Definition df-itg 25950

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 25948 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 25948 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 25945	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11204	. . . 4 class 0
6		c3 12402	. . . 4 class 3
7		cfz 13643	. . . 4 class ...
8	5, 6, 7	co 7421	. . 3 class (0...3)
9		ci 11206	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1568	. . . . 5 class 𝑘
12		cexp 14208	. . . . 5 class ↑
13	9, 11, 12	co 7421	. . . 4 class (i↑𝑘)
14		cr 11203	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11977	. . . . . . . . 9 class /
17	3, 13, 16	co 7421	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15270	. . . . . . . 8 class ℜ
19	17, 18	cfv 6539	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1568	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2144	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1568	. . . . . . . . . 10 class 𝑦
23		cle 11348	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5102	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 401	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4481	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3846	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5185	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25943	. . . . 5 class ∫2
30	28, 29	cfv 6539	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11209	. . . 4 class ·
32	13, 30, 31	co 7421	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15859	. 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 1569	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  26096  itgex  26097  itgeq1f  26098  itgeq1  26099  nfitg1  26100  cbvitgv  26103  itgeq12i  37088  itgeq12sdv  37101  cbvitgvw2  37130  cbvitgdavw  37163  cbvitgdavw2  37179