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Theorem tgbtwnouttr2 25814
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 787 . . 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 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐺 ∈ TarskiG)
7 tgbtwnintr.3 . . . . . 6 (𝜑𝐶𝑃)
87ad2antrr 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶𝑃)
9 tgbtwnintr.4 . . . . . 6 (𝜑𝐷𝑃)
109ad2antrr 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐷𝑃)
11 tgbtwnintr.2 . . . . . 6 (𝜑𝐵𝑃)
1211ad2antrr 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝑃)
13 simplr 785 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥𝑃)
14 tgbtwnouttr2.1 . . . . . 6 (𝜑𝐵𝐶)
1514ad2antrr 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵𝐶)
16 tgbtwnintr.1 . . . . . . 7 (𝜑𝐴𝑃)
1716ad2antrr 717 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐴𝑃)
18 tgbtwnouttr2.2 . . . . . . 7 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
1918ad2antrr 717 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐵 ∈ (𝐴𝐼𝐶))
202, 3, 4, 6, 17, 12, 8, 13, 19, 1tgbtwnexch3 25813 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝑥))
21 tgbtwnouttr2.3 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2221ad2antrr 717 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐵𝐼𝐷))
23 simprr 789 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐶 𝑥) = (𝐶 𝐷))
24 eqidd 2826 . . . . 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 25805 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝑥 = 𝐷)
2625oveq2d 6926 . . 3 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → (𝐴𝐼𝑥) = (𝐴𝐼𝐷))
271, 26eleqtrd 2908 . 2 (((𝜑𝑥𝑃) ∧ (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷))) → 𝐶 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 25783 . 2 (𝜑 → ∃𝑥𝑃 (𝐶 ∈ (𝐴𝐼𝑥) ∧ (𝐶 𝑥) = (𝐶 𝐷)))
2927, 28r19.29a 3288 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wne 2999  cfv 6127  (class class class)co 6910  Basecbs 16229  distcds 16321  TarskiGcstrkg 25749  Itvcitv 25755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-iota 6090  df-fv 6135  df-ov 6913  df-trkgc 25767  df-trkgb 25768  df-trkgcb 25769  df-trkg 25772
This theorem is referenced by:  tgbtwnexch2  25815  tgbtwnouttr  25816  tgbtwnconn22  25898  tglineeltr  25950  mirconn  25997  footex  26037
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