Theorem lnoppinn0 29231

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 4097	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → 𝑡 ∈ (𝐷 ∩ (𝑋𝐼𝑌)))
4	3	ne0d 4287	. 2 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)
5		lnoppinn0.1	. . . 4 ⊢ (𝜑 → 𝑋𝑂𝑌)
6		lnoppinn0.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
7		eqid 2760	. . . . 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 29215	. . . 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 3170	1 ⊢ (𝜑 → (𝐷 ∩ (𝑋𝐼𝑌)) ≠ ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  {copab 5166  ran crn 5651  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  distcds 17437  Itvcitv 28895  LineGclng 28896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423

This theorem is used by:  angmgmaddeu1  29379