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Theorem oppne3 25353
Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-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 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l 𝐿 = (LineG‘𝐺)
opphl.d (𝜑𝐷 ∈ ran 𝐿)
opphl.g (𝜑𝐺 ∈ TarskiG)
oppcom.a (𝜑𝐴𝑃)
oppcom.b (𝜑𝐵𝑃)
oppcom.o (𝜑𝐴𝑂𝐵)
Assertion
Ref Expression
oppne3 (𝜑𝐴𝐵)
Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   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)   𝐵(𝑎,𝑏)   𝐺(𝑎,𝑏)   𝐿(𝑎,𝑏)   (𝑎,𝑏)   𝑂(𝑎,𝑏)

Proof of Theorem oppne3
StepHypRef Expression
1 hpg.p . . . . . 6 𝑃 = (Base‘𝐺)
2 eqid 2609 . . . . . 6 (dist‘𝐺) = (dist‘𝐺)
3 hpg.i . . . . . 6 𝐼 = (Itv‘𝐺)
4 opphl.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
54ad3antrrr 761 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
6 oppcom.a . . . . . . 7 (𝜑𝐴𝑃)
76ad3antrrr 761 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
8 opphl.l . . . . . . 7 𝐿 = (LineG‘𝐺)
9 opphl.d . . . . . . . 8 (𝜑𝐷 ∈ ran 𝐿)
109ad3antrrr 761 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
11 simplr 787 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝐷)
121, 8, 3, 5, 10, 11tglnpt 25162 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
13 simpr 475 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
14 simpllr 794 . . . . . . . 8 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵)
1514oveq2d 6543 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵))
1613, 15eleqtrrd 2690 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴))
171, 2, 3, 5, 7, 12, 16axtgbtwnid 25082 . . . . 5 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡)
1817, 11eqeltrd 2687 . . . 4 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝐷)
19 oppcom.o . . . . . . 7 (𝜑𝐴𝑂𝐵)
20 hpg.d . . . . . . . 8 = (dist‘𝐺)
21 hpg.o . . . . . . . 8 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
22 oppcom.b . . . . . . . 8 (𝜑𝐵𝑃)
231, 20, 3, 21, 6, 22islnopp 25349 . . . . . . 7 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
2419, 23mpbid 220 . . . . . 6 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
2524simprd 477 . . . . 5 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2625adantr 479 . . . 4 ((𝜑𝐴 = 𝐵) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2718, 26r19.29a 3059 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴𝐷)
281, 20, 3, 21, 8, 9, 4, 6, 22, 19oppne1 25351 . . . 4 (𝜑 → ¬ 𝐴𝐷)
2928adantr 479 . . 3 ((𝜑𝐴 = 𝐵) → ¬ 𝐴𝐷)
3027, 29pm2.65da 597 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
3130neqned 2788 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wne 2779  wrex 2896  cdif 3536   class class class wbr 4577  {copab 4636  ran crn 5029  cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-iota 5754  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-trkgb 25065  df-trkg 25069
This theorem is referenced by:  colopp  25379  colhp  25380  trgcopyeulem  25415
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