Theorem flidz

Description: A real number equals its floor iff it is an integer. (Contributed by NM, 11-Nov-2008)

	Ref	Expression
Assertion	flidz	|- ( A e. RR -> ( ( |_ ` A ) = A <-> A e. ZZ ) )

Proof

Step	Hyp	Ref	Expression
1		flcl	|- ( A e. RR -> ( |_ ` A ) e. ZZ )
2		eleq1	|- ( ( |_ ` A ) = A -> ( ( |_ ` A ) e. ZZ <-> A e. ZZ ) )
3	1 2	syl5ibcom	|- ( A e. RR -> ( ( |_ ` A ) = A -> A e. ZZ ) )
4		flid	|- ( A e. ZZ -> ( |_ ` A ) = A )
5	3 4	impbid1	|- ( A e. RR -> ( ( |_ ` A ) = A <-> A e. ZZ ) )