Theorem fldivle

Description: The floor function of a division of a real number by a positive real number is less than or equal to the division. (Contributed by Alexander van der Vekens, 14-Apr-2018)

	Ref	Expression
Assertion	fldivle	⊢ ( ( 𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )

Proof

Step	Hyp	Ref	Expression
1		rerpdivcl	⊢ ( ( 𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+ ) → ( 𝐾 / 𝐿 ) ∈ ℝ )
2		fllelt	⊢ ( ( 𝐾 / 𝐿 ) ∈ ℝ → ( ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) ∧ ( 𝐾 / 𝐿 ) < ( ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) + 1 ) ) )
3		simpl	⊢ ( ( ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) ∧ ( 𝐾 / 𝐿 ) < ( ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) + 1 ) ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )
4	1 2 3	3syl	⊢ ( ( 𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )