Theorem sltaddsubd

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

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

Proof

Step	Hyp	Ref	Expression
1		sltsubadd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sltsubadd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sltsubadd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4	1 2	addscld	⊢ ( 𝜑 → ( 𝐴 +s 𝐵 ) ∈ No )
5	4 3 2	sltsub1d	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) <s 𝐶 ↔ ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) <s ( 𝐶 -s 𝐵 ) ) )
6		pncans	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) = 𝐴 )
7	1 2 6	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) = 𝐴 )
8	7	breq1d	⊢ ( 𝜑 → ( ( ( 𝐴 +s 𝐵 ) -s 𝐵 ) <s ( 𝐶 -s 𝐵 ) ↔ 𝐴 <s ( 𝐶 -s 𝐵 ) ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( ( 𝐴 +s 𝐵 ) <s 𝐶 ↔ 𝐴 <s ( 𝐶 -s 𝐵 ) ) )