Theorem cofcut1d

Description: If C is cofinal with A and D is coinitial with B and the cut of A and B lies between C and D , then the cut of C and D is equal to the cut of A and B . Theorem 2.6 of Gonshor p. 10. (Contributed by Scott Fenton, 23-Jan-2025)

	Ref	Expression
Hypotheses	cofcut1d.1	⊢ ( 𝜑 → 𝐴 <<s 𝐵 )
	cofcut1d.2	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 )
	cofcut1d.3	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 )
	cofcut1d.4	⊢ ( 𝜑 → 𝐶 <<s { ( 𝐴 |s 𝐵 ) } )
	cofcut1d.5	⊢ ( 𝜑 → { ( 𝐴 |s 𝐵 ) } <<s 𝐷 )
Assertion	cofcut1d	⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) = ( 𝐶 |s 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		cofcut1d.1	⊢ ( 𝜑 → 𝐴 <<s 𝐵 )
2		cofcut1d.2	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 )
3		cofcut1d.3	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 )
4		cofcut1d.4	⊢ ( 𝜑 → 𝐶 <<s { ( 𝐴 |s 𝐵 ) } )
5		cofcut1d.5	⊢ ( 𝜑 → { ( 𝐴 |s 𝐵 ) } <<s 𝐷 )
6		cofcut1	⊢ ( ( 𝐴 <<s 𝐵 ∧ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐶 𝑥 ≤s 𝑦 ∧ ∀ 𝑧 ∈ 𝐵 ∃ 𝑤 ∈ 𝐷 𝑤 ≤s 𝑧 ) ∧ ( 𝐶 <<s { ( 𝐴 |s 𝐵 ) } ∧ { ( 𝐴 |s 𝐵 ) } <<s 𝐷 ) ) → ( 𝐴 |s 𝐵 ) = ( 𝐶 |s 𝐷 ) )
7	1 2 3 4 5 6	syl122anc	⊢ ( 𝜑 → ( 𝐴 |s 𝐵 ) = ( 𝐶 |s 𝐷 ) )