Theorem ditgpos

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

	Ref	Expression
Hypothesis	ditgpos.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
Assertion	ditgpos	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		ditgpos.1	⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
2		df-ditg	⊢ ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = if ( 𝐴 ≤ 𝐵 , ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 , - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 )
3	1	iftrued	⊢ ( 𝜑 → if ( 𝐴 ≤ 𝐵 , ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 , - ∫ ( 𝐵 (,) 𝐴 ) 𝐶 d 𝑥 ) = ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 )
4	2 3	syl5eq	⊢ ( 𝜑 → ⨜ [ 𝐴 → 𝐵 ] 𝐶 d 𝑥 = ∫ ( 𝐴 (,) 𝐵 ) 𝐶 d 𝑥 )