Theorem zaddscld

Description: The surreal integers are closed under addition. (Contributed by Scott Fenton, 25-Jul-2025)

	Ref	Expression
Hypotheses	zaddscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤs )
	zaddscld.2	⊢ ( 𝜑 → 𝐵 ∈ ℤs )
Assertion	zaddscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ ℤs )

Proof

Step	Hyp	Ref	Expression
1		zaddscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤs )
2		zaddscld.2	⊢ ( 𝜑 → 𝐵 ∈ ℤs )
3		zaddscl	⊢ ( ( 𝐴 ∈ ℤs ∧ 𝐵 ∈ ℤs ) → ( 𝐴 +s 𝐵 ) ∈ ℤs )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ ℤs )