Theorem divscan2wd

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	divscan2wd	$$⊢\phi \to B{\cdot }_s\left(A{/}_{\mathrm{su}}B\right)=A$$

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		eqid	$$⊢A{/}_{\mathrm{su}}B=A{/}_{\mathrm{su}}B$$
6	1 2 3 4	divsclwd	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$
7	1 6 2 3 4	divmulswd	$$⊢\phi \to \left(A{/}_{\mathrm{su}}B=A{/}_{\mathrm{su}}B\leftrightarrow B{\cdot }_s\left(A{/}_{\mathrm{su}}B\right)=A\right)$$
8	5 7	mpbii	$$⊢\phi \to B{\cdot }_s\left(A{/}_{\mathrm{su}}B\right)=A$$