Theorem reim0bd

Description: A number is real iff its imaginary part is 0. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	reim0bd.2	⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) = 0 )
Assertion	reim0bd	⊢ ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		recld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		reim0bd.2	⊢ ( 𝜑 → ( ℑ ‘ 𝐴 ) = 0 )
3		reim0b	⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ↔ ( ℑ ‘ 𝐴 ) = 0 ) )
5	2 4	mpbird	⊢ ( 𝜑 → 𝐴 ∈ ℝ )