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Theorem ditgpos 23343
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 23334 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 ditgpos.1 . . 3 (𝜑𝐴𝐵)
32iftrued 4043 . 2 (𝜑 → if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
41, 3syl5eq 2655 1 (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  ifcif 4035   class class class wbr 4577  (class class class)co 6527  cle 9931  -cneg 10118  (,)cioo 12002  citg 23110  cdit 23333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-if 4036  df-ditg 23334
This theorem is referenced by:  ditgcl  23345  ditgswap  23346  ditgsplitlem  23347  ftc2ditglem  23529  itgsubstlem  23532  itgsubst  23533  ditgeqiooicc  38649  itgiccshift  38669  itgperiod  38670  fourierdlem82  38878
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