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	⊢ (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 25882	. 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 4533	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
4	1, 3	eqtrid 2789	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

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