Theorem oppnid 25859

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 709	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6		oppnid.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 709	. . . . 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 709	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11		simplr 752	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝐷)
12	1, 8, 3, 5, 10, 11	tglnpt 25665	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝑃)
13		simpr 471	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
14	1, 2, 3, 5, 7, 12, 13	axtgbtwnid 25586	. . . 4 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
15	14, 11	eqeltrd 2850	. . 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 25852	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
18	17	simplbda 487	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))
19	15, 18	r19.29a 3226	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → 𝐴 ∈ 𝐷)
20	17	simprbda 486	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷))
21	20	simpld 482	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ¬ 𝐴 ∈ 𝐷)
22	19, 21	pm2.65da 817	1 ⊢ (𝜑 → ¬ 𝐴𝑂𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∃wrex 3062   ∖ cdif 3720   class class class wbr 4786  {copab 4846  ran crn 5250  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-cnv 5257  df-dm 5259  df-rn 5260  df-iota 5994  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-trkgb 25569  df-trkg 25573

This theorem is referenced by:  lnoppnhpg  25877