Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppcom Structured version   Visualization version   GIF version

Theorem oppcom 26579
 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 26574 . . . . . 6 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
91, 8mpbid 235 . . . . 5 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
109simpld 498 . . . 4 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷))
1110simprd 499 . . 3 (𝜑 → ¬ 𝐵𝐷)
1210simpld 498 . . 3 (𝜑 → ¬ 𝐴𝐷)
139simprd 499 . . . 4 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14 opphl.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
1514ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
166ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
17 opphl.l . . . . . . . . 9 𝐿 = (LineG‘𝐺)
1814adantr 484 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐺 ∈ TarskiG)
19 opphl.d . . . . . . . . . 10 (𝜑𝐷 ∈ ran 𝐿)
2019adantr 484 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐷 ∈ ran 𝐿)
21 simpr 488 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝑡𝐷)
222, 17, 4, 18, 20, 21tglnpt 26384 . . . . . . . 8 ((𝜑𝑡𝐷) → 𝑡𝑃)
2322adantr 484 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
247ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
25 simpr 488 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 26323 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
2714ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
287ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵𝑃)
2922adantr 484 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡𝑃)
306ad2antrr 725 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴𝑃)
31 simpr 488 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 26323 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
3326, 32impbida 800 . . . . 5 ((𝜑𝑡𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
3433rexbidva 3255 . . . 4 (𝜑 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
3513, 34mpbid 235 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))
3611, 12, 35jca31 518 . 2 (𝜑 → ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
372, 3, 4, 5, 7, 6islnopp 26574 . 2 (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
3836, 37mpbird 260 1 (𝜑𝐵𝑂𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ∖ cdif 3879   class class class wbr 5033  {copab 5095  ran crn 5523  ‘cfv 6329  (class class class)co 7142  Basecbs 16492  distcds 16583  TarskiGcstrkg 26265  Itvcitv 26271  LineGclng 26272 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-br 5034  df-opab 5096  df-cnv 5530  df-dm 5532  df-rn 5533  df-iota 6288  df-fv 6337  df-ov 7145  df-oprab 7146  df-mpo 7147  df-trkgc 26283  df-trkgb 26284  df-trkgcb 26285  df-trkg 26288 This theorem is referenced by:  opphllem2  26583  opphllem4  26585  opphllem5  26586  opphllem6  26587  lnperpex  26638
 Copyright terms: Public domain W3C validator