Theorem sltsub1d

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

	Ref	Expression
Hypotheses	sltsubd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	sltsubd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	sltsubd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	sltsub1d	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sltsubd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sltsubd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sltsubd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		sltsub1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 <s 𝐵 ↔ ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ) )
5	1 2 3 4	syl3anc	⊢ ( 𝜑 → ( 𝐴 <s 𝐵 ↔ ( 𝐴 -s 𝐶 ) <s ( 𝐵 -s 𝐶 ) ) )