Theorem zaddscld

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

	Ref	Expression
Hypotheses	zaddscld.1	$$⊢\phi \to A\in {\mathbb{Z}}_s$$
	zaddscld.2	$$⊢\phi \to B\in {\mathbb{Z}}_s$$
Assertion	zaddscld	$$⊢\phi \to A{+}_{s}B\in {\mathbb{Z}}_s$$

Proof

Step	Hyp	Ref	Expression
1		zaddscld.1	$$⊢\phi \to A\in {\mathbb{Z}}_s$$
2		zaddscld.2	$$⊢\phi \to B\in {\mathbb{Z}}_s$$
3		zaddscl	$$⊢\left(A\in {\mathbb{Z}}_s\wedge B\in {\mathbb{Z}}_s\right)\to A{+}_{s}B\in {\mathbb{Z}}_s$$
4	1 2 3	syl2anc	$$⊢\phi \to A{+}_{s}B\in {\mathbb{Z}}_s$$