Theorem lenegsd

Description: Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 14-Mar-2025)

	Ref	Expression
Hypotheses	ltnegsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltnegsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
Assertion	lenegsd	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltnegsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltnegsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		lenegs	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ≤s 𝐵 ↔ ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ) )