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Metamath Proof Explorer

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 )