Theorem lnoppinn0 29106

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.)

Hypotheses

Ref	Expression
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 ∈ (𝑎𝐼𝑏))}
lnoppinn0.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
lnoppinn0.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
lnoppinn0.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
lnoppinn0.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
lnoppinn0.1	⊢ (𝜑 → 𝑋𝑂𝑌)

Assertion

Ref	Expression
lnoppinn0	⊢ (𝜑 → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)

Distinct variable groups:   𝐷,𝑎,𝑏,𝑡   𝐼,𝑎,𝑏,𝑡   𝑃,𝑎,𝑏   𝑡,𝑋   𝑡,𝑌   𝜑,𝑡

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝑃(𝑡)   𝐺(𝑡, 𝑎, 𝑏)   𝐿(𝑡, 𝑎, 𝑏)   𝑂(𝑡, 𝑎, 𝑏)   𝑉(𝑡, 𝑎, 𝑏)   𝑋(𝑎, 𝑏)   𝑌(𝑎, 𝑏)

Proof of Theorem lnoppinn0

Step	Hyp	Ref	Expression
1		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → 𝑡 ∈ 𝐷)
2		simpr 490	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → 𝑡 ∈ (𝑋𝐼𝑌))
3	1, 2	elind 4149	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → 𝑡 ∈ (𝐷 ∩ (𝑋𝐼𝑌)))
4	3	ne0d 4291	. 2 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)
5		lnoppinn0.1	. . . 4 ⊢ (𝜑 → 𝑋𝑂𝑌)
6		lnoppinn0.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
7		eqid 2762	. . . . 5 ⊢ (dist‘𝐺) = (dist‘𝐺)
8		lnoppinn0.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
9		lnoppinn0.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
10		lnoppinn0.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11		lnoppinn0.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
12	6, 7, 8, 9, 10, 11	islnopp 29090	. . . 4 ⊢ (𝜑 → (𝑋𝑂𝑌 ↔ ((¬ 𝑋 ∈ 𝐷 ∧ ¬ 𝑌 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑋𝐼𝑌))))
13	5, 12	mpbid 235	. . 3 ⊢ (𝜑 → ((¬ 𝑋 ∈ 𝐷 ∧ ¬ 𝑌 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑋𝐼𝑌)))
14	13	simprd 501	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑋𝐼𝑌))
15	4, 14	r19.29a 3172	1 ⊢ (𝜑 → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∃wrex 3088   ∖ cdif 3899   ∩ cin 3901  ∅c0 4282   class class class wbr 5107  {copab 5171  ran crn 5660  ‘cfv 6537  (class class class)co 7416  Basecbs 17303  distcds 17353  Itvcitv 28770  LineGclng 28771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  angmndaddeu1  29250