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Theorem ditgpos 24465
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 24456 . 2 ⨜[𝐴𝐵]𝐶 d𝑥 = if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2 ditgpos.1 . . 3 (𝜑𝐴𝐵)
32iftrued 4458 . 2 (𝜑 → if(𝐴𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
41, 3syl5eq 2871 1 (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ifcif 4450   class class class wbr 5052  (class class class)co 7149  cle 10674  -cneg 10869  (,)cioo 12735  citg 24228  cdit 24455
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-if 4451  df-ditg 24456
This theorem is referenced by:  ditgcl  24467  ditgswap  24468  ditgsplitlem  24469  ftc2ditglem  24654  itgsubstlem  24657  itgsubst  24658  ditgeqiooicc  42532  itgiccshift  42552  itgperiod  42553  fourierdlem82  42760
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