Theorem muls12d

Description: Commutative/associative law for surreal multiplication. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	muls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
	muls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
	muls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
Assertion	muls12d	$$⊢\phi \to A{\cdot }_s\left(B{\cdot }_{s}C\right)=B{\cdot }_s\left(A{\cdot }_{s}C\right)$$

Proof

Step	Hyp	Ref	Expression
1		muls12d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		muls12d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		muls12d.3	$$⊢\phi \to C\in \mathrm{No}$$
4	1 2	mulscomd	$$⊢\phi \to A{\cdot }_{s}B=B{\cdot }_{s}A$$
5	4	oveq1d	$$⊢\phi \to \left(A{\cdot }_{s}B\right){\cdot }_{s}C=\left(B{\cdot }_{s}A\right){\cdot }_{s}C$$
6	1 2 3	mulsassd	$$⊢\phi \to \left(A{\cdot }_{s}B\right){\cdot }_{s}C=A{\cdot }_s\left(B{\cdot }_{s}C\right)$$
7	2 1 3	mulsassd	$$⊢\phi \to \left(B{\cdot }_{s}A\right){\cdot }_{s}C=B{\cdot }_s\left(A{\cdot }_{s}C\right)$$
8	5 6 7	3eqtr3d	$$⊢\phi \to A{\cdot }_s\left(B{\cdot }_{s}C\right)=B{\cdot }_s\left(A{\cdot }_{s}C\right)$$