Theorem lnoppinn0

Description: The segment between two points X and Y on opposite sides of a line D intersects D . (Contributed by Thierry Arnoux, 23-Aug-2026)

	Ref	Expression
Hypotheses	lnoppinn0.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
	lnoppinn0.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
	lnoppinn0.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
	lnoppinn0.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
	lnoppinn0.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
	lnoppinn0.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
	lnoppinn0.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
	lnoppinn0.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
	lnoppinn0.1	⊢ ( 𝜑 → 𝑋 𝑂 𝑌 )
Assertion	lnoppinn0	⊢ ( 𝜑 → ( 𝐷 ∩ ( 𝑋 𝐼 𝑌 ) ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		lnoppinn0.p	⊢ 𝑃 = ( Base ‘ 𝐺 )
2		lnoppinn0.i	⊢ 𝐼 = ( Itv ‘ 𝐺 )
3		lnoppinn0.l	⊢ 𝐿 = ( LineG ‘ 𝐺 )
4		lnoppinn0.o	⊢ 𝑂 = { ⟨ 𝑎 , 𝑏 ⟩ ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) }
5		lnoppinn0.g	⊢ ( 𝜑 → 𝐺 ∈ 𝑉 )
6		lnoppinn0.d	⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 )
7		lnoppinn0.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑃 )
8		lnoppinn0.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑃 )
9		lnoppinn0.1	⊢ ( 𝜑 → 𝑋 𝑂 𝑌 )
10		simplr	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑡 ∈ 𝐷 )
11		simpr	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) )
12	10 11	elind	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) → 𝑡 ∈ ( 𝐷 ∩ ( 𝑋 𝐼 𝑌 ) ) )
13	12	ne0d	⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) → ( 𝐷 ∩ ( 𝑋 𝐼 𝑌 ) ) ≠ ∅ )
14		eqid	⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 )
15	1 14 2 4 7 8	islnopp	⊢ ( 𝜑 → ( 𝑋 𝑂 𝑌 ↔ ( ( ¬ 𝑋 ∈ 𝐷 ∧ ¬ 𝑌 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) ) )
16	9 15	mpbid	⊢ ( 𝜑 → ( ( ¬ 𝑋 ∈ 𝐷 ∧ ¬ 𝑌 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) ) )
17	16	simprd	⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑋 𝐼 𝑌 ) )
18	13 17	r19.29a	⊢ ( 𝜑 → ( 𝐷 ∩ ( 𝑋 𝐼 𝑌 ) ) ≠ ∅ )