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Definition df-itg 24151
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 24149 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 24149 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 24146 . 2 class 𝐴𝐵 d𝑥
5 cc0 10525 . . . 4 class 0
6 c3 11681 . . . 4 class 3
7 cfz 12880 . . . 4 class ...
85, 6, 7co 7145 . . 3 class (0...3)
9 ci 10527 . . . . 5 class i
10 vk . . . . . 6 setvar 𝑘
1110cv 1527 . . . . 5 class 𝑘
12 cexp 13417 . . . . 5 class
139, 11, 12co 7145 . . . 4 class (i↑𝑘)
14 cr 10524 . . . . . 6 class
15 vy . . . . . . 7 setvar 𝑦
16 cdiv 11285 . . . . . . . . 9 class /
173, 13, 16co 7145 . . . . . . . 8 class (𝐵 / (i↑𝑘))
18 cre 14444 . . . . . . . 8 class
1917, 18cfv 6348 . . . . . . 7 class (ℜ‘(𝐵 / (i↑𝑘)))
201cv 1527 . . . . . . . . . 10 class 𝑥
2120, 2wcel 2105 . . . . . . . . 9 wff 𝑥𝐴
2215cv 1527 . . . . . . . . . 10 class 𝑦
23 cle 10664 . . . . . . . . . 10 class
245, 22, 23wbr 5057 . . . . . . . . 9 wff 0 ≤ 𝑦
2521, 24wa 396 . . . . . . . 8 wff (𝑥𝐴 ∧ 0 ≤ 𝑦)
2625, 22, 5cif 4463 . . . . . . 7 class if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
2715, 19, 26csb 3880 . . . . . 6 class (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)
281, 14, 27cmpt 5137 . . . . 5 class (𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))
29 citg2 24144 . . . . 5 class 2
3028, 29cfv 6348 . . . 4 class (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0)))
31 cmul 10530 . . . 4 class ·
3213, 30, 31co 7145 . . 3 class ((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
338, 32, 10csu 15030 . 2 class Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ (ℜ‘(𝐵 / (i↑𝑘))) / 𝑦if((𝑥𝐴 ∧ 0 ≤ 𝑦), 𝑦, 0))))
344, 33wceq 1528 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  24297  itgex  24298  nfitg1  24301
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