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Theorem tgbtwnintr 25288
 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 761 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
76ad2antrr 761 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵𝑃)
8 simplr 791 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥𝑃)
9 simprr 795 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 25265 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
11 simprl 793 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶))
1210, 11eqeltrd 2698 . 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 25266 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
1912, 18r19.29a 3071 1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ‘cfv 5847  (class class class)co 6604  Basecbs 15781  distcds 15871  TarskiGcstrkg 25229  Itvcitv 25235 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-trkgb 25248  df-trkg 25252 This theorem is referenced by:  tgbtwnexch3  25289  tgbtwnexch2  25291  tgbtwnconn1lem3  25369  tgbtwnconn3  25372  tgbtwnconn22  25374  tglineeltr  25426  mirconn  25473
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