Theorem imcjd

Description: Imaginary part of a complex conjugate. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	imcjd	⊢ ( 𝜑 → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		imcj	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ℑ ‘ ( ∗ ‘ 𝐴 ) ) = - ( ℑ ‘ 𝐴 ) )