Theorem divscld

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

	Ref	Expression
Hypotheses	divscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	divscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	divscld.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
Assertion	divscld	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )

Proof

Step	Hyp	Ref	Expression
1		divscld.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		divscld.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		divscld.3	⊢ ( 𝜑 → 𝐵 ≠ 0s )
4		divscl	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → ( 𝐴 /su 𝐵 ) ∈ No )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 /su 𝐵 ) ∈ No )