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	$$⊢\phi \to A\in \mathrm{No}$$
	mulscld.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	mulscld	$$⊢\phi \to A{\cdot }_{s}B\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		mulscld.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulscld.2	$$⊢\phi \to B\in \mathrm{No}$$
3		mulscl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\right)\to A{\cdot }_{s}B\in \mathrm{No}$$
4	1 2 3	syl2anc	$$⊢\phi \to A{\cdot }_{s}B\in \mathrm{No}$$