Theorem flcl

Description: The floor (greatest integer) function is an integer (closure law). (Contributed by NM, 15-Nov-2004) (Revised by Mario Carneiro, 2-Nov-2013)

	Ref	Expression
Assertion	flcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		flval	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) )
2		rebtwnz	⊢ ( 𝐴 ∈ ℝ → ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) )
3		riotacl	⊢ ( ∃! 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) → ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) ∈ ℤ )
4	2 3	syl	⊢ ( 𝐴 ∈ ℝ → ( ℩ 𝑥 ∈ ℤ ( 𝑥 ≤ 𝐴 ∧ 𝐴 < ( 𝑥 + 1 ) ) ) ∈ ℤ )
5	1 4	eqeltrd	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℤ )