Theorem fsumcj

Description: The complex conjugate 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	fsumcj	⊢ ( 𝜑 → ( ∗ ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( ∗ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fsumre.1	⊢ ( 𝜑 → 𝐴 ∈ Fin )
2		fsumre.2	⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3		cjf	⊢ ∗ : ℂ ⟶ ℂ
4		cjadd	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( ∗ ‘ ( 𝑥 + 𝑦 ) ) = ( ( ∗ ‘ 𝑥 ) + ( ∗ ‘ 𝑦 ) ) )
5	1 2 3 4	fsumrelem	⊢ ( 𝜑 → ( ∗ ‘ Σ 𝑘 ∈ 𝐴 𝐵 ) = Σ 𝑘 ∈ 𝐴 ( ∗ ‘ 𝐵 ) )