Theorem divscan1d

Description: A cancellation law for surreal division. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
	divscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
	divscan2d.3	$$⊢\phi \to B\ne 0_s$$
Assertion	divscan1d	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B\right){\cdot }_{s}B=A$$

Proof

Step	Hyp	Ref	Expression
1		divscan2d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divscan2d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divscan2d.3	$$⊢\phi \to B\ne 0_s$$
4	2 3	recsexd	$$⊢\phi \to \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s$$
5	1 2 3 4	divscan1wd	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B\right){\cdot }_{s}B=A$$