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| Mirrors > Home > MPE Home > Th. List > ditgpos | Structured version Visualization version GIF version | ||
| Description: Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.) |
| Ref | Expression |
|---|---|
| ditgpos.1 | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
| Ref | Expression |
|---|---|
| ditgpos | ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ditg 25882 | . 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) | |
| 2 | ditgpos.1 | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
| 3 | 2 | iftrued 4533 | . 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
| 4 | 1, 3 | eqtrid 2789 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ifcif 4525 class class class wbr 5143 (class class class)co 7431 ≤ cle 11296 -cneg 11493 (,)cioo 13387 ∫citg 25653 ⨜cdit 25881 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-if 4526 df-ditg 25882 |
| This theorem is referenced by: ditgcl 25893 ditgswap 25894 ditgsplitlem 25895 ftc2ditglem 26086 itgsubstlem 26089 itgsubst 26090 ditgeqiooicc 45975 itgiccshift 45995 itgperiod 45996 fourierdlem82 46203 |
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