Theorem zaddscld

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

	Ref	Expression
Hypotheses	zaddscld.1	|- ( ph -> A e. ZZ_s )
	zaddscld.2	|- ( ph -> B e. ZZ_s )
Assertion	zaddscld	|- ( ph -> ( A +s B ) e. ZZ_s )

Proof

Step	Hyp	Ref	Expression
1		zaddscld.1	|- ( ph -> A e. ZZ_s )
2		zaddscld.2	|- ( ph -> B e. ZZ_s )
3		zaddscl	|- ( ( A e. ZZ_s /\ B e. ZZ_s ) -> ( A +s B ) e. ZZ_s )
4	1 2 3	syl2anc	|- ( ph -> ( A +s B ) e. ZZ_s )