Theorem addsassd

Description: Surreal addition is associative. Part of theorem 3 of Conway p. 17. (Contributed by Scott Fenton, 22-Jan-2025)

	Ref	Expression
Hypotheses	addsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
	addsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
	addsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	addsassd	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=A{+}_s\left(B{+}_{s}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		addsassd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsassd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsassd.3	$$⊢\phi \to C\in \mathrm{No}$$
4		addsass	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to \left(A{+}_{s}B\right){+}_{s}C=A{+}_s\left(B{+}_{s}C\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=A{+}_s\left(B{+}_{s}C\right)$$