Theorem immul2d

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

	Ref	Expression
Hypotheses	crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	remul2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	immul2d	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		remul2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		immul2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℂ ) → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐴 · 𝐵 ) ) = ( 𝐴 · ( ℑ ‘ 𝐵 ) ) )