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Definition df-itg 23591
Description: Define the full Lebesgue integral, for complex-valued functions to . 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 23589 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 23589 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))) / 𝑦if((𝑥𝐴 ∧ 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
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
41, 2, 3citg 23586 . 2 class 𝐴𝐵 d𝑥
5 cc0 10128 . . . 4 class 0
6 c3 11263 . . . 4 class 3
7 cfz 12519 . . . 4 class ...
85, 6, 7co 6813 . . 3 class (0...3)
9 ci 10130 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1631 . . . . 5 class 𝑘
12 cexp 13054 . . . . 5 class
139, 11, 12co 6813 . . . 4 class (i↑𝑘)
14 cr 10127 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 10876 . . . . . . . . 9 class /
173, 13, 16co 6813 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 14036 . . . . . . . 8 class
1917, 18cfv 6049 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1631 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2139 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1631 . . . . . . . . . 10 class 𝑦
23 cle 10267 . . . . . . . . . 10 class
245, 22, 23wbr 4804 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 383 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4230 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3674 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 4881 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 23584 . . . . 5 class 2
3028, 29cfv 6049 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 10133 . . . 4 class ·
3213, 30, 31co 6813 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 14615 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1632 1 wff 𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
Colors of variables: wff setvar class
This definition is referenced by:  dfitg  23735  itgex  23736  nfitg1  23739
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