Metamath Proof Explorer


Theorem imaddi

Description: Imaginary part distributes over addition. (Contributed by NM, 28-Jul-1999)

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

Proof

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