Theorem flcld

Description: The floor (greatest integer) function is an integer (closure law). (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypothesis	flcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion	flcld	⊢ ( 𝜑 → ( ⌊ ‘ 𝐴 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		flcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		flcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ )
3	1 2	syl	⊢ ( 𝜑 → ( ⌊ ‘ 𝐴 ) ∈ ℤ )