Theorem zsubcld

Description: Closure of subtraction of integers. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	zred.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
	zaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
Assertion	zsubcld	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		zred.1	⊢ ( 𝜑 → 𝐴 ∈ ℤ )
2		zaddcld.1	⊢ ( 𝜑 → 𝐵 ∈ ℤ )
3		zsubcl	⊢ ( ( 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝐴 − 𝐵 ) ∈ ℤ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℤ )