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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 25596 | . 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 4535 | . 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
4 | 1, 3 | eqtrid 2782 | 1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ifcif 4527 class class class wbr 5147 (class class class)co 7411 ≤ cle 11253 -cneg 11449 (,)cioo 13328 ∫citg 25367 ⨜cdit 25595 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-if 4528 df-ditg 25596 |
This theorem is referenced by: ditgcl 25607 ditgswap 25608 ditgsplitlem 25609 ftc2ditglem 25797 itgsubstlem 25800 itgsubst 25801 ditgeqiooicc 44974 itgiccshift 44994 itgperiod 44995 fourierdlem82 45202 |
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