Definition df-itg 25763

Description: Define the full Lebesgue integral, for complex-valued functions to ℝ. The syntax is designed to be suggestive of the standard notation for integrals. For example, our notation for the integral of 𝑥↑2 from 0 to 1 is ∫(0[,]1)(𝑥↑2) d𝑥 = (1 / 3). The only real function of this definition is to break the integral up into nonnegative real parts and send it off to df-itg2 25761 for further processing. Note that this definition cannot handle integrals which evaluate to infinity, because addition and multiplication are not currently defined on extended reals. (You can use df-itg2 25761 directly for this use-case.) (Contributed by Mario Carneiro, 28-Jun-2014.)

Assertion

Ref	Expression
df-itg	⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))

Distinct variable groups:   𝑦,𝑘,𝐴   𝐵,𝑘,𝑦   𝑥,𝑘,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Detailed syntax breakdown of Definition df-itg

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		cA	. . 3 class 𝐴
3		cB	. . 3 class 𝐵
4	1, 2, 3	citg 25758	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11101	. . . 4 class 0
6		c3 12297	. . . 4 class 3
7		cfz 13536	. . . 4 class ...
8	5, 6, 7	co 7412	. . 3 class (0...3)
9		ci 11103	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1569	. . . . 5 class 𝑘
12		cexp 14099	. . . . 5 class ↑
13	9, 11, 12	co 7412	. . . 4 class (i↑𝑘)
14		cr 11100	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11872	. . . . . . . . 9 class /
17	3, 13, 16	co 7412	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15150	. . . . . . . 8 class ℜ
19	17, 18	cfv 6538	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1569	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2143	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1569	. . . . . . . . . 10 class 𝑦
23		cle 11245	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5110	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 400	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4488	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3854	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5193	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25756	. . . . 5 class ∫2
30	28, 29	cfv 6538	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11106	. . . 4 class ·
32	13, 30, 31	co 7412	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15739	. 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
34	4, 33	wceq 1570	1 wff ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))

This definition is referenced by:  dfitg  25909  itgex  25910  itgeq1f  25911  itgeq1  25913  nfitg1  25914  cbvitgv  25917  itgeq12i  36699  itgeq12sdv  36712  cbvitgvw2  36741  cbvitgdavw  36774  cbvitgdavw2  36790