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	⊢ ( 𝜑 → 𝐴 ∈ No )
	mulscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	mulscld	⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		mulscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		mulscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		mulscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) ∈ No )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) ∈ No )