Theorem mulscomd

Description: Surreal multiplication commutes. Part of theorem 7 of Conway p. 19. (Contributed by Scott Fenton, 6-Mar-2025)

	Ref	Expression
Hypotheses	mulscomd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	mulscomd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	mulscomd	⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) = ( 𝐵 ·s 𝐴 ) )

Proof

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