Theorem flidz

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

	Ref	Expression
Assertion	flidz	$$⊢A\in ℝ\to \left(⌊A⌋=A\leftrightarrow A\in \mathbb{Z}\right)$$

Proof

Step	Hyp	Ref	Expression
1		flcl	$$⊢A\in ℝ\to ⌊A⌋\in \mathbb{Z}$$
2		eleq1	$$⊢⌊A⌋=A\to \left(⌊A⌋\in \mathbb{Z}\leftrightarrow A\in \mathbb{Z}\right)$$
3	1 2	syl5ibcom	$$⊢A\in ℝ\to \left(⌊A⌋=A\to A\in \mathbb{Z}\right)$$
4		flid	$$⊢A\in \mathbb{Z}\to ⌊A⌋=A$$
5	3 4	impbid1	$$⊢A\in ℝ\to \left(⌊A⌋=A\leftrightarrow A\in \mathbb{Z}\right)$$