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Definition df-itg 23684
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 23682 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 23682 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 23679 . 2 class 𝐴𝐵 d𝑥
5 cc0 10191 . . . 4 class 0
6 c3 11330 . . . 4 class 3
7 cfz 12536 . . . 4 class ...
85, 6, 7co 6844 . . 3 class (0...3)
9 ci 10193 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1651 . . . . 5 class 𝑘
12 cexp 13070 . . . . 5 class
139, 11, 12co 6844 . . . 4 class (i↑𝑘)
14 cr 10190 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 10940 . . . . . . . . 9 class /
173, 13, 16co 6844 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 14125 . . . . . . . 8 class
1917, 18cfv 6070 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1651 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2155 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1651 . . . . . . . . . 10 class 𝑦
23 cle 10331 . . . . . . . . . 10 class
245, 22, 23wbr 4811 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 384 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4245 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3693 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 4890 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 23677 . . . . 5 class 2
3028, 29cfv 6070 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 10196 . . . 4 class ·
3213, 30, 31co 6844 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 14704 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1652 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  23830  itgex  23831  nfitg1  23834
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