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Mirrors > Home > MPE Home > Th. List > df-ditg | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
df-ditg | ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . 3 setvar 𝑥 | |
2 | cA | . . 3 class 𝐴 | |
3 | cB | . . 3 class 𝐵 | |
4 | cC | . . 3 class 𝐶 | |
5 | 1, 2, 3, 4 | cdit 23416 | . 2 class ⨜[𝐴 → 𝐵]𝐶 d𝑥 |
6 | cle 9954 | . . . 4 class ≤ | |
7 | 2, 3, 6 | wbr 4583 | . . 3 wff 𝐴 ≤ 𝐵 |
8 | cioo 12046 | . . . . 5 class (,) | |
9 | 2, 3, 8 | co 6549 | . . . 4 class (𝐴(,)𝐵) |
10 | 1, 9, 4 | citg 23193 | . . 3 class ∫(𝐴(,)𝐵)𝐶 d𝑥 |
11 | 3, 2, 8 | co 6549 | . . . . 5 class (𝐵(,)𝐴) |
12 | 1, 11, 4 | citg 23193 | . . . 4 class ∫(𝐵(,)𝐴)𝐶 d𝑥 |
13 | 12 | cneg 10146 | . . 3 class -∫(𝐵(,)𝐴)𝐶 d𝑥 |
14 | 7, 10, 13 | cif 4036 | . 2 class if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) |
15 | 5, 14 | wceq 1475 | 1 wff ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) |
Colors of variables: wff setvar class |
This definition is referenced by: ditgeq1 23418 ditgeq2 23419 ditgeq3 23420 ditgex 23422 ditg0 23423 cbvditg 23424 ditgpos 23426 ditgneg 23427 ditgeq3d 38856 |
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