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

Theorem tgtrisegint 25194
Description: A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwnintr.1 (𝜑𝐴𝑃)
tgbtwnintr.2 (𝜑𝐵𝑃)
tgbtwnintr.3 (𝜑𝐶𝑃)
tgbtwnintr.4 (𝜑𝐷𝑃)
tgtrisegint.e (𝜑𝐸𝑃)
tgtrisegint.p (𝜑𝐹𝑃)
tgtrisegint.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgtrisegint.2 (𝜑𝐸 ∈ (𝐷𝐼𝐶))
tgtrisegint.3 (𝜑𝐹 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgtrisegint (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
Distinct variable groups:   ,𝑞   𝐴,𝑞   𝐵,𝑞   𝐶,𝑞   𝐷,𝑞   𝐸,𝑞   𝐹,𝑞   𝐼,𝑞   𝑃,𝑞   𝜑,𝑞
Allowed substitution hint:   𝐺(𝑞)

Proof of Theorem tgtrisegint
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 758 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐺 ∈ TarskiG)
6 tgtrisegint.e . . . . 5 (𝜑𝐸𝑃)
76ad2antrr 758 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐸𝑃)
8 tgbtwnintr.3 . . . . 5 (𝜑𝐶𝑃)
98ad2antrr 758 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐶𝑃)
10 tgbtwnintr.1 . . . . 5 (𝜑𝐴𝑃)
1110ad2antrr 758 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐴𝑃)
12 simplr 788 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟𝑃)
13 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
1413ad2antrr 758 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵𝑃)
15 simprl 790 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐸𝐼𝐴))
16 tgtrisegint.1 . . . . . 6 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1716ad2antrr 758 . . . . 5 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐴𝐼𝐶))
181, 2, 3, 5, 11, 14, 9, 17tgbtwncom 25183 . . . 4 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 5, 7, 9, 11, 12, 14, 15, 18axtgpasch 25166 . . 3 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
205ad2antrr 758 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐺 ∈ TarskiG)
21 tgtrisegint.p . . . . . . . . 9 (𝜑𝐹𝑃)
2221ad2antrr 758 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝐹𝑃)
2322ad2antrr 758 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐹𝑃)
2412ad2antrr 758 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟𝑃)
25 simplr 788 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞𝑃)
269ad2antrr 758 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝐶𝑃)
27 simprr 792 . . . . . . . 8 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → 𝑟 ∈ (𝐹𝐼𝐶))
2827ad2antrr 758 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑟 ∈ (𝐹𝐼𝐶))
29 simpr 476 . . . . . . 7 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝑟𝐼𝐶))
301, 2, 3, 20, 23, 24, 25, 26, 28, 29tgbtwnexch2 25191 . . . . . 6 (((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) ∧ 𝑞 ∈ (𝑟𝐼𝐶)) → 𝑞 ∈ (𝐹𝐼𝐶))
3130ex 449 . . . . 5 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → (𝑞 ∈ (𝑟𝐼𝐶) → 𝑞 ∈ (𝐹𝐼𝐶)))
3231anim1d 586 . . . 4 ((((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) ∧ 𝑞𝑃) → ((𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
3332reximdva 3000 . . 3 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → (∃𝑞𝑃 (𝑞 ∈ (𝑟𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)) → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸))))
3419, 33mpd 15 . 2 (((𝜑𝑟𝑃) ∧ (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶))) → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
35 tgbtwnintr.4 . . 3 (𝜑𝐷𝑃)
36 tgtrisegint.2 . . . 4 (𝜑𝐸 ∈ (𝐷𝐼𝐶))
371, 2, 3, 4, 35, 6, 8, 36tgbtwncom 25183 . . 3 (𝜑𝐸 ∈ (𝐶𝐼𝐷))
38 tgtrisegint.3 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐷))
391, 2, 3, 4, 8, 10, 35, 6, 21, 37, 38axtgpasch 25166 . 2 (𝜑 → ∃𝑟𝑃 (𝑟 ∈ (𝐸𝐼𝐴) ∧ 𝑟 ∈ (𝐹𝐼𝐶)))
4034, 39r19.29a 3060 1 (𝜑 → ∃𝑞𝑃 (𝑞 ∈ (𝐹𝐼𝐶) ∧ 𝑞 ∈ (𝐵𝐼𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wrex 2897  cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135
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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  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-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152
This theorem is referenced by:  krippenlem  25385  colperpexlem3  25424
  Copyright terms: Public domain W3C validator