Theorem fldivnn0

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

	Ref	Expression
Assertion	fldivnn0	⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		nn0nndivcl	⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ) → ( 𝐾 / 𝐿 ) ∈ ℝ )
2		nn0ge0div	⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ) → 0 ≤ ( 𝐾 / 𝐿 ) )
3		flge0nn0	⊢ ( ( ( 𝐾 / 𝐿 ) ∈ ℝ ∧ 0 ≤ ( 𝐾 / 𝐿 ) ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ∈ ℕ0 )
4	1 2 3	syl2anc	⊢ ( ( 𝐾 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ) → ( ⌊ ‘ ( 𝐾 / 𝐿 ) ) ∈ ℕ0 )