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Theorem ditgpos 25597
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 25588 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 ditgpos.1 . . 3 (𝜑𝐴𝐵)
32iftrued 4536 . 2 (𝜑 → if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
41, 3eqtrid 2784 1 (𝜑 → ⨜[𝐴𝐵]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
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