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Theorem oppcom 28464
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 28459 . . . . . 6 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
91, 8mpbid 231 . . . . 5 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
109simpld 494 . . . 4 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷))
1110simprd 495 . . 3 (𝜑 → ¬ 𝐵𝐷)
1210simpld 494 . . 3 (𝜑 → ¬ 𝐴𝐷)
139simprd 495 . . . 4 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14 opphl.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
1514ad2antrr 723 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
166ad2antrr 723 . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝑡𝐷)
222, 17, 4, 18, 20, 21tglnpt 28269 . . . . . . . 8 ((𝜑𝑡𝐷) → 𝑡𝑃)
2322adantr 480 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
247ad2antrr 723 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
25 simpr 484 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 28208 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
2714ad2antrr 723 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
287ad2antrr 723 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵𝑃)
2922adantr 480 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡𝑃)
306ad2antrr 723 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴𝑃)
31 simpr 484 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 28208 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
3326, 32impbida 798 . . . . 5 ((𝜑𝑡𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
3433rexbidva 3168 . . . 4 (𝜑 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
3513, 34mpbid 231 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))
3611, 12, 35jca31 514 . 2 (𝜑 → ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
372, 3, 4, 5, 7, 6islnopp 28459 . 2 (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
3836, 37mpbird 257 1 (𝜑𝐵𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1533  wcel 2098  wrex 3062  cdif 3937   class class class wbr 5138  {copab 5200  ran crn 5667  cfv 6533  (class class class)co 7401  Basecbs 17143  distcds 17205  TarskiGcstrkg 28147  Itvcitv 28153  LineGclng 28154
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 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-cnv 5674  df-dm 5676  df-rn 5677  df-iota 6485  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-trkgc 28168  df-trkgb 28169  df-trkgcb 28170  df-trkg 28173
This theorem is referenced by:  opphllem2  28468  opphllem4  28470  opphllem5  28471  opphllem6  28472  lnperpex  28523
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