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	$$⊢\phi \to A\in ℂ$$
	reim0bd.2	$$⊢\phi \to \Im \left(A\right)=0$$
Assertion	reim0bd	$$⊢\phi \to A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		reim0bd.2	$$⊢\phi \to \Im \left(A\right)=0$$
3		reim0b	$$⊢A\in ℂ\to \left(A\in ℝ\leftrightarrow \Im \left(A\right)=0\right)$$
4	1 3	syl	$$⊢\phi \to \left(A\in ℝ\leftrightarrow \Im \left(A\right)=0\right)$$
5	2 4	mpbird	$$⊢\phi \to A\in ℝ$$