Theorem subscl

Description: Closure law for surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	subscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		subsval	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )
2		negscl	⊢ ( 𝐵 ∈ No → ( -us ‘ 𝐵 ) ∈ No )
3		addscl	⊢ ( ( 𝐴 ∈ No ∧ ( -us ‘ 𝐵 ) ∈ No ) → ( 𝐴 +s ( -us ‘ 𝐵 ) ) ∈ No )
4	2 3	sylan2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 +s ( -us ‘ 𝐵 ) ) ∈ No )
5	1 4	eqeltrd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) ∈ No )