Definition df-itg 25590

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 25588 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 25588 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 25585	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11038	. . . 4 class 0
6		c3 12237	. . . 4 class 3
7		cfz 13461	. . . 4 class ...
8	5, 6, 7	co 7367	. . 3 class (0...3)
9		ci 11040	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1541	. . . . 5 class 𝑘
12		cexp 14023	. . . . 5 class ↑
13	9, 11, 12	co 7367	. . . 4 class (i↑𝑘)
14		cr 11037	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11807	. . . . . . . . 9 class /
17	3, 13, 16	co 7367	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15059	. . . . . . . 8 class ℜ
19	17, 18	cfv 6499	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1541	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2114	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1541	. . . . . . . . . 10 class 𝑦
23		cle 11180	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5086	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 395	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4467	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3838	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5167	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25583	. . . . 5 class ∫2
30	28, 29	cfv 6499	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11043	. . . 4 class ·
32	13, 30, 31	co 7367	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15648	. 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 1542	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  25736  itgex  25737  itgeq1f  25738  itgeq1  25740  nfitg1  25741  cbvitgv  25744  itgeq12i  36388  itgeq12sdv  36401  cbvitgvw2  36430  cbvitgdavw  36463  cbvitgdavw2  36479