Theorem subsvald

Description: The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	subsvald.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	subsvald.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	subsvald	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		subsvald.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		subsvald.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		subsval	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) = ( 𝐴 +s ( -us ‘ 𝐵 ) ) )