Theorem addscomd

Description: Surreal addition commutes. Part of Theorem 3 of Conway p. 17. (Contributed by Scott Fenton, 20-Aug-2024)

	Ref	Expression
Hypotheses	addscomd.1	$$⊢\phi \to A\in \mathrm{No}$$
	addscomd.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	addscomd	$$⊢\phi \to A{+}_{s}B=B{+}_{s}A$$

Proof

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