Definition df-itg 25677

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 25675 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 25675 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 25672	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 11184	. . . 4 class 0
6		c3 12349	. . . 4 class 3
7		cfz 13567	. . . 4 class ...
8	5, 6, 7	co 7448	. . 3 class (0...3)
9		ci 11186	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1536	. . . . 5 class 𝑘
12		cexp 14112	. . . . 5 class ↑
13	9, 11, 12	co 7448	. . . 4 class (i↑𝑘)
14		cr 11183	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 11947	. . . . . . . . 9 class /
17	3, 13, 16	co 7448	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 15146	. . . . . . . 8 class ℜ
19	17, 18	cfv 6573	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1536	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 2108	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1536	. . . . . . . . . 10 class 𝑦
23		cle 11325	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 5166	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 395	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4548	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3921	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 5249	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 25670	. . . . 5 class ∫2
30	28, 29	cfv 6573	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 11189	. . . 4 class ·
32	13, 30, 31	co 7448	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 15734	. 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 1537	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  25824  itgex  25825  itgeq1f  25826  itgeq1  25828  nfitg1  25829  cbvitgv  25832  itgeq12i  36170  itgeq12sdv  36185  cbvitgvw2  36214  cbvitgdavw  36247  cbvitgdavw2  36263