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Theorem tgbtwnexch2 25611
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 471 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
2 tgbtwnexch2.1 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
32adantr 466 . . 3 ((𝜑𝐵 = 𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
41, 3eqeltrrd 2850 . 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 466 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
10 tgbtwnintr.1 . . . 4 (𝜑𝐴𝑃)
1110adantr 466 . . 3 ((𝜑𝐵𝐶) → 𝐴𝑃)
12 tgbtwnintr.2 . . . 4 (𝜑𝐵𝑃)
1312adantr 466 . . 3 ((𝜑𝐵𝐶) → 𝐵𝑃)
14 tgbtwnintr.3 . . . 4 (𝜑𝐶𝑃)
1514adantr 466 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
16 tgbtwnintr.4 . . . 4 (𝜑𝐷𝑃)
1716adantr 466 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
18 simpr 471 . . 3 ((𝜑𝐵𝐶) → 𝐵𝐶)
19 tgbtwnexch2.2 . . . . . 6 (𝜑𝐶 ∈ (𝐵𝐼𝐷))
2019adantr 466 . . . . 5 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐷))
212adantr 466 . . . . 5 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐷))
225, 6, 7, 9, 15, 13, 11, 17, 20, 21tgbtwnintr 25608 . . . 4 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐶𝐼𝐴))
235, 6, 7, 9, 15, 13, 11, 22tgbtwncom 25603 . . 3 ((𝜑𝐵𝐶) → 𝐵 ∈ (𝐴𝐼𝐶))
245, 6, 7, 9, 11, 13, 15, 17, 18, 23, 20tgbtwnouttr2 25610 . 2 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐴𝐼𝐷))
254, 24pm2.61dane 3029 1 (𝜑𝐶 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wne 2942  cfv 6031  (class class class)co 6792  Basecbs 16063  distcds 16157  TarskiGcstrkg 25549  Itvcitv 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-trkgc 25567  df-trkgb 25568  df-trkgcb 25569  df-trkg 25572
This theorem is referenced by:  tgbtwnexch  25613  tgtrisegint  25614  tgbtwnconn1lem3  25689  legtri3  25705  miriso  25785
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