Definition df-itg 25615

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 25613 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 25613 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 25610	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11036	. . . 4 class 0
6		c3 12235	. . . 4 class 3
7		cfz 13459	. . . 4 class ...
8	5, 6, 7	co 7363	. . 3 class (0...3)
9		ci 11038	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1546	. . . . 5 class 𝑘
12		cexp 14021	. . . . 5 class ↑
13	9, 11, 12	co 7363	. . . 4 class (i↑𝑘)
14		cr 11035	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11805	. . . . . . . . 9 class /
17	3, 13, 16	co 7363	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15057	. . . . . . . 8 class ℜ
19	17, 18	cfv 6492	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1546	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2119	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1546	. . . . . . . . . 10 class 𝑦
23		cle 11178	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5079	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 396	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4461	. . . . . . 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 5160	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25608	. . . . 5 class ∫2
30	28, 29	cfv 6492	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11041	. . . 4 class ·
32	13, 30, 31	co 7363	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15646	. 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 1547	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  25761  itgex  25762  itgeq1f  25763  itgeq1  25765  nfitg1  25766  cbvitgv  25769  itgeq12i  36441  itgeq12sdv  36454  cbvitgvw2  36483  cbvitgdavw  36516  cbvitgdavw2  36532