Theorem ltsubnn0

Description: Subtracting a nonnegative integer from a nonnegative integer which is greater than the first one results in a nonnegative integer. (Contributed by Alexander van der Vekens, 6-Apr-2018)

	Ref	Expression
Assertion	ltsubnn0	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 < 𝐴 → ( 𝐴 − 𝐵 ) ∈ ℕ0 ) )

Proof

Step	Hyp	Ref	Expression
1		nn0re	⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ )
2		nn0re	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
3		ltle	⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝐵 < 𝐴 → 𝐵 ≤ 𝐴 ) )
4	1 2 3	syl2anr	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 < 𝐴 → 𝐵 ≤ 𝐴 ) )
5		nn0sub	⊢ ( ( 𝐵 ∈ ℕ0 ∧ 𝐴 ∈ ℕ0 ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐴 − 𝐵 ) ∈ ℕ0 ) )
6	5	ancoms	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 ≤ 𝐴 ↔ ( 𝐴 − 𝐵 ) ∈ ℕ0 ) )
7	4 6	sylibd	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐵 < 𝐴 → ( 𝐴 − 𝐵 ) ∈ ℕ0 ) )