Definition df-itg 25819

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 25817 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 25817 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 25814	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11118	. . . 4 class 0
6		c3 12314	. . . 4 class 3
7		cfz 13553	. . . 4 class ...
8	5, 6, 7	co 7423	. . 3 class (0...3)
9		ci 11120	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1569	. . . . 5 class 𝑘
12		cexp 14117	. . . . 5 class ↑
13	9, 11, 12	co 7423	. . . 4 class (i↑𝑘)
14		cr 11117	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11889	. . . . . . . . 9 class /
17	3, 13, 16	co 7423	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15174	. . . . . . . 8 class ℜ
19	17, 18	cfv 6543	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1569	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2146	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1569	. . . . . . . . . 10 class 𝑦
23		cle 11262	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5114	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 401	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4492	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3856	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5197	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25812	. . . . 5 class ∫2
30	28, 29	cfv 6543	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11123	. . . 4 class ·
32	13, 30, 31	co 7423	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15763	. 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  25965  itgex  25966  itgeq1f  25967  itgeq1  25969  nfitg1  25970  cbvitgv  25973  itgeq12i  36759  itgeq12sdv  36772  cbvitgvw2  36801  cbvitgdavw  36834  cbvitgdavw2  36850