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	|- ( ph -> A e. CC )
	reim0bd.2	|- ( ph -> ( Im ` A ) = 0 )
Assertion	reim0bd	|- ( ph -> A e. RR )

Proof

Step	Hyp	Ref	Expression
1		recld.1	|- ( ph -> A e. CC )
2		reim0bd.2	|- ( ph -> ( Im ` A ) = 0 )
3		reim0b	|- ( A e. CC -> ( A e. RR <-> ( Im ` A ) = 0 ) )
4	1 3	syl	|- ( ph -> ( A e. RR <-> ( Im ` A ) = 0 ) )
5	2 4	mpbird	|- ( ph -> A e. RR )