Theorem ditgpos 25823

Description: Value of the directed integral in the forward direction. (Contributed by Mario Carneiro, 13-Aug-2014.)

Hypothesis

Ref	Expression
ditgpos.1	⊢ (𝜑 → 𝐴 ≤ 𝐵)

Assertion

Ref	Expression
ditgpos	⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem ditgpos

Step	Hyp	Ref	Expression
1		df-ditg 25814	. 2 ⊢ ⨜[𝐴 → 𝐵]𝐶 d𝑥 = if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥)
2		ditgpos.1	. . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
3	2	iftrued 4474	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
4	1, 3	eqtrid 2783	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1542  ifcif 4466   class class class wbr 5085  (class class class)co 7367   ≤ cle 11180  -cneg 11378  (,)cioo 13298  ∫citg 25585  ⨜cdit 25813

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-if 4467  df-ditg 25814

This theorem is referenced by:  ditgcl  25825  ditgswap  25826  ditgsplitlem  25827  ftc2ditglem  26012  itgsubstlem  26015  itgsubst  26016  ditgeqiooicc  46388  itgiccshift  46408  itgperiod  46409  fourierdlem82  46616