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Metamath Proof Explorer

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