF version

Theorem itgsin0pilem1 45951

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 6822	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (cos‘𝑡) = (cos‘𝑥))
3	2	negeqd 11357	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → -(cos‘𝑡) = -(cos‘𝑥))
4	3	cbvmptv 5196	. . . . . . . . . . 11 ⊢ (𝑡 ∈ (0[,]π) ↦ -(cos‘𝑡)) = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
5	1, 4	eqtri 2752	. . . . . . . . . 10 ⊢ 𝐶 = (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))
6	5	oveq2i 7360	. . . . . . . . 9 ⊢ (ℝ D 𝐶) = (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)))
7		ax-resscn 11066	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ⊆ ℂ)
9		0re 11117	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ
10		pire 26364	. . . . . . . . . . . . 13 ⊢ π ∈ ℝ
11		iccssre 13332	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → (0[,]π) ⊆ ℝ)
12	9, 10, 11	mp2an 692	. . . . . . . . . . . 12 ⊢ (0[,]π) ⊆ ℝ
13	12	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0[,]π) ⊆ ℝ)
14	12, 7	sstri 3945	. . . . . . . . . . . . . . 15 ⊢ (0[,]π) ⊆ ℂ
15	14	sseli 3931	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,]π) → 𝑥 ∈ ℂ)
16	15	coscld 16040	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (0[,]π) → (cos‘𝑥) ∈ ℂ)
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → (cos‘𝑥) ∈ ℂ)
18	17	negcld 11462	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ (0[,]π)) → -(cos‘𝑥) ∈ ℂ)
19		tgioo4 24691	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
20		eqid 2729	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
21		iccntr 24708	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ ∧ π ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(0[,]π)) = (0(,)π))
22	9, 10, 21	mp2an 692	. . . . . . . . . . . 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 25873	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))))
25	24	mptru 1547	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥))) = (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥)))
26		reelprrecn 11101	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
27	26	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ℝ ∈ {ℝ, ℂ})
28		recn 11099	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ)
29	28	coscld 16040	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → (cos‘𝑥) ∈ ℂ)
30	29	adantl 481	. . . . . . . . . . . 12 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (cos‘𝑥) ∈ ℂ)
31	30	negcld 11462	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(cos‘𝑥) ∈ ℂ)
32	28	sincld 16039	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ)
33	32	adantl 481	. . . . . . . . . . 11 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → (sin‘𝑥) ∈ ℂ)
34	32	negcld 11462	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ)
35	34	adantl 481	. . . . . . . . . . . . 13 ⊢ ((⊤ ∧ 𝑥 ∈ ℝ) → -(sin‘𝑥) ∈ ℂ)
36		dvcosre 45913	. . . . . . . . . . . . . 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 25868	. . . . . . . . . . . 12 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ --(sin‘𝑥)))
39	32	negnegd 11466	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ → --(sin‘𝑥) = (sin‘𝑥))
40	39	mpteq2ia 5187	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ ↦ --(sin‘𝑥)) = (𝑥 ∈ ℝ ↦ (sin‘𝑥))
41	38, 40	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (⊤ → (ℝ D (𝑥 ∈ ℝ ↦ -(cos‘𝑥))) = (𝑥 ∈ ℝ ↦ (sin‘𝑥)))
42		ioossre 13310	. . . . . . . . . . . 12 ⊢ (0(,)π) ⊆ ℝ
43	42	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ⊆ ℝ)
