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 24445 | . 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 4475 | . 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
4 | 1, 3 | syl5eq 2868 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ifcif 4467 class class class wbr 5066 (class class class)co 7156 ≤ cle 10676 -cneg 10871 (,)cioo 12739 ∫citg 24219 ⨜cdit 24444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-if 4468 df-ditg 24445 |
This theorem is referenced by: ditgcl 24456 ditgswap 24457 ditgsplitlem 24458 ftc2ditglem 24642 itgsubstlem 24645 itgsubst 24646 ditgeqiooicc 42265 itgiccshift 42285 itgperiod 42286 fourierdlem82 42493 |
Copyright terms: Public domain | W3C validator |