Metamath Proof Explorer


Theorem imcjd

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

Ref Expression
Hypothesis recld.1
|- ( ph -> A e. CC )
Assertion imcjd
|- ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )

Proof

Step Hyp Ref Expression
1 recld.1
 |-  ( ph -> A e. CC )
2 imcj
 |-  ( A e. CC -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )
3 1 2 syl
 |-  ( ph -> ( Im ` ( * ` A ) ) = -u ( Im ` A ) )