Definition df-itg 25665

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 25663 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 25663 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 25660	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11070	. . . 4 class 0
6		c3 12270	. . . 4 class 3
7		cfz 13509	. . . 4 class ...
8	5, 6, 7	co 7392	. . 3 class (0...3)
9		ci 11072	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1558	. . . . 5 class 𝑘
12		cexp 14071	. . . . 5 class ↑
13	9, 11, 12	co 7392	. . . 4 class (i↑𝑘)
14		cr 11069	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11841	. . . . . . . . 9 class /
17	3, 13, 16	co 7392	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15107	. . . . . . . 8 class ℜ
19	17, 18	cfv 6517	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1558	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2141	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1558	. . . . . . . . . 10 class 𝑦
23		cle 11214	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5099	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 399	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4479	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3852	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5180	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25658	. . . . 5 class ∫2
30	28, 29	cfv 6517	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11075	. . . 4 class ·
32	13, 30, 31	co 7392	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15696	. 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 1559	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  25811  itgex  25812  itgeq1f  25813  itgeq1  25815  nfitg1  25816  cbvitgv  25819  itgeq12i  36530  itgeq12sdv  36543  cbvitgvw2  36572  cbvitgdavw  36605  cbvitgdavw2  36621