Theorem ditgpos 25784

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 25775	. 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 4480	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
4	1, 3	eqtrid 2778	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1541  ifcif 4472   class class class wbr 5089  (class class class)co 7346   ≤ cle 11147  -cneg 11345  (,)cioo 13245  ∫citg 25546  ⨜cdit 25774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-if 4473  df-ditg 25775

This theorem is referenced by:  ditgcl  25786  ditgswap  25787  ditgsplitlem  25788  ftc2ditglem  25979  itgsubstlem  25982  itgsubst  25983  ditgeqiooicc  46006  itgiccshift  46026  itgperiod  46027  fourierdlem82  46234