Theorem fsumre

Description: The real 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	fsumre	⊢ ( 𝜑 → ( ℜ ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( ℜ ‘ 𝐵 ) )

Proof

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