Definition df-itg 25658

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 25656 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 25656 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 25653	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11155	. . . 4 class 0
6		c3 12322	. . . 4 class 3
7		cfz 13547	. . . 4 class ...
8	5, 6, 7	co 7431	. . 3 class (0...3)
9		ci 11157	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1539	. . . . 5 class 𝑘
12		cexp 14102	. . . . 5 class ↑
13	9, 11, 12	co 7431	. . . 4 class (i↑𝑘)
14		cr 11154	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11920	. . . . . . . . 9 class /
17	3, 13, 16	co 7431	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15136	. . . . . . . 8 class ℜ
19	17, 18	cfv 6561	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1539	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2108	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1539	. . . . . . . . . 10 class 𝑦
23		cle 11296	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5143	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 395	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4525	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3899	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5225	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25651	. . . . 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 11160	. . . 4 class ·
32	13, 30, 31	co 7431	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15722	. 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 1540	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  25804  itgex  25805  itgeq1f  25806  itgeq1  25808  nfitg1  25809  cbvitgv  25812  itgeq12i  36207  itgeq12sdv  36220  cbvitgvw2  36249  cbvitgdavw  36282  cbvitgdavw2  36298