Theorem divscld

Description: Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divscld.1	|- ( ph -> A e. No )
	divscld.2	|- ( ph -> B e. No )
	divscld.3	|- ( ph -> B =/= 0s )
Assertion	divscld	|- ( ph -> ( A /su B ) e. No )

Proof

Step	Hyp	Ref	Expression
1		divscld.1	|- ( ph -> A e. No )
2		divscld.2	|- ( ph -> B e. No )
3		divscld.3	|- ( ph -> B =/= 0s )
4		divscl	|- ( ( A e. No /\ B e. No /\ B =/= 0s ) -> ( A /su B ) e. No )
5	1 2 3 4	syl3anc	|- ( ph -> ( A /su B ) e. No )