Theorem ltsubadds2d

Description: Surreal less-than relationship between subtraction and addition. (Contributed by Scott Fenton, 27-Feb-2025)

	Ref	Expression
Hypotheses	ltsubadds.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	ltsubadds.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	ltsubadds.3	⊢ ( 𝜑 → 𝐶 ∈ No )
Assertion	ltsubadds2d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) <s 𝐶 ↔ 𝐴 <s ( 𝐵 +s 𝐶 ) ) )

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 3	ltsubaddsd	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) <s 𝐶 ↔ 𝐴 <s ( 𝐶 +s 𝐵 ) ) )
5	2 3	addscomd	⊢ ( 𝜑 → ( 𝐵 +s 𝐶 ) = ( 𝐶 +s 𝐵 ) )
6	5	breq2d	⊢ ( 𝜑 → ( 𝐴 <s ( 𝐵 +s 𝐶 ) ↔ 𝐴 <s ( 𝐶 +s 𝐵 ) ) )
7	4 6	bitr4d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) <s 𝐶 ↔ 𝐴 <s ( 𝐵 +s 𝐶 ) ) )