Definition df-itg 25524

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 25522 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 25522 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 25519	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11068	. . . 4 class 0
6		c3 12242	. . . 4 class 3
7		cfz 13468	. . . 4 class ...
8	5, 6, 7	co 7387	. . 3 class (0...3)
9		ci 11070	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1539	. . . . 5 class 𝑘
12		cexp 14026	. . . . 5 class ↑
13	9, 11, 12	co 7387	. . . 4 class (i↑𝑘)
14		cr 11067	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11835	. . . . . . . . 9 class /
17	3, 13, 16	co 7387	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15063	. . . . . . . 8 class ℜ
19	17, 18	cfv 6511	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1539	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2109	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1539	. . . . . . . . . 10 class 𝑦
23		cle 11209	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5107	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 395	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4488	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3862	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5188	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25517	. . . . 5 class ∫2
30	28, 29	cfv 6511	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11073	. . . 4 class ·
32	13, 30, 31	co 7387	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15652	. 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  25670  itgex  25671  itgeq1f  25672  itgeq1  25674  nfitg1  25675  cbvitgv  25678  itgeq12i  36194  itgeq12sdv  36207  cbvitgvw2  36236  cbvitgdavw  36269  cbvitgdavw2  36285