Theorem itgitg2

Description: Transfer an integral using S.2 to an equivalent integral using S. . (Contributed by Mario Carneiro, 6-Aug-2014)

	Ref	Expression
Hypotheses	itgitg2.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐴 ∈ ℝ )
	itgitg2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 0 ≤ 𝐴 )
	itgitg2.3	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ 𝐴 ) ∈ 𝐿1 )
Assertion	itgitg2	⊢ ( 𝜑 → ∫ ℝ 𝐴 d 𝑥 = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		itgitg2.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐴 ∈ ℝ )
2		itgitg2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 0 ≤ 𝐴 )
3		itgitg2.3	⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ 𝐴 ) ∈ 𝐿1 )
4	1 3 2	itgposval	⊢ ( 𝜑 → ∫ ℝ 𝐴 d 𝑥 = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ℝ , 𝐴 , 0 ) ) ) )
5		iftrue	⊢ ( 𝑥 ∈ ℝ → if ( 𝑥 ∈ ℝ , 𝐴 , 0 ) = 𝐴 )
6	5	mpteq2ia	⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ℝ , 𝐴 , 0 ) ) = ( 𝑥 ∈ ℝ ↦ 𝐴 )
7	6	fveq2i	⊢ ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ℝ , 𝐴 , 0 ) ) ) = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ 𝐴 ) )
8	4 7	eqtrdi	⊢ ( 𝜑 → ∫ ℝ 𝐴 d 𝑥 = ( ∫2 ‘ ( 𝑥 ∈ ℝ ↦ 𝐴 ) ) )