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Metamath Proof Explorer

Theorem fldivnn0le

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

		Ref	Expression
	Assertion	fldivnn0le	⊢ ( ( 𝐾 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0re	⊢ ( 𝐾 ∈ ℕ₀ → 𝐾 ∈ ℝ )
2		nnrp	⊢ ( 𝐿 ∈ ℕ → 𝐿 ∈ ℝ⁺ )
3		fldivle	⊢ ( ( 𝐾 ∈ ℝ ∧ 𝐿 ∈ ℝ⁺ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )
4	1 2 3	syl2an	⊢ ( ( 𝐾 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ≤ ( 𝐾 / 𝐿 ) )