Theorem mulscan1d

Description: Cancellation of surreal multiplication when the left term is non-zero. (Contributed by Scott Fenton, 10-Mar-2025)

	Ref	Expression
Hypotheses	mulscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
	mulscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
	mulscan2d.3	$$⊢\phi \to C\in \mathrm{No}$$
	mulscan2d.4	$$⊢\phi \to C\ne 0_s$$
Assertion	mulscan1d	$$⊢\phi \to \left(C{\cdot }_{s}A=C{\cdot }_{s}B\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		mulscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		mulscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		mulscan2d.3	$$⊢\phi \to C\in \mathrm{No}$$
4		mulscan2d.4	$$⊢\phi \to C\ne 0_s$$
5	1 3	mulscomd	$$⊢\phi \to A{\cdot }_{s}C=C{\cdot }_{s}A$$
6	2 3	mulscomd	$$⊢\phi \to B{\cdot }_{s}C=C{\cdot }_{s}B$$
7	5 6	eqeq12d	$$⊢\phi \to \left(A{\cdot }_{s}C=B{\cdot }_{s}C\leftrightarrow C{\cdot }_{s}A=C{\cdot }_{s}B\right)$$
8	1 2 3 4	mulscan2d	$$⊢\phi \to \left(A{\cdot }_{s}C=B{\cdot }_{s}C\leftrightarrow A=B\right)$$
9	7 8	bitr3d	$$⊢\phi \to \left(C{\cdot }_{s}A=C{\cdot }_{s}B\leftrightarrow A=B\right)$$