Theorem adds42d

Description: Rearrangement of four terms in a surreal sum. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	adds4d.1	$$⊢\phi \to A\in \mathrm{No}$$
	adds4d.2	$$⊢\phi \to B\in \mathrm{No}$$
	adds4d.3	$$⊢\phi \to C\in \mathrm{No}$$
	adds4d.4	$$⊢\phi \to D\in \mathrm{No}$$
Assertion	adds42d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(D{+}_{s}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		adds4d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		adds4d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		adds4d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		adds4d.4	$$⊢\phi \to D\in \mathrm{No}$$
5	1 2 3 4	adds4d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(B{+}_{s}D\right)$$
6	2 4	addscomd	$$⊢\phi \to B{+}_{s}D=D{+}_{s}B$$
7	6	oveq2d	$$⊢\phi \to \left(A{+}_{s}C\right){+}_s\left(B{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(D{+}_{s}B\right)$$
8	5 7	eqtrd	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(D{+}_{s}B\right)$$