Theorem ltaddsubs2d

Description: Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 28-Feb-2025)

	Ref	Expression
Hypotheses	ltsubadds.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltsubadds.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltsubadds.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	ltaddsubs2d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) <s 𝐶 ↔ 𝐵 <s ( 𝐶 -s 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltsubadds.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		ltsubadds.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		ltsubadds.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	1 2	addscomd	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) = ( 𝐵 +s 𝐴 ) )
5	4	breq1d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) <s 𝐶 ↔ ( 𝐵 +s 𝐴 ) <s 𝐶 ) )
6	2 1 3	ltaddsubsd	⊢ ( 𝜑 → ( ( 𝐵 +s 𝐴 ) <s 𝐶 ↔ 𝐵 <s ( 𝐶 -s 𝐴 ) ) )
7	5 6	bitrd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) <s 𝐶 ↔ 𝐵 <s ( 𝐶 -s 𝐴 ) ) )