MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnouttr2 Structured version   Visualization version   GIF version

Theorem tgbtwnouttr2 28013
Description: Outer transitivity law for betweenness. Left-hand side of Theorem 3.7 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 18-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgbtwnintr.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgbtwnintr.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
tgbtwnintr.3 (πœ‘ β†’ 𝐢 ∈ 𝑃)
tgbtwnintr.4 (πœ‘ β†’ 𝐷 ∈ 𝑃)
tgbtwnouttr2.1 (πœ‘ β†’ 𝐡 β‰  𝐢)
tgbtwnouttr2.2 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
tgbtwnouttr2.3 (πœ‘ β†’ 𝐢 ∈ (𝐡𝐼𝐷))
Assertion
Ref Expression
tgbtwnouttr2 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐷))

Proof of Theorem tgbtwnouttr2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simprl 767 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐴𝐼π‘₯))
2 tkgeom.p . . . . 5 𝑃 = (Baseβ€˜πΊ)
3 tkgeom.d . . . . 5 βˆ’ = (distβ€˜πΊ)
4 tkgeom.i . . . . 5 𝐼 = (Itvβ€˜πΊ)
5 tkgeom.g . . . . . 6 (πœ‘ β†’ 𝐺 ∈ TarskiG)
65ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐺 ∈ TarskiG)
7 tgbtwnintr.3 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ 𝑃)
87ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ 𝑃)
9 tgbtwnintr.4 . . . . . 6 (πœ‘ β†’ 𝐷 ∈ 𝑃)
109ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐷 ∈ 𝑃)
11 tgbtwnintr.2 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ 𝑃)
1211ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐡 ∈ 𝑃)
13 simplr 765 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ π‘₯ ∈ 𝑃)
14 tgbtwnouttr2.1 . . . . . 6 (πœ‘ β†’ 𝐡 β‰  𝐢)
1514ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐡 β‰  𝐢)
16 tgbtwnintr.1 . . . . . . 7 (πœ‘ β†’ 𝐴 ∈ 𝑃)
1716ad2antrr 722 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐴 ∈ 𝑃)
18 tgbtwnouttr2.2 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐢))
1918ad2antrr 722 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐡 ∈ (𝐴𝐼𝐢))
202, 3, 4, 6, 17, 12, 8, 13, 19, 1tgbtwnexch3 28012 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐡𝐼π‘₯))
21 tgbtwnouttr2.3 . . . . . 6 (πœ‘ β†’ 𝐢 ∈ (𝐡𝐼𝐷))
2221ad2antrr 722 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐡𝐼𝐷))
23 simprr 769 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))
24 eqidd 2731 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ (𝐢 βˆ’ 𝐷) = (𝐢 βˆ’ 𝐷))
252, 3, 4, 6, 8, 8, 10, 12, 13, 10, 15, 20, 22, 23, 24tgsegconeq 28004 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ π‘₯ = 𝐷)
2625oveq2d 7427 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ (𝐴𝐼π‘₯) = (𝐴𝐼𝐷))
271, 26eleqtrd 2833 . 2 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 27982 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷)))
2927, 28r19.29a 3160 1 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGcstrkg 27945  Itvcitv 27951
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6494  df-fv 6550  df-ov 7414  df-trkgc 27966  df-trkgb 27967  df-trkgcb 27968  df-trkg 27971
This theorem is referenced by:  tgbtwnexch2  28014  tgbtwnouttr  28015  tgbtwnconn22  28097  tglineeltr  28149  mirconn  28196  footexALT  28236  footexlem1  28237  footexlem2  28238
  Copyright terms: Public domain W3C validator