Theorem itgim

Description: Imaginary part of an integral. (Contributed by Mario Carneiro, 14-Aug-2014)

	Ref	Expression
Hypotheses	itgcnval.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	itgcnval.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 )
Assertion	itgim	⊢ ( 𝜑 → ( ℑ ‘ ∫ 𝐴 𝐵 d 𝑥 ) = ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		itgcnval.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
2		itgcnval.2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 )
3	1 2	itgcnval	⊢ ( 𝜑 → ∫ 𝐴 𝐵 d 𝑥 = ( ∫ 𝐴 ( ℜ ‘ 𝐵 ) d 𝑥 + ( i · ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 ) ) )
4	3	fveq2d	⊢ ( 𝜑 → ( ℑ ‘ ∫ 𝐴 𝐵 d 𝑥 ) = ( ℑ ‘ ( ∫ 𝐴 ( ℜ ‘ 𝐵 ) d 𝑥 + ( i · ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 ) ) ) )
5		iblmbf	⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn )
6	2 5	syl	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ MblFn )
7	6 1	mbfmptcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
8	7	recld	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℜ ‘ 𝐵 ) ∈ ℝ )
9	7	iblcn	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝐿1 ↔ ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 ) ) )
10	2 9	mpbid	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 ∧ ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 ) )
11	10	simpld	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ∈ 𝐿1 )
12	8 11	itgrecl	⊢ ( 𝜑 → ∫ 𝐴 ( ℜ ‘ 𝐵 ) d 𝑥 ∈ ℝ )
13	7	imcld	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ℑ ‘ 𝐵 ) ∈ ℝ )
14	10	simprd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( ℑ ‘ 𝐵 ) ) ∈ 𝐿1 )
15	13 14	itgrecl	⊢ ( 𝜑 → ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 ∈ ℝ )
16	12 15	crimd	⊢ ( 𝜑 → ( ℑ ‘ ( ∫ 𝐴 ( ℜ ‘ 𝐵 ) d 𝑥 + ( i · ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 ) ) ) = ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 )
17	4 16	eqtrd	⊢ ( 𝜑 → ( ℑ ‘ ∫ 𝐴 𝐵 d 𝑥 ) = ∫ 𝐴 ( ℑ ‘ 𝐵 ) d 𝑥 )