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Theorem tgbtwnexch 26276
Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.6 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)
tgbtwnexch.1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnexch.2 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch (𝜑𝐵 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch
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 tgbtwnexch.2 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
101, 2, 3, 4, 7, 8, 5, 9tgbtwncom 26266 . . 3 (𝜑𝐶 ∈ (𝐷𝐼𝐴))
11 tgbtwnexch.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
121, 2, 3, 4, 7, 6, 8, 11tgbtwncom 26266 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
131, 2, 3, 4, 5, 8, 6, 7, 10, 12tgbtwnexch2 26274 . 2 (𝜑𝐵 ∈ (𝐷𝐼𝐴))
141, 2, 3, 4, 5, 6, 7, 13tgbtwncom 26266 1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cfv 6348  (class class class)co 7148  Basecbs 16475  distcds 16566  TarskiGcstrkg 26208  Itvcitv 26214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-trkgc 26226  df-trkgb 26227  df-trkgcb 26228  df-trkg 26231
This theorem is referenced by:  tgcgrxfr  26296  tgbtwnconn1lem1  26350  tgbtwnconn1lem3  26352  legtrd  26367  hltr  26388  hlbtwn  26389  tglineeltr  26409  miriso  26448  outpasch  26533
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