Theorem flid

Description: An integer is its own floor. (Contributed by NM, 15-Nov-2004)

	Ref	Expression
Assertion	flid	|- ( A e. ZZ -> ( |_ ` A ) = A )

Proof

Step	Hyp	Ref	Expression
1		zre	|- ( A e. ZZ -> A e. RR )
2		flle	|- ( A e. RR -> ( |_ ` A ) <_ A )
3	1 2	syl	|- ( A e. ZZ -> ( |_ ` A ) <_ A )
4	1	leidd	|- ( A e. ZZ -> A <_ A )
5		flge	|- ( ( A e. RR /\ A e. ZZ ) -> ( A <_ A <-> A <_ ( |_ ` A ) ) )
6	1 5	mpancom	|- ( A e. ZZ -> ( A <_ A <-> A <_ ( |_ ` A ) ) )
7	4 6	mpbid	|- ( A e. ZZ -> A <_ ( |_ ` A ) )
8		reflcl	|- ( A e. RR -> ( |_ ` A ) e. RR )
9	1 8	syl	|- ( A e. ZZ -> ( |_ ` A ) e. RR )
10	9 1	letri3d	|- ( A e. ZZ -> ( ( |_ ` A ) = A <-> ( ( |_ ` A ) <_ A /\ A <_ ( |_ ` A ) ) ) )
11	3 7 10	mpbir2and	|- ( A e. ZZ -> ( |_ ` A ) = A )