Definition df-itg 25743

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 25741 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 25741 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 25738	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11088	. . . 4 class 0
6		c3 12287	. . . 4 class 3
7		cfz 13526	. . . 4 class ...
8	5, 6, 7	co 7400	. . 3 class (0...3)
9		ci 11090	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1562	. . . . 5 class 𝑘
12		cexp 14088	. . . . 5 class ↑
13	9, 11, 12	co 7400	. . . 4 class (i↑𝑘)
14		cr 11087	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11859	. . . . . . . . 9 class /
17	3, 13, 16	co 7400	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15138	. . . . . . . 8 class ℜ
19	17, 18	cfv 6525	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1562	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2145	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1562	. . . . . . . . . 10 class 𝑦
23		cle 11232	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5105	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 400	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4483	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3855	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5186	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25736	. . . . 5 class ∫2
30	28, 29	cfv 6525	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11093	. . . 4 class ·
32	13, 30, 31	co 7400	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15727	. 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 1563	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  25889  itgex  25890  itgeq1f  25891  itgeq1  25893  nfitg1  25894  cbvitgv  25897  itgeq12i  36579  itgeq12sdv  36592  cbvitgvw2  36621  cbvitgdavw  36654  cbvitgdavw2  36670