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Theorem tgbtwnouttr2 28518
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 771 . . 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 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐺 ∈ TarskiG)
7 tgbtwnintr.3 . . . . . 6 (𝜑𝐶𝑃)
87ad2antrr 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶𝑃)
9 tgbtwnintr.4 . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐷𝑃)
11 tgbtwnintr.2 . . . . . 6 (𝜑𝐵𝑃)
1211ad2antrr 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝑃)
13 simplr 769 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥𝑃)
14 tgbtwnouttr2.1 . . . . . 6 (𝜑𝐵𝐶)
1514ad2antrr 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝐶)
16 tgbtwnintr.1 . . . . . . 7 (𝜑𝐴𝑃)
1716ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐴𝑃)
18 tgbtwnouttr2.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1918ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
202, 3, 4, 6, 17, 12, 8, 13, 19, 1tgbtwnexch3 28517 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21 tgbtwnouttr2.3 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2221ad2antrr 726 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23 simprr 773 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐶 𝑥) = (𝐶 𝐷))
24 eqidd 2736 . . . . 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 28509 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥 = 𝐷)
2625oveq2d 7447 . . 3 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
271, 26eleqtrd 2841 . 2 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 28487 . 2 (𝜑 → ∃𝑥𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷)))
2927, 28r19.29a 3160 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  distcds 17307  TarskiGcstrkg 28450  Itvcitv 28456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-trkgc 28471  df-trkgb 28472  df-trkgcb 28473  df-trkg 28476
This theorem is referenced by:  tgbtwnexch2  28519  tgbtwnouttr  28520  tgbtwnconn22  28602  tglineeltr  28654  mirconn  28701  footexALT  28741  footexlem1  28742  footexlem2  28743
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