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Metamath Proof Explorer

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 ) )