Theorem adds4d

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	adds4d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(B{+}_{s}D\right)$$

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	adds32d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_{s}C=\left(A{+}_{s}C\right){+}_{s}B$$
6	5	oveq1d	$$⊢\phi \to \left(\left(A{+}_{s}B\right){+}_{s}C\right){+}_{s}D=\left(\left(A{+}_{s}C\right){+}_{s}B\right){+}_{s}D$$
7	1 2	addscld	$$⊢\phi \to A{+}_{s}B\in \mathrm{No}$$
8	7 3 4	addsassd	$$⊢\phi \to \left(\left(A{+}_{s}B\right){+}_{s}C\right){+}_{s}D=\left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)$$
9	1 3	addscld	$$⊢\phi \to A{+}_{s}C\in \mathrm{No}$$
10	9 2 4	addsassd	$$⊢\phi \to \left(\left(A{+}_{s}C\right){+}_{s}B\right){+}_{s}D=\left(A{+}_{s}C\right){+}_s\left(B{+}_{s}D\right)$$
11	6 8 10	3eqtr3d	$$⊢\phi \to \left(A{+}_{s}B\right){+}_s\left(C{+}_{s}D\right)=\left(A{+}_{s}C\right){+}_s\left(B{+}_{s}D\right)$$