Theorem oppnid 29002

Description: The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-Mar-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))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
oppnid.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)

Assertion

Ref	Expression
oppnid	⊢ (𝜑 → ¬ 𝐴𝑂𝐴)

Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐷   𝑡,𝐺   𝑡,𝐿   𝑡,𝐼   𝑡,𝑂   𝑡,𝑃   𝜑,𝑡   𝑡, −   𝑡,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐴(𝑎,𝑏)   𝐺(𝑎,𝑏)   𝐿(𝑎,𝑏)   − (𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem oppnid

Step	Hyp	Ref	Expression
1		hpg.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		hpg.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		opphl.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6		oppnid.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 742	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 ∈ 𝑃)
8		opphl.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
9		opphl.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
10	9	ad3antrrr 742	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11		simplr 780	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝐷)
12	1, 8, 3, 5, 10, 11	tglnpt 28794	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝑃)
13		simpr 489	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
14	1, 2, 3, 5, 7, 12, 13	axtgbtwnid 28711	. . . 4 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
15	14, 11	eqeltrd 2861	. . 3 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 ∈ 𝐷)
16		hpg.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
17	1, 2, 3, 16, 6, 6	islnopp 28995	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
18	17	simplbda 504	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))
19	15, 18	r19.29a 3171	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → 𝐴 ∈ 𝐷)
20	17	simprbda 503	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷))
21	20	simpld 499	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ¬ 𝐴 ∈ 𝐷)
22	19, 21	pm2.65da 828	1 ⊢ (𝜑 → ¬ 𝐴𝑂𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∃wrex 3087   ∖ cdif 3901   class class class wbr 5108  {copab 5172  ran crn 5662  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  distcds 17318  TarskiGcstrkg 28672  Itvcitv 28678  LineGclng 28679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-cnv 5669  df-dm 5671  df-rn 5672  df-iota 6492  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-trkgb 28694  df-trkg 28698

This theorem is referenced by:  lnoppnhpg  29021