MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnintr Structured version   Visualization version   GIF version

Theorem tgbtwnintr 28501
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 726 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐺 ∈ TarskiG)
6 tgbtwnintr.2 . . . . 5 (𝜑𝐵𝑃)
76ad2antrr 726 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵𝑃)
8 simplr 769 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥𝑃)
9 simprr 773 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐵𝐼𝐵))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 28474 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝐵 = 𝑥)
11 simprl 771 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵))) → 𝑥 ∈ (𝐴𝐼𝐶))
1210, 11eqeltrd 2841 . 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 28475 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐴𝐼𝐶) ∧ 𝑥 ∈ (𝐵𝐼𝐵)))
1912, 18r19.29a 3162 1 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  Basecbs 17247  distcds 17306  TarskiGcstrkg 28435  Itvcitv 28441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-trkgb 28457  df-trkg 28461
This theorem is referenced by:  tgbtwnexch3  28502  tgbtwnexch2  28504  tgbtwnconn1lem3  28582  tgbtwnconn3  28585  tgbtwnconn22  28587  tglineeltr  28639  mirconn  28686
  Copyright terms: Public domain W3C validator