Definition df-itg 25977

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 25975 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 25975 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 25972	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11230	. . . 4 class 0
6		c3 12428	. . . 4 class 3
7		cfz 13669	. . . 4 class ...
8	5, 6, 7	co 7446	. . 3 class (0...3)
9		ci 11232	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1569	. . . . 5 class 𝑘
12		cexp 14234	. . . . 5 class ↑
13	9, 11, 12	co 7446	. . . 4 class (i↑𝑘)
14		cr 11229	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 12003	. . . . . . . . 9 class /
17	3, 13, 16	co 7446	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15296	. . . . . . . 8 class ℜ
19	17, 18	cfv 6561	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1569	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2147	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1569	. . . . . . . . . 10 class 𝑦
23		cle 11374	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5106	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 401	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4485	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3851	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5190	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25970	. . . . 5 class ∫2
30	28, 29	cfv 6561	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11235	. . . 4 class ·
32	13, 30, 31	co 7446	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15885	. 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 used by:  dfitg  26123  itgex  26124  itgeq1f  26125  itgeq1  26126  nfitg1  26127  cbvitgv  26130  itgeq12i  37186  itgeq12sdv  37199  cbvitgvw2  37228  cbvitgdavw  37261  cbvitgdavw2  37277