Metamath Proof Explorer

Theorem reim0bi

Description: A number is real iff its imaginary part is 0. (Contributed by NM, 29-May-1999)

Ref Expression
Hypothesis recl.1 
|- A e. CC
Assertion reim0bi 
|- ( A e. RR <-> ( Im ` A ) = 0 )

Proof

Step Hyp Ref Expression
1 recl.1 
 |-  A e. CC
2 reim0b 
 |-  ( A e. CC -> ( A e. RR <-> ( Im ` A ) = 0 ) )
3 1 2 ax-mp 
 |-  ( A e. RR <-> ( Im ` A ) = 0 )