Definition df-itg 23198

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 23196 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 23196 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 23193	. 2 class ∫𝐴𝐵 d𝑥
5		cc0 9815	. . . 4 class 0
6		c3 10948	. . . 4 class 3
7		cfz 12197	. . . 4 class ...
8	5, 6, 7	co 6549	. . 3 class (0...3)
9		ci 9817	. . . . 5 class i
10		vk	. . . . . 6 setvar 𝑘
11	10	cv 1474	. . . . 5 class 𝑘
12		cexp 12722	. . . . 5 class ↑
13	9, 11, 12	co 6549	. . . 4 class (i↑𝑘)
14		cr 9814	. . . . . 6 class ℝ
15		vy	. . . . . . 7 setvar 𝑦
16		cdiv 10563	. . . . . . . . 9 class /
17	3, 13, 16	co 6549	. . . . . . . 8 class (𝐵 / (i↑𝑘))
18		cre 13685	. . . . . . . 8 class ℜ
19	17, 18	cfv 5804	. . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
20	1	cv 1474	. . . . . . . . . 10 class 𝑥
21	20, 2	wcel 1977	. . . . . . . . 9 wff 𝑥 ∈ 𝐴
22	15	cv 1474	. . . . . . . . . 10 class 𝑦
23		cle 9954	. . . . . . . . . 10 class ≤
24	5, 22, 23	wbr 4583	. . . . . . . . 9 wff 0 ≤ 𝑦
25	21, 24	wa 383	. . . . . . . 8 wff (𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦)
26	25, 22, 5	cif 4036	. . . . . . 7 class if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
27	15, 19, 26	csb 3499	. . . . . 6 class ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
28	1, 14, 27	cmpt 4643	. . . . 5 class (𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29		citg2 23191	. . . . 5 class ∫2
30	28, 29	cfv 5804	. . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31		cmul 9820	. . . 4 class ·
32	13, 30, 31	co 6549	. . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑦⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
33	8, 32, 10	csu 14264	. 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 1475	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  23342  itgex  23343  nfitg1  23346