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Theorem oppne3 28567
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 . . . 4 𝑃 = (Base‘𝐺)
2 hpg.d . . . 4 = (dist‘𝐺)
3 hpg.i . . . 4 𝐼 = (Itv‘𝐺)
4 hpg.o . . . 4 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5 opphl.l . . . 4 𝐿 = (LineG‘𝐺)
6 opphl.d . . . 4 (𝜑𝐷 ∈ ran 𝐿)
7 opphl.g . . . 4 (𝜑𝐺 ∈ TarskiG)
8 oppcom.a . . . 4 (𝜑𝐴𝑃)
9 oppcom.b . . . 4 (𝜑𝐵𝑃)
10 oppcom.o . . . 4 (𝜑𝐴𝑂𝐵)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10oppne1 28565 . . 3 (𝜑 → ¬ 𝐴𝐷)
127ad3antrrr 728 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
138ad3antrrr 728 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
146ad3antrrr 728 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
15 simplr 767 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝐷)
161, 5, 3, 12, 14, 15tglnpt 28373 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
17 simpr 483 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
18 simpllr 774 . . . . . . . 8 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵)
1918oveq2d 7442 . . . . . . 7 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵))
2017, 19eleqtrrd 2832 . . . . . 6 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴))
211, 2, 3, 12, 13, 16, 20axtgbtwnid 28290 . . . . 5 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡)
2221, 15eqeltrd 2829 . . . 4 ((((𝜑𝐴 = 𝐵) ∧ 𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝐷)
231, 2, 3, 4, 8, 9islnopp 28563 . . . . . . 7 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
2410, 23mpbid 231 . . . . . 6 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
2524simprd 494 . . . . 5 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2625adantr 479 . . . 4 ((𝜑𝐴 = 𝐵) → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
2722, 26r19.29a 3159 . . 3 ((𝜑𝐴 = 𝐵) → 𝐴𝐷)
2811, 27mtand 814 . 2 (𝜑 → ¬ 𝐴 = 𝐵)
2928neqned 2944 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wne 2937  wrex 3067  cdif 3946   class class class wbr 5152  {copab 5214  ran crn 5683  cfv 6553  (class class class)co 7426  Basecbs 17187  distcds 17249  TarskiGcstrkg 28251  Itvcitv 28257  LineGclng 28258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-cnv 5690  df-dm 5692  df-rn 5693  df-iota 6505  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-trkgb 28273  df-trkg 28277
This theorem is referenced by:  colopp  28593  trgcopyeulem  28629
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