Theorem ditgpos 24459

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 24450	. 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 4433	. 2 ⊢ (𝜑 → if(𝐴 ≤ 𝐵, ∫(𝐴(,)𝐵)𝐶 d𝑥, -∫(𝐵(,)𝐴)𝐶 d𝑥) = ∫(𝐴(,)𝐵)𝐶 d𝑥)
4	1, 3	syl5eq 2845	1 ⊢ (𝜑 → ⨜[𝐴 → 𝐵]𝐶 d𝑥 = ∫(𝐴(,)𝐵)𝐶 d𝑥)

Syntax hints:   → wi 4   = wceq 1538  ifcif 4425   class class class wbr 5030  (class class class)co 7135   ≤ cle 10665  -cneg 10860  (,)cioo 12726  ∫citg 24222  ⨜cdit 24449

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-if 4426  df-ditg 24450

This theorem is referenced by:  ditgcl  24461  ditgswap  24462  ditgsplitlem  24463  ftc2ditglem  24648  itgsubstlem  24651  itgsubst  24652  ditgeqiooicc  42602  itgiccshift  42622  itgperiod  42623  fourierdlem82  42830