Definition df-itg 25671

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 25669 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 25669 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 25666	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11152	. . . 4 class 0
6		c3 12319	. . . 4 class 3
7		cfz 13543	. . . 4 class ...
8	5, 6, 7	co 7430	. . 3 class (0...3)
9		ci 11154	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1535	. . . . 5 class 𝑘
12		cexp 14098	. . . . 5 class ↑
13	9, 11, 12	co 7430	. . . 4 class (i↑𝑘)
14		cr 11151	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11917	. . . . . . . . 9 class /
17	3, 13, 16	co 7430	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15132	. . . . . . . 8 class ℜ
19	17, 18	cfv 6562	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1535	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2105	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1535	. . . . . . . . . 10 class 𝑦
23		cle 11293	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5147	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 395	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4530	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3907	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5230	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25664	. . . . 5 class ∫2
30	28, 29	cfv 6562	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11157	. . . 4 class ·
32	13, 30, 31	co 7430	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15718	. 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 1536	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  25818  itgex  25819  itgeq1f  25820  itgeq1  25822  nfitg1  25823  cbvitgv  25826  itgeq12i  36187  itgeq12sdv  36201  cbvitgvw2  36230  cbvitgdavw  36263  cbvitgdavw2  36279