Theorem ltsubsubs2bd

Description: Equivalence for the surreal less-than relationship between differences. (Contributed by Scott Fenton, 21-Feb-2025)

	Ref	Expression
Hypotheses	ltsubsubsbd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltsubsubsbd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltsubsubsbd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	ltsubsubsbd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
Assertion	ltsubsubs2bd	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) <s ( 𝐶 -s 𝐷 ) ↔ ( 𝐷 -s 𝐶 ) <s ( 𝐵 -s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubsubsbd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltsubsubsbd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltsubsubsbd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		ltsubsubsbd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
5	4 3	subscld	⊢ ( 𝜑 → ( 𝐷 -s 𝐶 ) ∈ No )
6	2 1	subscld	⊢ ( 𝜑 → ( 𝐵 -s 𝐴 ) ∈ No )
7	5 6	ltnegsd	⊢ ( 𝜑 → ( ( 𝐷 -s 𝐶 ) <s ( 𝐵 -s 𝐴 ) ↔ ( -us ‘ ( 𝐵 -s 𝐴 ) ) <s ( -us ‘ ( 𝐷 -s 𝐶 ) ) ) )
8	2 1	negsubsdi2d	⊢ ( 𝜑 → ( -us ‘ ( 𝐵 -s 𝐴 ) ) = ( 𝐴 -s 𝐵 ) )
9	4 3	negsubsdi2d	⊢ ( 𝜑 → ( -us ‘ ( 𝐷 -s 𝐶 ) ) = ( 𝐶 -s 𝐷 ) )
10	8 9	breq12d	⊢ ( 𝜑 → ( ( -us ‘ ( 𝐵 -s 𝐴 ) ) <s ( -us ‘ ( 𝐷 -s 𝐶 ) ) ↔ ( 𝐴 -s 𝐵 ) <s ( 𝐶 -s 𝐷 ) ) )
11	7 10	bitr2d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) <s ( 𝐶 -s 𝐷 ) ↔ ( 𝐷 -s 𝐶 ) <s ( 𝐵 -s 𝐴 ) ) )