Theorem ltsubs2d

Description: Subtraction from both sides of surreal less-than. (Contributed by Scott Fenton, 5-Feb-2025)

	Ref	Expression
Hypotheses	ltsubsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltsubsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltsubsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	ltsubs2d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 -s 𝐵 ) <s ( 𝐶 -s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubsd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltsubsd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltsubsd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		ltsubs2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐶 -s 𝐵 ) <s ( 𝐶 -s 𝐴 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐶 -s 𝐵 ) <s ( 𝐶 -s 𝐴 ) ) )