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Theorem btwnhl2 26972
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 2740 . . . 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 26847 . . 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 26961 . 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 1542  wcel 2110  wne 2945   class class class wbr 5079  cfv 6432  (class class class)co 7271  Basecbs 16910  distcds 16969  TarskiGcstrkg 26786  Itvcitv 26792  hlGchlg 26959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-trkgc 26807  df-trkgb 26808  df-trkgcb 26809  df-trkg 26812  df-hlg 26960
This theorem is referenced by:  hlpasch  27115
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