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

Theorem oppcom 26458
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 26453 . . . . . 6 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
91, 8mpbid 233 . . . . 5 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
109simpld 495 . . . 4 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷))
1110simprd 496 . . 3 (𝜑 → ¬ 𝐵𝐷)
1210simpld 495 . . 3 (𝜑 → ¬ 𝐴𝐷)
139simprd 496 . . . 4 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14 opphl.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
1514ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
166ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
17 opphl.l . . . . . . . . 9 𝐿 = (LineG‘𝐺)
1814adantr 481 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐺 ∈ TarskiG)
19 opphl.d . . . . . . . . . 10 (𝜑𝐷 ∈ ran 𝐿)
2019adantr 481 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐷 ∈ ran 𝐿)
21 simpr 485 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝑡𝐷)
222, 17, 4, 18, 20, 21tglnpt 26263 . . . . . . . 8 ((𝜑𝑡𝐷) → 𝑡𝑃)
2322adantr 481 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
247ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
25 simpr 485 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 26202 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
2714ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
287ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵𝑃)
2922adantr 481 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡𝑃)
306ad2antrr 722 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴𝑃)
31 simpr 485 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 26202 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
3326, 32impbida 797 . . . . 5 ((𝜑𝑡𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
3433rexbidva 3296 . . . 4 (𝜑 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
3513, 34mpbid 233 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))
3611, 12, 35jca31 515 . 2 (𝜑 → ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
372, 3, 4, 5, 7, 6islnopp 26453 . 2 (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
3836, 37mpbird 258 1 (𝜑𝐵𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wrex 3139  cdif 3932   class class class wbr 5058  {copab 5120  ran crn 5550  cfv 6349  (class class class)co 7145  Basecbs 16473  distcds 16564  TarskiGcstrkg 26144  Itvcitv 26150  LineGclng 26151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-cnv 5557  df-dm 5559  df-rn 5560  df-iota 6308  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-trkgc 26162  df-trkgb 26163  df-trkgcb 26164  df-trkg 26167
This theorem is referenced by:  opphllem2  26462  opphllem4  26464  opphllem5  26465  opphllem6  26466  lnperpex  26517
  Copyright terms: Public domain W3C validator