Theorem divscan3d

Description: A cancellation law for surreal division. (Contributed by Scott Fenton, 13-Aug-2025)

	Ref	Expression
Hypotheses	divscan3d.1	$$⊢\phi \to A\in \mathrm{No}$$
	divscan3d.2	$$⊢\phi \to B\in \mathrm{No}$$
	divscan3d.3	$$⊢\phi \to B\ne 0_s$$
Assertion	divscan3d	$$⊢\phi \to \left(B{\cdot }_{s}A\right){/}_{\mathrm{su}}B=A$$

Proof

Step	Hyp	Ref	Expression
1		divscan3d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divscan3d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divscan3d.3	$$⊢\phi \to B\ne 0_s$$
4		eqid	$$⊢B{\cdot }_{s}A=B{\cdot }_{s}A$$
5	2 1	mulscld	$$⊢\phi \to B{\cdot }_{s}A\in \mathrm{No}$$
6	5 1 2 3	divmulsd	$$⊢\phi \to \left(\left(B{\cdot }_{s}A\right){/}_{\mathrm{su}}B=A\leftrightarrow B{\cdot }_{s}A=B{\cdot }_{s}A\right)$$
7	4 6	mpbiri	$$⊢\phi \to \left(B{\cdot }_{s}A\right){/}_{\mathrm{su}}B=A$$