Theorem imdivd

Description: Imaginary part of a division. Related to remul2 . (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	remul2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	redivd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
Assertion	imdivd	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℑ ‘ 𝐵 ) / 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		crred.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		remul2d.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		redivd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
4		imdiv	⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ) → ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℑ ‘ 𝐵 ) / 𝐴 ) )
5	2 1 3 4	syl3anc	⊢ ( 𝜑 → ( ℑ ‘ ( 𝐵 / 𝐴 ) ) = ( ( ℑ ‘ 𝐵 ) / 𝐴 ) )