Theorem divscan1wd

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

	Ref	Expression
Hypotheses	divscan2wd.1	$$⊢\phi \to A\in \mathrm{No}$$
	divscan2wd.2	$$⊢\phi \to B\in \mathrm{No}$$
	divscan2wd.3	$$⊢\phi \to B\ne 0_s$$
	divscan2wd.4	$$⊢\phi \to \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s$$
Assertion	divscan1wd	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B\right){\cdot }_{s}B=A$$

Proof

Step	Hyp	Ref	Expression
1		divscan2wd.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divscan2wd.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divscan2wd.3	$$⊢\phi \to B\ne 0_s$$
4		divscan2wd.4	$$⊢\phi \to \exists x\in \mathrm{No}B{\cdot }_{s}x=1_s$$
5	1 2 3 4	divsclwd	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$
6	5 2	mulscomd	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B\right){\cdot }_{s}B=B{\cdot }_s\left(A{/}_{\mathrm{su}}B\right)$$
7	1 2 3 4	divscan2wd	$$⊢\phi \to B{\cdot }_s\left(A{/}_{\mathrm{su}}B\right)=A$$
8	6 7	eqtrd	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B\right){\cdot }_{s}B=A$$