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Definition df-ditg 23330
Description: Define the directed integral, which is just a regular integral but with a sign change when the limits are interchanged. The 𝐴 and 𝐵 here are the lower and upper limits of the integral, usually written as a subscript and superscript next to the integral sign. We define the region of integration to be an open interval instead of closed so that we can use +∞, -∞ for limits and also integrate up to a singularity at an endpoint. (Contributed by Mario Carneiro, 13-Aug-2014.)
Assertion
Ref Expression
df-ditg ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)

Detailed syntax breakdown of Definition df-ditg
StepHypRef Expression
1 vx . . 3 setvar 𝑥
2 cA . . 3 class 𝐴
3 cB . . 3 class 𝐵
4 cC . . 3 class 𝐶
51, 2, 3, 4cdit 23329 . 2 class ⨜[𝐴𝐵]𝐶 d𝑥
6 cle 9927 . . . 4 class
72, 3, 6wbr 4573 . . 3 wff 𝐴𝐵
8 cioo 11998 . . . . 5 class (,)
92, 3, 8co 6523 . . . 4 class (𝐴(,)𝐵)
101, 9, 4citg 23106 . . 3 class ∫(𝐴(,)𝐵)𝐶 d𝑥
113, 2, 8co 6523 . . . . 5 class (𝐵(,)𝐴)
121, 11, 4citg 23106 . . . 4 class ∫(𝐵(,)𝐴)𝐶 d𝑥
1312cneg 10114 . . 3 class -∫(𝐵(,)𝐴)𝐶 d𝑥
147, 10, 13cif 4031 . 2 class if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
155, 14wceq 1474 1 wff ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
Colors of variables: wff setvar class
This definition is referenced by:  ditgeq1  23331  ditgeq2  23332  ditgeq3  23333  ditgex  23335  ditg0  23336  cbvditg  23337  ditgpos  23339  ditgneg  23340  ditgeq3d  38656
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