MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  btwnhl2 Structured version   Visualization version   GIF version

Theorem btwnhl2 26974
Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.)
Hypotheses
Ref Expression
ishlg.p 𝑃 = (Base‘𝐺)
ishlg.i 𝐼 = (Itv‘𝐺)
ishlg.k 𝐾 = (hlG‘𝐺)
ishlg.a (𝜑𝐴𝑃)
ishlg.b (𝜑𝐵𝑃)
ishlg.c (𝜑𝐶𝑃)
hlln.1 (𝜑𝐺 ∈ TarskiG)
hltr.d (𝜑𝐷𝑃)
btwnhl1.1 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
btwnhl1.2 (𝜑𝐴𝐵)
btwnhl2.3 (𝜑𝐶𝐵)
Assertion
Ref Expression
btwnhl2 (𝜑𝐶(𝐾𝐵)𝐴)

Proof of Theorem btwnhl2
StepHypRef Expression
1 btwnhl2.3 . 2 (𝜑𝐶𝐵)
2 btwnhl1.2 . 2 (𝜑𝐴𝐵)
3 ishlg.p . . . 4 𝑃 = (Base‘𝐺)
4 eqid 2738 . . . 4 (dist‘𝐺) = (dist‘𝐺)
5 ishlg.i . . . 4 𝐼 = (Itv‘𝐺)
6 hlln.1 . . . 4 (𝜑𝐺 ∈ TarskiG)
7 ishlg.a . . . 4 (𝜑𝐴𝑃)
8 ishlg.c . . . 4 (𝜑𝐶𝑃)
9 ishlg.b . . . 4 (𝜑𝐵𝑃)
10 btwnhl1.1 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
113, 4, 5, 6, 7, 8, 9, 10tgbtwncom 26849 . . 3 (𝜑𝐶 ∈ (𝐵𝐼𝐴))
1211orcd 870 . 2 (𝜑 → (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))
13 ishlg.k . . 3 𝐾 = (hlG‘𝐺)
143, 5, 13, 8, 7, 9, 6ishlg 26963 . 2 (𝜑 → (𝐶(𝐾𝐵)𝐴 ↔ (𝐶𝐵𝐴𝐵 ∧ (𝐶 ∈ (𝐵𝐼𝐴) ∨ 𝐴 ∈ (𝐵𝐼𝐶)))))
151, 2, 12, 14mpbir3and 1341 1 (𝜑𝐶(𝐾𝐵)𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1539  wcel 2106  wne 2943   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  distcds 16971  TarskiGcstrkg 26788  Itvcitv 26794  hlGchlg 26961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-trkgc 26809  df-trkgb 26810  df-trkgcb 26811  df-trkg 26814  df-hlg 26962
This theorem is referenced by:  hlpasch  27117
  Copyright terms: Public domain W3C validator