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Theorem oppcom 25354
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 25349 . . . . . 6 (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
91, 8mpbid 220 . . . . 5 (𝜑 → ((¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
109simpld 473 . . . 4 (𝜑 → (¬ 𝐴𝐷 ∧ ¬ 𝐵𝐷))
1110simprd 477 . . 3 (𝜑 → ¬ 𝐵𝐷)
1210simpld 473 . . 3 (𝜑 → ¬ 𝐴𝐷)
139simprd 477 . . . 4 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵))
14 opphl.g . . . . . . . 8 (𝜑𝐺 ∈ TarskiG)
1514ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG)
166ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴𝑃)
17 opphl.l . . . . . . . . 9 𝐿 = (LineG‘𝐺)
1814adantr 479 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐺 ∈ TarskiG)
19 opphl.d . . . . . . . . . 10 (𝜑𝐷 ∈ ran 𝐿)
2019adantr 479 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝐷 ∈ ran 𝐿)
21 simpr 475 . . . . . . . . 9 ((𝜑𝑡𝐷) → 𝑡𝐷)
222, 17, 4, 18, 20, 21tglnpt 25162 . . . . . . . 8 ((𝜑𝑡𝐷) → 𝑡𝑃)
2322adantr 479 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡𝑃)
247ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐵𝑃)
25 simpr 475 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵))
262, 3, 4, 15, 16, 23, 24, 25tgbtwncom 25100 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐵𝐼𝐴))
2714ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐺 ∈ TarskiG)
287ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐵𝑃)
2922adantr 479 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡𝑃)
306ad2antrr 757 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝐴𝑃)
31 simpr 475 . . . . . . 7 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐵𝐼𝐴))
322, 3, 4, 27, 28, 29, 30, 31tgbtwncom 25100 . . . . . 6 (((𝜑𝑡𝐷) ∧ 𝑡 ∈ (𝐵𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐵))
3326, 32impbida 872 . . . . 5 ((𝜑𝑡𝐷) → (𝑡 ∈ (𝐴𝐼𝐵) ↔ 𝑡 ∈ (𝐵𝐼𝐴)))
3433rexbidva 3030 . . . 4 (𝜑 → (∃𝑡𝐷 𝑡 ∈ (𝐴𝐼𝐵) ↔ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
3513, 34mpbid 220 . . 3 (𝜑 → ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))
3611, 12, 35jca31 554 . 2 (𝜑 → ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴)))
372, 3, 4, 5, 7, 6islnopp 25349 . 2 (𝜑 → (𝐵𝑂𝐴 ↔ ((¬ 𝐵𝐷 ∧ ¬ 𝐴𝐷) ∧ ∃𝑡𝐷 𝑡 ∈ (𝐵𝐼𝐴))))
3836, 37mpbird 245 1 (𝜑𝐵𝑂𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wcel 1976  wrex 2896  cdif 3536   class class class wbr 4577  {copab 4636  ran crn 5029  cfv 5790  (class class class)co 6527  Basecbs 15641  distcds 15723  TarskiGcstrkg 25046  Itvcitv 25052  LineGclng 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-cnv 5036  df-dm 5038  df-rn 5039  df-iota 5754  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-trkgc 25064  df-trkgb 25065  df-trkgcb 25066  df-trkg 25069
This theorem is referenced by:  opphllem2  25358  opphllem4  25360  opphllem5  25361  opphllem6  25362  lnperpex  25413
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