Theorem lenegs

Description: Negative of both sides of surreal less-than or equal. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	lenegs	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltnegs	⊢ ( ( 𝐵 ∈ No ∧ 𝐴 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s ( -us ‘ 𝐵 ) ) )
2	1	ancoms	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐵 <s 𝐴 ↔ ( -us ‘ 𝐴 ) <s ( -us ‘ 𝐵 ) ) )
3	2	notbid	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ¬ 𝐵 <s 𝐴 ↔ ¬ ( -us ‘ 𝐴 ) <s ( -us ‘ 𝐵 ) ) )
4		lenlts	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ¬ 𝐵 <s 𝐴 ) )
5		negscl	⊢ ( 𝐵 ∈ No → ( -us ‘ 𝐵 ) ∈ No )
6		negscl	⊢ ( 𝐴 ∈ No → ( -us ‘ 𝐴 ) ∈ No )
7		lenlts	⊢ ( ( ( -us ‘ 𝐵 ) ∈ No ∧ ( -us ‘ 𝐴 ) ∈ No ) → ( ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ↔ ¬ ( -us ‘ 𝐴 ) <s ( -us ‘ 𝐵 ) ) )
8	5 6 7	syl2anr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ↔ ¬ ( -us ‘ 𝐴 ) <s ( -us ‘ 𝐵 ) ) )
9	3 4 8	3bitr4d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ≤s 𝐵 ↔ ( -us ‘ 𝐵 ) ≤s ( -us ‘ 𝐴 ) ) )