Theorem slesubsubbd

Description: Equivalence for the surreal less-than or equal relationship between differences. (Contributed by Scott Fenton, 7-Mar-2025)

	Ref	Expression
Hypotheses	sltsubsubbd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
	sltsubsubbd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
	sltsubsubbd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
	sltsubsubbd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
Assertion	slesubsubbd	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐶 ) ≤s ( 𝐵 -s 𝐷 ) ↔ ( 𝐴 -s 𝐵 ) ≤s ( 𝐶 -s 𝐷 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sltsubsubbd.1	⊢ ( 𝜑 → 𝐴 ∈ No )
2		sltsubsubbd.2	⊢ ( 𝜑 → 𝐵 ∈ No )
3		sltsubsubbd.3	⊢ ( 𝜑 → 𝐶 ∈ No )
4		sltsubsubbd.4	⊢ ( 𝜑 → 𝐷 ∈ No )
5	2 1 4 3	sltsubsub3bd	⊢ ( 𝜑 → ( ( 𝐵 -s 𝐷 ) <s ( 𝐴 -s 𝐶 ) ↔ ( 𝐶 -s 𝐷 ) <s ( 𝐴 -s 𝐵 ) ) )
6	5	notbid	⊢ ( 𝜑 → ( ¬ ( 𝐵 -s 𝐷 ) <s ( 𝐴 -s 𝐶 ) ↔ ¬ ( 𝐶 -s 𝐷 ) <s ( 𝐴 -s 𝐵 ) ) )
7	1 3	subscld	⊢ ( 𝜑 → ( 𝐴 -s 𝐶 ) ∈ No )
8	2 4	subscld	⊢ ( 𝜑 → ( 𝐵 -s 𝐷 ) ∈ No )
9		slenlt	⊢ ( ( ( 𝐴 -s 𝐶 ) ∈ No ∧ ( 𝐵 -s 𝐷 ) ∈ No ) → ( ( 𝐴 -s 𝐶 ) ≤s ( 𝐵 -s 𝐷 ) ↔ ¬ ( 𝐵 -s 𝐷 ) <s ( 𝐴 -s 𝐶 ) ) )
10	7 8 9	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐶 ) ≤s ( 𝐵 -s 𝐷 ) ↔ ¬ ( 𝐵 -s 𝐷 ) <s ( 𝐴 -s 𝐶 ) ) )
11	1 2	subscld	⊢ ( 𝜑 → ( 𝐴 -s 𝐵 ) ∈ No )
12	3 4	subscld	⊢ ( 𝜑 → ( 𝐶 -s 𝐷 ) ∈ No )
13		slenlt	⊢ ( ( ( 𝐴 -s 𝐵 ) ∈ No ∧ ( 𝐶 -s 𝐷 ) ∈ No ) → ( ( 𝐴 -s 𝐵 ) ≤s ( 𝐶 -s 𝐷 ) ↔ ¬ ( 𝐶 -s 𝐷 ) <s ( 𝐴 -s 𝐵 ) ) )
14	11 12 13	syl2anc	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐵 ) ≤s ( 𝐶 -s 𝐷 ) ↔ ¬ ( 𝐶 -s 𝐷 ) <s ( 𝐴 -s 𝐵 ) ) )
15	6 10 14	3bitr4d	⊢ ( 𝜑 → ( ( 𝐴 -s 𝐶 ) ≤s ( 𝐵 -s 𝐷 ) ↔ ( 𝐴 -s 𝐵 ) ≤s ( 𝐶 -s 𝐷 ) ) )