Theorem zsubscld

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

	Ref	Expression
Hypotheses	zsubscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤs )
	zsubscld.2	⊢ ( 𝜑 → 𝐵 ∈ ℤs )
Assertion	zsubscld	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) ∈ ℤs )

Proof

Step	Hyp	Ref	Expression
1		zsubscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℤs )
2		zsubscld.2	⊢ ( 𝜑 → 𝐵 ∈ ℤs )
3	1	znod	⊢ ( 𝜑 → 𝐴 ∈ No )
4	2	znod	⊢ ( 𝜑 → 𝐵 ∈ No )
5	3 4	subsvald	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )
6	2	znegscld	⊢ ( 𝜑 → ( -us ‘ 𝐵 ) ∈ ℤs )
7	1 6	zaddscld	⊢ ( 𝜑 → ( 𝐴 +s ( -us ‘ 𝐵 ) ) ∈ ℤs )
8	5 7	eqeltrd	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) ∈ ℤs )