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Theorem tgbtwnouttr 26300
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (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)
tgbtwnouttr.1 (𝜑𝐵𝐶)
tgbtwnouttr.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnouttr.3 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Assertion
Ref Expression
tgbtwnouttr (𝜑𝐵 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnouttr
StepHypRef Expression
1 tkgeom.p . 2 𝑃 = (Base‘𝐺)
2 tkgeom.d . 2 = (dist‘𝐺)
3 tkgeom.i . 2 𝐼 = (Itv‘𝐺)
4 tkgeom.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tgbtwnintr.4 . 2 (𝜑𝐷𝑃)
6 tgbtwnintr.2 . 2 (𝜑𝐵𝑃)
7 tgbtwnintr.1 . 2 (𝜑𝐴𝑃)
8 tgbtwnintr.3 . . 3 (𝜑𝐶𝑃)
9 tgbtwnouttr.1 . . . 4 (𝜑𝐵𝐶)
109necomd 3069 . . 3 (𝜑𝐶𝐵)
11 tgbtwnouttr.3 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
121, 2, 3, 4, 6, 8, 5, 11tgbtwncom 26291 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐵))
13 tgbtwnouttr.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
141, 2, 3, 4, 7, 6, 8, 13tgbtwncom 26291 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
151, 2, 3, 4, 5, 8, 6, 7, 10, 12, 14tgbtwnouttr2 26298 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
161, 2, 3, 4, 5, 6, 7, 15tgbtwncom 26291 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wne 3014  cfv 6345  (class class class)co 7151  Basecbs 16485  distcds 16576  TarskiGcstrkg 26233  Itvcitv 26239
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5197
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-iota 6304  df-fv 6353  df-ov 7154  df-trkgc 26251  df-trkgb 26252  df-trkgcb 26253  df-trkg 26256
This theorem is referenced by:  btwnhl  26417  tglineeltr  26434  outpasch  26558
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