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 ) )