Theorem islnoppd 27101

Description: Deduce that 𝐴 and 𝐵 lie on opposite sides of line 𝐿. (Contributed by Thierry Arnoux, 16-Aug-2020.)

Hypotheses

Ref	Expression
hpg.p	⊢ 𝑃 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
islnoppd.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
islnoppd.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
islnoppd.c	⊢ (𝜑 → 𝐶 ∈ 𝐷)
islnoppd.1	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
islnoppd.2	⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
islnoppd.3	⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))

Assertion

Ref	Expression
islnoppd	⊢ (𝜑 → 𝐴𝑂𝐵)

Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝐶   𝑡,𝐷,𝑎,𝑏   𝑡,𝐼   𝜑,𝑡

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑎,𝑏)   𝑃(𝑡)   𝐺(𝑡,𝑎,𝑏)   − (𝑡,𝑎,𝑏)   𝑂(𝑡,𝑎,𝑏)

Proof of Theorem islnoppd

Step	Hyp	Ref	Expression
1		islnoppd.1	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
2		islnoppd.2	. . 3 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)
3		islnoppd.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐷)
4		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → 𝑡 = 𝐶)
5	4	eleq1d 2823	. . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6		islnoppd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
7	3, 5, 6	rspcedvd 3563	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
8	1, 2, 7	jca31 515	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
9		hpg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
10		hpg.d	. . 3 ⊢ − = (dist‘𝐺)
11		hpg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
12		hpg.o	. . 3 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
13		islnoppd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
14		islnoppd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
15	9, 10, 11, 12, 13, 14	islnopp 27100	. 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
16	8, 15	mpbird 256	1 ⊢ (𝜑 → 𝐴𝑂𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∖ cdif 3884   class class class wbr 5074  {copab 5136  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  Itvcitv 26794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441  df-ov 7278

This theorem is referenced by:  opphllem2  27109  opphllem4  27111  outpasch  27116  lmiopp  27163