Theorem nn0resubcl

Description: Closure law for subtraction of reals, restricted to nonnegative integers. (Contributed by Alexander van der Vekens, 6-Apr-2018)

	Ref	Expression
Assertion	nn0resubcl	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 − 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		nn0re	⊢ ( 𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ )
2		nn0re	⊢ ( 𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ )
3		resubcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 − 𝐵 ) ∈ ℝ )
4	1 2 3	syl2an	⊢ ( ( 𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ) → ( 𝐴 − 𝐵 ) ∈ ℝ )