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Theorem btwnhl1 26409
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)
btwnhl1.3 (𝜑𝐶𝐴)
Assertion
Ref Expression
btwnhl1 (𝜑𝐶(𝐾𝐴)𝐵)

Proof of Theorem btwnhl1
StepHypRef Expression
1 btwnhl1.3 . 2 (𝜑𝐶𝐴)
2 btwnhl1.2 . . 3 (𝜑𝐴𝐵)
32necomd 3069 . 2 (𝜑𝐵𝐴)
4 btwnhl1.1 . . 3 (𝜑𝐶 ∈ (𝐴𝐼𝐵))
54orcd 870 . 2 (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))
6 ishlg.p . . 3 𝑃 = (Base‘𝐺)
7 ishlg.i . . 3 𝐼 = (Itv‘𝐺)
8 ishlg.k . . 3 𝐾 = (hlG‘𝐺)
9 ishlg.c . . 3 (𝜑𝐶𝑃)
10 ishlg.b . . 3 (𝜑𝐵𝑃)
11 ishlg.a . . 3 (𝜑𝐴𝑃)
12 hlln.1 . . 3 (𝜑𝐺 ∈ TarskiG)
136, 7, 8, 9, 10, 11, 12ishlg 26399 . 2 (𝜑 → (𝐶(𝐾𝐴)𝐵 ↔ (𝐶𝐴𝐵𝐴 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶)))))
141, 3, 5, 13mpbir3and 1339 1 (𝜑𝐶(𝐾𝐴)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  TarskiGcstrkg 26227  Itvcitv 26233  hlGchlg 26397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-hlg 26398
This theorem is referenced by:  outpasch  26552  hlpasch  26553  lnopp2hpgb  26560  dfcgra2  26627
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