Theorem divscld

Description: Surreal division closure law. (Contributed by Scott Fenton, 16-Mar-2025)

	Ref	Expression
Hypotheses	divscld.1	$$⊢\phi \to A\in \mathrm{No}$$
	divscld.2	$$⊢\phi \to B\in \mathrm{No}$$
	divscld.3	$$⊢\phi \to B\ne 0_s$$
Assertion	divscld	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$

Proof

Step	Hyp	Ref	Expression
1		divscld.1	$$⊢\phi \to A\in \mathrm{No}$$
2		divscld.2	$$⊢\phi \to B\in \mathrm{No}$$
3		divscld.3	$$⊢\phi \to B\ne 0_s$$
4		divscl	$$⊢\left(A\in \mathrm{No}\wedge B\in \mathrm{No}\wedge B\ne 0_s\right)\to A{/}_{\mathrm{su}}B\in \mathrm{No}$$
5	1 2 3 4	syl3anc	$$⊢\phi \to A{/}_{\mathrm{su}}B\in \mathrm{No}$$