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Theorem tgbtwnouttr2 26401
 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 770 . . 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 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐺 ∈ TarskiG)
7 tgbtwnintr.3 . . . . . 6 (𝜑𝐶𝑃)
87ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶𝑃)
9 tgbtwnintr.4 . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐷𝑃)
11 tgbtwnintr.2 . . . . . 6 (𝜑𝐵𝑃)
1211ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝑃)
13 simplr 768 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥𝑃)
14 tgbtwnouttr2.1 . . . . . 6 (𝜑𝐵𝐶)
1514ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝐶)
16 tgbtwnintr.1 . . . . . . 7 (𝜑𝐴𝑃)
1716ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐴𝑃)
18 tgbtwnouttr2.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1918ad2antrr 725 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
202, 3, 4, 6, 17, 12, 8, 13, 19, 1tgbtwnexch3 26400 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21 tgbtwnouttr2.3 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2221ad2antrr 725 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23 simprr 772 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐶 𝑥) = (𝐶 𝐷))
24 eqidd 2759 . . . . 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 26392 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥 = 𝐷)
2625oveq2d 7172 . . 3 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
271, 26eleqtrd 2854 . 2 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 26370 . 2 (𝜑 → ∃𝑥𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷)))
2927, 28r19.29a 3213 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  distcds 16645  TarskiGcstrkg 26336  Itvcitv 26342 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-trkgc 26354  df-trkgb 26355  df-trkgcb 26356  df-trkg 26359 This theorem is referenced by:  tgbtwnexch2  26402  tgbtwnouttr  26403  tgbtwnconn22  26485  tglineeltr  26537  mirconn  26584  footexALT  26624  footexlem1  26625  footexlem2  26626
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