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Theorem tgbtwnouttr2 25190
Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-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)
tgbtwnouttr2.1 (𝜑𝐵𝐶)
tgbtwnouttr2.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnouttr2.3 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Assertion
Ref Expression
tgbtwnouttr2 (𝜑𝐶 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnouttr2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 790 . . 3 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐴𝐼𝑥))
2 tkgeom.p . . . . 5 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . . 5 = (dist‘𝐺)
4 tkgeom.i . . . . 5 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . . . 6 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐺 ∈ TarskiG)
7 tgbtwnintr.3 . . . . . 6 (𝜑𝐶𝑃)
87ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶𝑃)
9 tgbtwnintr.4 . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐷𝑃)
11 tgbtwnintr.2 . . . . . 6 (𝜑𝐵𝑃)
1211ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝑃)
13 simplr 788 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥𝑃)
14 tgbtwnouttr2.1 . . . . . 6 (𝜑𝐵𝐶)
1514ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝐶)
16 tgbtwnintr.1 . . . . . . 7 (𝜑𝐴𝑃)
1716ad2antrr 758 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐴𝑃)
18 tgbtwnouttr2.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1918ad2antrr 758 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
202, 3, 4, 6, 17, 12, 8, 13, 19, 1tgbtwnexch3 25189 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21 tgbtwnouttr2.3 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2221ad2antrr 758 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23 simprr 792 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐶 𝑥) = (𝐶 𝐷))
24 eqidd 2611 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐶 𝐷) = (𝐶 𝐷))
252, 3, 4, 6, 8, 8, 10, 12, 13, 10, 15, 20, 22, 23, 24tgsegconeq 25181 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥 = 𝐷)
2625oveq2d 6565 . . 3 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
271, 26eleqtrd 2690 . 2 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 25163 . 2 (𝜑 → ∃𝑥𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷)))
2927, 28r19.29a 3060 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wne 2780  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:  tgbtwnexch2  25191  tgbtwnouttr  25192  tgbtwnconn22  25274  tglineeltr  25326  mirconn  25373  footex  25413
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