![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 25588 | . 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 4536 | . 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
4 | 1, 3 | eqtrid 2784 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ifcif 4528 class class class wbr 5148 (class class class)co 7411 ≤ cle 11253 -cneg 11449 (,)cioo 13328 ∫citg 25359 ⨜cdit 25587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-if 4529 df-ditg 25588 |
This theorem is referenced by: ditgcl 25599 ditgswap 25600 ditgsplitlem 25601 ftc2ditglem 25786 itgsubstlem 25789 itgsubst 25790 ditgeqiooicc 44975 itgiccshift 44995 itgperiod 44996 fourierdlem82 45203 |
Copyright terms: Public domain | W3C validator |