Theorem fsumim

Description: The imaginary part of a sum. (Contributed by Paul Chapman, 9-Nov-2007) (Revised by Mario Carneiro, 25-Jul-2014)

	Ref	Expression
Hypotheses	fsumre.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
	fsumre.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	fsumim	⊢ ( 𝜑 → ( ℑ ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( ℑ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fsumre.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumre.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		imf	⊢ ℑ : ℂ ⟶ ℝ
4		ax-resscn	⊢ ℝ ⊆ ℂ
5		fss	⊢ ( ( ℑ : ℂ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → ℑ : ℂ ⟶ ℂ )
6	3 4 5	mp2an	⊢ ℑ : ℂ ⟶ ℂ
7		imadd	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ℑ ‘ ( 𝑥 + 𝑦 ) ) = ( ( ℑ ‘ 𝑥 ) + ( ℑ ‘ 𝑦 ) ) )
8	1 2 6 7	fsumrelem	⊢ ( 𝜑 → ( ℑ ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( ℑ ‘ 𝐵 ) )