Theorem imcld

Description: The imaginary part of a complex number is real (closure law). (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	imcld	⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		imcl	⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ )
3	1 2	syl	⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) ∈ ℝ )