Metamath Proof Explorer


Theorem immuli

Description: Imaginary part of a product. (Contributed by NM, 28-Jul-1999)

Ref Expression
Hypotheses recl.1 𝐴 ∈ ℂ
readdi.2 𝐵 ∈ ℂ
Assertion immuli ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 recl.1 𝐴 ∈ ℂ
2 readdi.2 𝐵 ∈ ℂ
3 immul ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) ) )
4 1 2 3 mp2an ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( ( ( ℜ ‘ 𝐴 ) · ( ℑ ‘ 𝐵 ) ) + ( ( ℑ ‘ 𝐴 ) · ( ℜ ‘ 𝐵 ) ) )