Theorem mulscld

Description: The surreals are closed under multiplication. Theorem 8(i) of Conway p. 19. (Contributed by Scott Fenton, 6-Mar-2025)

	Ref	Expression
Hypotheses	mulscld.1	|- ( ph -> A e. No )
	mulscld.2	|- ( ph -> B e. No )
Assertion	mulscld	|- ( ph -> ( A x.s B ) e. No )

Proof

Step	Hyp	Ref	Expression
1		mulscld.1	|- ( ph -> A e. No )
2		mulscld.2	|- ( ph -> B e. No )
3		mulscl	|- ( ( A e. No /\ B e. No ) -> ( A x.s B ) e. No )
4	1 2 3	syl2anc	|- ( ph -> ( A x.s B ) e. No )