Metamath Proof Explorer


Theorem remimd

Description: Value of the conjugate of a complex number. The value is the real part minus _i times the imaginary part. Definition 10-3.2 of Gleason p. 132. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis recld.1
|- ( ph -> A e. CC )
Assertion remimd
|- ( ph -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )

Proof

Step Hyp Ref Expression
1 recld.1
 |-  ( ph -> A e. CC )
2 remim
 |-  ( A e. CC -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )
3 1 2 syl
 |-  ( ph -> ( * ` A ) = ( ( Re ` A ) - ( _i x. ( Im ` A ) ) ) )