44		iooretop 24651	. . . . . . . . . . . 12 ⊢ (0(,)π) ∈ (topGen‘ran (,))
45	44	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → (0(,)π) ∈ (topGen‘ran (,)))
46	27, 31, 33, 41, 43, 19, 20, 45	dvmptres 25865	. . . . . . . . . 10 ⊢ (⊤ → (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)))
47	46	mptru 1547	. . . . . . . . 9 ⊢ (ℝ D (𝑥 ∈ (0(,)π) ↦ -(cos‘𝑥))) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
48	6, 25, 47	3eqtri 2756	. . . . . . . 8 ⊢ (ℝ D 𝐶) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
49	48	fveq1i 6823	. . . . . . 7 ⊢ ((ℝ D 𝐶)‘𝑥) = ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥)
50	42, 7	sstri 3945	. . . . . . . . . 10 ⊢ (0(,)π) ⊆ ℂ
51	50	sseli 3931	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) → 𝑥 ∈ ℂ)
52	51	sincld 16039	. . . . . . . 8 ⊢ (𝑥 ∈ (0(,)π) → (sin‘𝑥) ∈ ℂ)
53		eqid 2729	. . . . . . . . 9 ⊢ (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) = (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))
54	53	fvmpt2 6941	. . . . . . . 8 ⊢ ((𝑥 ∈ (0(,)π) ∧ (sin‘𝑥) ∈ ℂ) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
55	52, 54	mpdan 687	. . . . . . 7 ⊢ (𝑥 ∈ (0(,)π) → ((𝑥 ∈ (0(,)π) ↦ (sin‘𝑥))‘𝑥) = (sin‘𝑥))
56	49, 55	eqtrid 2776	. . . . . 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 25681	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ∫(0(,)π)(sin‘𝑥) d𝑥)
59	58	mptru 1547	. . 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 26366	. . . . . . 7 ⊢ 0 < π
63	9, 10, 62	ltleii 11239	. . . . . 6 ⊢ 0 ≤ π
64	63	a1i 11	. . . . 5 ⊢ (⊤ → 0 ≤ π)
65		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑥sin
66		sincn 26352	. . . . . . . 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 45588	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ ((0(,)π)–cn→ℂ))
70	48, 69	eqeltrid 2832	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ ((0(,)π)–cn→ℂ))
71		ioossicc 13336	. . . . . . . 8 ⊢ (0(,)π) ⊆ (0[,]π)
72	71	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ⊆ (0[,]π))
73		ioombl 25464	. . . . . . . 8 ⊢ (0(,)π) ∈ dom vol
74	73	a1i 11	. . . . . . 7 ⊢ (⊤ → (0(,)π) ∈ dom vol)
75	15	sincld 16039	. . . . . . . 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 45588	. . . . . . . . . 10 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
79	78	mptru 1547	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
80		cniccibl 25740	. . . . . . . . 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 1463	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹
82	81	a1i 11	. . . . . . 7 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
83	72, 74, 76, 82	iblss 25704	. . . . . 6 ⊢ (⊤ → (𝑥 ∈ (0(,)π) ↦ (sin‘𝑥)) ∈ 𝐿¹)
84	48, 83	eqeltrid 2832	. . . . 5 ⊢ (⊤ → (ℝ D 𝐶) ∈ 𝐿¹)
85	16	negcld 11462	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) ∈ ℂ)
86		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ ↦ -(cos‘𝑥)) = (𝑥 ∈ ℂ ↦ -(cos‘𝑥))
87	86	fvmpt2 6941	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℂ ∧ -(cos‘𝑥) ∈ ℂ) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
88	15, 85, 87	syl2anc 584	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]π) → ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥) = -(cos‘𝑥))
89	88	eqcomd 2735	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,]π) → -(cos‘𝑥) = ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
90	89	mpteq2ia 5187	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) = (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥))
91		nfmpt1 5191	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑥 ∈ ℂ ↦ -(cos‘𝑥))
