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Theorem tgbtwnouttr2 28014
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 768 . . 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 723 . . . . 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 723 . . . . 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 723 . . . . 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 723 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐡 ∈ 𝑃)
13 simplr 766 . . . . 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 723 . . . . 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 723 . . . . . 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 723 . . . . . 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 28013 . . . . 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 723 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐡𝐼𝐷))
23 simprr 770 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))
24 eqidd 2732 . . . . 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 28005 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ π‘₯ = 𝐷)
2625oveq2d 7428 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ (𝐴𝐼π‘₯) = (𝐴𝐼𝐷))
271, 26eleqtrd 2834 . 2 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷))) β†’ 𝐢 ∈ (𝐴𝐼𝐷))
282, 3, 4, 5, 16, 7, 7, 9axtgsegcon 27983 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (𝐢 ∈ (𝐴𝐼π‘₯) ∧ (𝐢 βˆ’ π‘₯) = (𝐢 βˆ’ 𝐷)))
2927, 28r19.29a 3161 1 (πœ‘ β†’ 𝐢 ∈ (𝐴𝐼𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  β€˜cfv 6543  (class class class)co 7412  Basecbs 17149  distcds 17211  TarskiGcstrkg 27946  Itvcitv 27952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-trkgc 27967  df-trkgb 27968  df-trkgcb 27969  df-trkg 27972
This theorem is referenced by:  tgbtwnexch2  28015  tgbtwnouttr  28016  tgbtwnconn22  28098  tglineeltr  28150  mirconn  28197  footexALT  28237  footexlem1  28238  footexlem2  28239
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