Theorem islnoppd 29076

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 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → 𝑡 = 𝐶)
5	4	eleq1d 2850	. . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝐶) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝐶 ∈ (𝐴𝐼𝐵)))
6		islnoppd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵))
7	3, 5, 6	rspcedvd 3585	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))
8	1, 2, 7	jca31 524	. 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 29075	. 2 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
16	8, 15	mpbird 260	1 ⊢ (𝜑 → 𝐴𝑂𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ∖ cdif 3903   class class class wbr 5111  {copab 5175  ‘cfv 6540  (class class class)co 7419  Basecbs 17295  distcds 17345  Itvcitv 28757

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  opphllem2  29084  opphllem4  29086  oppmir  29091  outpasch  29092  lnincplng  29121  plngrotlem1  29124  plngrotlem2  29125  lmiopp  29167  tgaaddcpbllem1  29207  tgaaddcpbllem2  29208  tgaaddcpbl  29210  prlngmolem1  29261  prlngsymquadopp  29274