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Theorem oppcom 25436
 Description: Commutativity rule for "opposite" Theorem 9.2 of [Schwabhauser] p. 67. (Contributed by Thierry Arnoux, 19-Dec-2019.)
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
oppcom (𝜑𝐵𝑂𝐴)
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 oppcom
StepHypRef Expression
1 oppcom.o . . . . . 6 (𝜑𝐴𝑂𝐵)
2 hpg.p . . . . . . 7 𝑃 = (Base‘𝐺)
3 hpg.d . . . . . . 7 = (dist‘𝐺)
4 hpg.i . . . . . . 7 𝐼 = (Itv‘𝐺)
5 hpg.o . . . . . . 7 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
6 oppcom.a . . . . . . 7 (𝜑𝐴𝑃)
7 oppcom.b . . . . . . 7 (𝜑𝐵𝑃)
82, 3, 4, 5, 6, 7islnopp 25431 . . . . . 6 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
91, 8mpbid 221 . . . . 5 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
109simpld 474 . . . 4 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷))
1110simprd 478 . . 3 (𝜑 → ¬ 𝐵𝐷)
1210simpld 474 . . 3 (𝜑 → ¬ 𝐴𝐷)
139simprd 478 . . . 4 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14 opphl.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
1514ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
166ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
17 opphl.l . . . . . . . . 9 𝐿 = (LineG‘𝐺)
1814adantr 480 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐺 ∈ TarskiG)
19 opphl.d . . . . . . . . . 10 (𝜑𝐷 ∈ ran 𝐿)
2019adantr 480 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐷 ∈ ran 𝐿)
21 simpr 476 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝑡𝐷)
222, 17, 4, 18, 20, 21tglnpt 25244 . . . . . . . 8 ((𝜑𝑡𝐷) → 𝑡𝑃)
2322adantr 480 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
247ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
25 simpr 476 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 25183 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
2714ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
287ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵𝑃)
2922adantr 480 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡𝑃)
306ad2antrr 758 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴𝑃)
31 simpr 476 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 25183 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
3326, 32impbida 873 . . . . 5 ((𝜑𝑡𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
3433rexbidva 3031 . . . 4 (𝜑 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
3513, 34mpbid 221 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))
3611, 12, 35jca31 555 . 2 (𝜑 → ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
372, 3, 4, 5, 7, 6islnopp 25431 . 2 (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
3836, 37mpbird 246 1 (𝜑𝐵𝑂𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537   class class class wbr 4583  {copab 4642  ran crn 5039  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135  LineGclng 25136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152 This theorem is referenced by:  opphllem2  25440  opphllem4  25442  opphllem5  25443  opphllem6  25444  lnperpex  25495
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