Theorem addsdird

Description: Distributive law for surreal numbers. Part of theorem 7 of Conway p. 19. (Contributed by Scott Fenton, 9-Mar-2025)

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

Proof

Step	Hyp	Ref	Expression
1		addsdid.1	$$⊢\phi \to A\in \mathrm{No}$$
2		addsdid.2	$$⊢\phi \to B\in \mathrm{No}$$
3		addsdid.3	$$⊢\phi \to C\in \mathrm{No}$$
4	3 1 2	addsdid	$$⊢\phi \to C{\cdot }_s\left(A{+}_{s}B\right)=\left(C{\cdot }_{s}A\right){+}_s\left(C{\cdot }_{s}B\right)$$
5	1 2	addscld	$$⊢\phi \to A{+}_{s}B\in \mathrm{No}$$
6	5 3	mulscomd	$$⊢\phi \to \left(A{+}_{s}B\right){\cdot }_{s}C=C{\cdot }_s\left(A{+}_{s}B\right)$$
7	1 3	mulscomd	$$⊢\phi \to A{\cdot }_{s}C=C{\cdot }_{s}A$$
8	2 3	mulscomd	$$⊢\phi \to B{\cdot }_{s}C=C{\cdot }_{s}B$$
9	7 8	oveq12d	$$⊢\phi \to \left(A{\cdot }_{s}C\right){+}_s\left(B{\cdot }_{s}C\right)=\left(C{\cdot }_{s}A\right){+}_s\left(C{\cdot }_{s}B\right)$$
10	4 6 9	3eqtr4d	$$⊢\phi \to \left(A{+}_{s}B\right){\cdot }_{s}C=\left(A{\cdot }_{s}C\right){+}_s\left(B{\cdot }_{s}C\right)$$