Theorem zsubscld

Description: The surreal integers are closed under subtraction. (Contributed by Scott Fenton, 25-Jul-2025)

	Ref	Expression
Hypotheses	zsubscld.1	$$⊢\phi \to A\in {\mathbb{Z}}_s$$
	zsubscld.2	$$⊢\phi \to B\in {\mathbb{Z}}_s$$
Assertion	zsubscld	$$⊢\phi \to A{-}_{s}B\in {\mathbb{Z}}_s$$

Proof

Step	Hyp	Ref	Expression
1		zsubscld.1	$$⊢\phi \to A\in {\mathbb{Z}}_s$$
2		zsubscld.2	$$⊢\phi \to B\in {\mathbb{Z}}_s$$
3	1	znod	$$⊢\phi \to A\in \mathrm{No}$$
4	2	znod	$$⊢\phi \to B\in \mathrm{No}$$
5	3 4	subsvald	$$⊢\phi \to A{-}_{s}B=A{+}_s{+}_s\left(B\right)$$
6	2	znegscld	$$⊢\phi \to {+}_s\left(B\right)\in {\mathbb{Z}}_s$$
7	1 6	zaddscld	$$⊢\phi \to A{+}_s{+}_s\left(B\right)\in {\mathbb{Z}}_s$$
8	5 7	eqeltrd	$$⊢\phi \to A{-}_{s}B\in {\mathbb{Z}}_s$$