Theorem lestr

Description: Surreal transitive law. (Contributed by Scott Fenton, 8-Dec-2021)

	Ref	Expression
Assertion	lestr	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ltlestr	⊢ ( ( 𝐶 ∈ No ∧ 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( 𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) )
2	1	3coml	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐶 <s 𝐴 ∧ 𝐴 ≤s 𝐵 ) → 𝐶 <s 𝐵 ) )
3	2	expcomd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐵 → ( 𝐶 <s 𝐴 → 𝐶 <s 𝐵 ) ) )
4	3	imp	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵 ) → ( 𝐶 <s 𝐴 → 𝐶 <s 𝐵 ) )
5	4	con3d	⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) ∧ 𝐴 ≤s 𝐵 ) → ( ¬ 𝐶 <s 𝐵 → ¬ 𝐶 <s 𝐴 ) )
6	5	expimpd	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) → ¬ 𝐶 <s 𝐴 ) )
7		lenlts	⊢ ( ( 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
8	7	3adant1	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐵 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐵 ) )
9	8	anbi2d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) ↔ ( 𝐴 ≤s 𝐵 ∧ ¬ 𝐶 <s 𝐵 ) ) )
10		lenlts	⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) )
11	10	3adant2	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ≤s 𝐶 ↔ ¬ 𝐶 <s 𝐴 ) )
12	6 9 11	3imtr4d	⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No ) → ( ( 𝐴 ≤s 𝐵 ∧ 𝐵 ≤s 𝐶 ) → 𝐴 ≤s 𝐶 ) )