Theorem ditgpos 25836

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 25827	. 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 4475	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
4	1, 3	eqtrid 2784	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1542  ifcif 4467   class class class wbr 5086  (class class class)co 7361   ≤ cle 11174  -cneg 11372  (,)cioo 13292  ∫citg 25598  ⨜cdit 25826

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-if 4468  df-ditg 25827

This theorem is referenced by:  ditgcl  25838  ditgswap  25839  ditgsplitlem  25840  ftc2ditglem  26025  itgsubstlem  26028  itgsubst  26029  ditgeqiooicc  46409  itgiccshift  46429  itgperiod  46430  fourierdlem82  46637