Metamath Proof Explorer
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 ) ) |