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Theorem tgbtwnexch2 25815
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 479 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
2 tgbtwnexch2.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32adantr 474 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
41, 3eqeltrrd 2907 . 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 474 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
10 tgbtwnintr.1 . . . 4 (𝜑𝐴𝑃)
1110adantr 474 . . 3 ((𝜑𝐵𝐶) → 𝐴𝑃)
12 tgbtwnintr.2 . . . 4 (𝜑𝐵𝑃)
1312adantr 474 . . 3 ((𝜑𝐵𝐶) → 𝐵𝑃)
14 tgbtwnintr.3 . . . 4 (𝜑𝐶𝑃)
1514adantr 474 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
16 tgbtwnintr.4 . . . 4 (𝜑𝐷𝑃)
1716adantr 474 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
18 simpr 479 . . 3 ((𝜑𝐵𝐶) → 𝐵𝐶)
19 tgbtwnexch2.2 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2019adantr 474 . . . . 5 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
212adantr 474 . . . . 5 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
225, 6, 7, 9, 15, 13, 11, 17, 20, 21tgbtwnintr 25812 . . . 4 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
235, 6, 7, 9, 15, 13, 11, 22tgbtwncom 25807 . . 3 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
245, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20tgbtwnouttr2 25814 . 2 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
254, 24pm2.61dane 3086 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:  tgbtwnexch  25817  tgtrisegint  25818  tgbtwnconn1lem3  25893  legtri3  25909  miriso  25989
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