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

Proof

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