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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 23417 | . 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 4044 | . 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
4 | 1, 3 | syl5eq 2656 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ifcif 4036 class class class wbr 4583 (class class class)co 6549 ≤ cle 9954 -cneg 10146 (,)cioo 12046 ∫citg 23193 ⨜cdit 23416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-if 4037 df-ditg 23417 |
This theorem is referenced by: ditgcl 23428 ditgswap 23429 ditgsplitlem 23430 ftc2ditglem 23612 itgsubstlem 23615 itgsubst 23616 ditgeqiooicc 38852 itgiccshift 38872 itgperiod 38873 fourierdlem82 39081 |
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