Metamath Proof Explorer


Theorem imcjd

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

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

Proof

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