Theorem hlgrcl1

Description: Reverse closure for rays. (Contributed by Thierry Arnoux, 13-Jul-2026)

	Ref	Expression
Hypotheses	ishlg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	ishlg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	ishlg.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
	ishlg2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
	ishlg2.1	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
	hlgrcl1.1	⊢ ( 𝜑 → 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 )
Assertion	hlgrcl1	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		ishlg.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		ishlg.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		ishlg.k	⊢ 𝐾 = ( hlG ‘ 𝐺 )
4		ishlg2.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
5		ishlg2.1	⊢ ( 𝜑 → 𝐶 ∈ 𝑃 )
6		hlgrcl1.1	⊢ ( 𝜑 → 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 )
7	1 2 3 4 5	ishlg2	⊢ ( 𝜑 → ( 𝐴 ( 𝐾 ‘ 𝐶 ) 𝐵 ↔ ( ( 𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃 ) ∧ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ) ) )
8	6 7	mpbid	⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑃 ∧ 𝐵 ∈ 𝑃 ) ∧ ( 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶 ∧ ( 𝐴 ∈ ( 𝐶 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝐶 𝐼 𝐴 ) ) ) ) )
9	8	simplld	⊢ ( 𝜑 → 𝐴 ∈ 𝑃 )