Theorem lnoppinn0 29110

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 4146	. . 3 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝑋𝐼𝑌)) → 𝑡 ∈ (𝐷 ∩ (𝑋𝐼𝑌)))
4	3	ne0d 4288	. 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 29094	. . . 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 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   ∖ cdif 3896   ∩ cin 3898  ∅c0 4279   class class class wbr 5103  {copab 5167  ran crn 5656  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  distcds 17351  Itvcitv 28774  LineGclng 28775

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 2732  ax-sep 5251  ax-pr 5398

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-iota 6489  df-fv 6541  df-ov 7416

This theorem is used by:  angmgmaddeu1  29258