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 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ ℤ ) )

Proof

Step	Hyp	Ref	Expression
1		flcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ )
2		eleq1	⊢ ( ( ⌊ ‘ 𝐴 ) = 𝐴 → ( ( ⌊ ‘ 𝐴 ) ∈ ℤ ↔ 𝐴 ∈ ℤ ) )
3	1 2	syl5ibcom	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 𝐴 → 𝐴 ∈ ℤ ) )
4		flid	⊢ ( 𝐴 ∈ ℤ → ( ⌊ ‘ 𝐴 ) = 𝐴 )
5	3 4	impbid1	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 𝐴 ↔ 𝐴 ∈ ℤ ) )