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Theorem tgbtwnexch2 26209
Description: Exchange the outer point of two betweenness statements. Right-hand side of Theorem 3.5 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)
tgbtwnexch2.1 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnexch2.2 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
Assertion
Ref Expression
tgbtwnexch2 (𝜑𝐶 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnexch2
StepHypRef Expression
1 simpr 485 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
2 tgbtwnexch2.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32adantr 481 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
41, 3eqeltrrd 2911 . 2 ((𝜑𝐵 = 𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
5 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
6 tkgeom.d . . 3 = (dist‘𝐺)
7 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
8 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
98adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
10 tgbtwnintr.1 . . . 4 (𝜑𝐴𝑃)
1110adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐴𝑃)
12 tgbtwnintr.2 . . . 4 (𝜑𝐵𝑃)
1312adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐵𝑃)
14 tgbtwnintr.3 . . . 4 (𝜑𝐶𝑃)
1514adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
16 tgbtwnintr.4 . . . 4 (𝜑𝐷𝑃)
1716adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
18 simpr 485 . . 3 ((𝜑𝐵𝐶) → 𝐵𝐶)
19 tgbtwnexch2.2 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2019adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
212adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
225, 6, 7, 9, 15, 13, 11, 17, 20, 21tgbtwnintr 26206 . . . 4 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
235, 6, 7, 9, 15, 13, 11, 22tgbtwncom 26201 . . 3 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
245, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20tgbtwnouttr2 26208 . 2 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
254, 24pm2.61dane 3101 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  cfv 6348  (class class class)co 7145  Basecbs 16471  distcds 16562  TarskiGcstrkg 26143  Itvcitv 26149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  tgbtwnexch  26211  tgtrisegint  26212  tgbtwnconn1lem3  26287  legtri3  26303  miriso  26383
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