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Theorem tgbtwnintr 26290
Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-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)
tgbtwnintr.5 (𝜑𝐴 ∈ (𝐵𝐼𝐷))
tgbtwnintr.6 (𝜑𝐵 ∈ (𝐶𝐼𝐷))
Assertion
Ref Expression
tgbtwnintr (𝜑𝐵 ∈ (𝐴𝐼𝐶))

Proof of Theorem tgbtwnintr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 725 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
76ad2antrr 725 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵𝑃)
8 simplr 768 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥𝑃)
9 simprr 772 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 26263 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
11 simprl 770 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶))
1210, 11eqeltrd 2893 . 2 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 ∈ (𝐴𝐼𝐶))
13 tgbtwnintr.3 . . 3 (𝜑𝐶𝑃)
14 tgbtwnintr.4 . . 3 (𝜑𝐷𝑃)
15 tgbtwnintr.1 . . 3 (𝜑𝐴𝑃)
16 tgbtwnintr.5 . . 3 (𝜑𝐴 ∈ (𝐵𝐼𝐷))
17 tgbtwnintr.6 . . 3 (𝜑𝐵 ∈ (𝐶𝐼𝐷))
181, 2, 3, 4, 6, 13, 14, 15, 6, 16, 17axtgpasch 26264 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
1912, 18r19.29a 3251 1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cfv 6328  (class class class)co 7139  Basecbs 16478  distcds 16569  TarskiGcstrkg 26227  Itvcitv 26233
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-trkgb 26246  df-trkg 26250
This theorem is referenced by:  tgbtwnexch3  26291  tgbtwnexch2  26293  tgbtwnconn1lem3  26371  tgbtwnconn3  26374  tgbtwnconn22  26376  tglineeltr  26428  mirconn  26475
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