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Theorem ditgpos 23426
 Description: Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)
Hypothesis
Ref Expression
ditgpos.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
ditgpos (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ditgpos
StepHypRef Expression
1 df-ditg 23417 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 ditgpos.1 . . 3 (𝜑𝐴𝐵)
32iftrued 4044 . 2 (𝜑 → if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
41, 3syl5eq 2656 1 (𝜑 → ⨜[𝐴𝐵]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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