92		coscn 26353	. . . . . . . . . . . 12 ⊢ cos ∈ (ℂ–cn→ℂ)
93	86	negfcncf 24815	. . . . . . . . . . . 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 45588	. . . . . . . . 9 ⊢ (⊤ → (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ))
97	96	mptru 1547	. . . . . . . 8 ⊢ (𝑥 ∈ (0[,]π) ↦ ((𝑥 ∈ ℂ ↦ -(cos‘𝑥))‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
98	90, 97	eqeltri 2824	. . . . . . 7 ⊢ (𝑥 ∈ (0[,]π) ↦ -(cos‘𝑥)) ∈ ((0[,]π)–cn→ℂ)
99	5, 98	eqeltri 2824	. . . . . 6 ⊢ 𝐶 ∈ ((0[,]π)–cn→ℂ)
100	99	a1i 11	. . . . 5 ⊢ (⊤ → 𝐶 ∈ ((0[,]π)–cn→ℂ))
101	60, 61, 64, 70, 84, 100	ftc2 25949	. . . 4 ⊢ (⊤ → ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0)))
102	101	mptru 1547	. . 3 ⊢ ∫(0(,)π)((ℝ D 𝐶)‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
103	59, 102	eqtr3i 2754	. 2 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = ((𝐶‘π) − (𝐶‘0))
104		0xr 11162	. . . . 5 ⊢ 0 ∈ ℝ^*
105	10	rexri 11173	. . . . 5 ⊢ π ∈ ℝ^*
106		ubicc2 13368	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → π ∈ (0[,]π))
107	104, 105, 63, 106	mp3an 1463	. . . 4 ⊢ π ∈ (0[,]π)
108		fveq2 6822	. . . . . . . 8 ⊢ (𝑡 = π → (cos‘𝑡) = (cos‘π))
109		cospi 26379	. . . . . . . 8 ⊢ (cos‘π) = -1
110	108, 109	eqtrdi 2780	. . . . . . 7 ⊢ (𝑡 = π → (cos‘𝑡) = -1)
111	110	negeqd 11357	. . . . . 6 ⊢ (𝑡 = π → -(cos‘𝑡) = --1)
112		ax-1cn 11067	. . . . . . . 8 ⊢ 1 ∈ ℂ
113	112	a1i 11	. . . . . . 7 ⊢ (𝑡 = π → 1 ∈ ℂ)
114	113	negnegd 11466	. . . . . 6 ⊢ (𝑡 = π → --1 = 1)
115	111, 114	eqtrd 2764	. . . . 5 ⊢ (𝑡 = π → -(cos‘𝑡) = 1)
116		1ex 11111	. . . . 5 ⊢ 1 ∈ V
117	115, 1, 116	fvmpt 6930	. . . 4 ⊢ (π ∈ (0[,]π) → (𝐶‘π) = 1)
118	107, 117	ax-mp 5	. . 3 ⊢ (𝐶‘π) = 1
119		lbicc2 13367	. . . . . 6 ⊢ ((0 ∈ ℝ^* ∧ π ∈ ℝ^* ∧ 0 ≤ π) → 0 ∈ (0[,]π))
120	104, 105, 63, 119	mp3an 1463	. . . . 5 ⊢ 0 ∈ (0[,]π)
121		fveq2 6822	. . . . . . 7 ⊢ (𝑡 = 0 → (cos‘𝑡) = (cos‘0))
122	121	negeqd 11357	. . . . . 6 ⊢ (𝑡 = 0 → -(cos‘𝑡) = -(cos‘0))
123		negex 11361	. . . . . 6 ⊢ -(cos‘0) ∈ V
124	122, 1, 123	fvmpt 6930	. . . . 5 ⊢ (0 ∈ (0[,]π) → (𝐶‘0) = -(cos‘0))
125	120, 124	ax-mp 5	. . . 4 ⊢ (𝐶‘0) = -(cos‘0)
126		cos0 16059	. . . . 5 ⊢ (cos‘0) = 1
127	126	negeqi 11356	. . . 4 ⊢ -(cos‘0) = -1
128	125, 127	eqtri 2752	. . 3 ⊢ (𝐶‘0) = -1
129	118, 128	oveq12i 7361	. 2 ⊢ ((𝐶‘π) − (𝐶‘0)) = (1 − -1)
130	112, 112	subnegi 11443	. . 3 ⊢ (1 − -1) = (1 + 1)
131		1p1e2 12248	. . 3 ⊢ (1 + 1) = 2
132	130, 131	eqtri 2752	. 2 ⊢ (1 − -1) = 2
133	103, 129, 132	3eqtri 2756	1 ⊢ ∫(0(,)π)(sin‘𝑥) d𝑥 = 2

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ⊆ wss 3903 {cpr 4579 class class class wbr 5092 ↦ cmpt 5173 dom cdm 5619 ran crn 5620 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 ℝ^*cxr 11148 ≤ cle 11150 − cmin 11347 -cneg 11348 2c2 12183 (,)cioo 13248 [,]cicc 13251 sincsin 15970 cosccos 15971 πcpi 15973 TopOpenctopn 17325 topGenctg 17341 ℂ_fldccnfld 21261 intcnt 22902 –cn→ccncf 24767 volcvol 25362 𝐿¹cibl 25516 ∫citg 25517 D cdv 25762

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-symdif 4204 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-fac 14181 df-bc 14210 df-hash 14238 df-shft 14974 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-ef 15974 df-sin 15976 df-cos 15977 df-pi 15979 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-lp 23021 df-perf 23022 df-cn 23112 df-cnp 23113 df-haus 23200 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-limc 25765 df-dv 25766

This theorem is referenced by: itgsin0pi 45953

W3C validator