Metamath Proof Explorer


Theorem immuld

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

Ref Expression
Hypotheses recld.1 ( 𝜑𝐴 ∈ ℂ )
readdd.2 ( 𝜑𝐵 ∈ ℂ )
Assertion immuld ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) )

Proof

Step Hyp Ref Expression
1 recld.1 ( 𝜑𝐴 ∈ ℂ )
2 readdd.2 ( 𝜑𝐵 ∈ ℂ )
3 immul ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) )
4 1 2 3 syl2anc ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) )