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Theorem ditgpos 25891
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 25882 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 ditgpos.1 . . 3 (𝜑𝐴𝐵)
32iftrued 4533 . 2 (𝜑 → if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
41, 3eqtrid 2789 1 (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ifcif 4525   class class class wbr 5143  (class class class)co 7431  cle 11296  -cneg 11493  (,)cioo 13387  citg 25653  cdit 25881
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-if 4526  df-ditg 25882
This theorem is referenced by:  ditgcl  25893  ditgswap  25894  ditgsplitlem  25895  ftc2ditglem  26086  itgsubstlem  26089  itgsubst  26090  ditgeqiooicc  45975  itgiccshift  45995  itgperiod  45996  fourierdlem82  46203
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