Theorem addsdid

Description: Distributive law for surreal numbers. Commuted form of 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	addsdid	$$⊢\phi \to A{\cdot }_s\left(B{+}_{s}C\right)=\left(A{\cdot }_{s}B\right){+}_s\left(A{\cdot }_{s}C\right)$$

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		addsdi	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge C\in \mathrm{No}\right)\to A{\cdot }_s\left(B{+}_{s}C\right)=\left(A{\cdot }_{s}B\right){+}_s\left(A{\cdot }_{s}C\right)$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A{\cdot }_s\left(B{+}_{s}C\right)=\left(A{\cdot }_{s}B\right){+}_s\left(A{\cdot }_{s}C\right)$$