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Theorem tgbtwncom 25604
Description: Betweenness commutes. Theorem 3.2 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
tgbtwncom.3 (𝜑𝐶𝑃)
tgbtwncom.4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
Assertion
Ref Expression
tgbtwncom (𝜑𝐵 ∈ (𝐶𝐼𝐴))

Proof of Theorem tgbtwncom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 tkgeom.p . . . 4 𝑃 = (Base‘𝐺)
2 tkgeom.d . . . 4 = (dist‘𝐺)
3 tkgeom.i . . . 4 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
54ad2antrr 705 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.2 . . . . 5 (𝜑𝐵𝑃)
76ad2antrr 705 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵𝑃)
8 simplr 752 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥𝑃)
9 simprl 754 . . . 4 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵))
101, 2, 3, 5, 7, 8, 9axtgbtwnid 25586 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥)
11 simprr 756 . . 3 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴))
1210, 11eqeltrd 2850 . 2 (((𝜑𝑥𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ (𝐶𝐼𝐴))
13 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
14 tgbtwncom.3 . . 3 (𝜑𝐶𝑃)
15 tgbtwncom.4 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
161, 2, 3, 4, 6, 14tgbtwntriv2 25603 . . 3 (𝜑𝐶 ∈ (𝐵𝐼𝐶))
171, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16axtgpasch 25587 . 2 (𝜑 → ∃𝑥𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴)))
1812, 17r19.29a 3226 1 (𝜑𝐵 ∈ (𝐶𝐼𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6793  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tgbtwncomb  25605  tgbtwntriv1  25607  tgbtwnexch3  25610  tgbtwnexch2  25612  tgbtwnouttr  25613  tgbtwnexch  25614  tgtrisegint  25615  tgifscgr  25624  tgcgrxfr  25634  tgbtwnconn1lem1  25688  tgbtwnconn1lem2  25689  tgbtwnconn1lem3  25690  tgbtwnconn1  25691  tgbtwnconn3  25693  tgbtwnconn22  25695  tgbtwnconnln1  25696  tgbtwnconnln2  25697  legtri3  25706  legtrid  25707  legbtwn  25710  tgcgrsub2  25711  hlln  25723  btwnhl2  25729  btwnhl  25730  hlcgrex  25732  hlcgreulem  25733  tglineeltr  25747  mirreu3  25770  mirmir  25778  mireq  25781  miriso  25786  mirconn  25794  mirbtwnhl  25796  mirhl2  25797  mircgrextend  25798  miduniq  25801  colmid  25804  krippenlem  25806  krippen  25807  midexlem  25808  ragflat  25820  ragcgr  25823  footex  25834  colperpexlem1  25843  colperpexlem3  25845  mideulem2  25847  opphllem  25848  midex  25850  oppcom  25857  opphllem5  25864  opphllem6  25865  outpasch  25868  hlpasch  25869  lnopp2hpgb  25876  colhp  25883  midbtwn  25892  hypcgrlem1  25912  hypcgrlem2  25913  cgrabtwn  25938  cgracol  25940  dfcgra2  25942  sacgr  25943  oacgr  25944  inagswap  25951  inaghl  25952
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