Definition df-itg 25924

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 25922 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 25922 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 25919	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11181	. . . 4 class 0
6		c3 12379	. . . 4 class 3
7		cfz 13620	. . . 4 class ...
8	5, 6, 7	co 7412	. . 3 class (0...3)
9		ci 11183	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1569	. . . . 5 class 𝑘
12		cexp 14184	. . . . 5 class ↑
13	9, 11, 12	co 7412	. . . 4 class (i↑𝑘)
14		cr 11180	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11954	. . . . . . . . 9 class /
17	3, 13, 16	co 7412	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15244	. . . . . . . 8 class ℜ
19	17, 18	cfv 6531	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1569	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2145	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1569	. . . . . . . . . 10 class 𝑦
23		cle 11325	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5103	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 401	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4482	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3847	. . . . . 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 25917	. . . . 5 class ∫2
30	28, 29	cfv 6531	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11186	. . . 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 15833	. 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  26070  itgex  26071  itgeq1f  26072  itgeq1  26073  nfitg1  26074  cbvitgv  26077  itgeq12i  36965  itgeq12sdv  36978  cbvitgvw2  37007  cbvitgdavw  37040  cbvitgdavw2  37056