Theorem tgbtwncom 28437

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	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgbtwntriv2.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgbtwntriv2.2	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgbtwncom.3	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgbtwncom.4	⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))

Assertion

Ref	Expression
tgbtwncom	⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))

Proof of Theorem tgbtwncom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	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 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ 𝑃)
8		simplr 768	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ 𝑃)
9		simprl 770	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵))
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 28415	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥)
11		simprr 772	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴))
12	10, 11	eqeltrd 2828	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ (𝐶𝐼𝐴))
13		tgbtwntriv2.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
14		tgbtwncom.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
15		tgbtwncom.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶))
16	1, 2, 3, 4, 6, 14	tgbtwntriv2 28436	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶))
17	1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16	axtgpasch 28416	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴)))
18	12, 17	r19.29a 3137	1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  distcds 17170  TarskiGcstrkg 28376  Itvcitv 28382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-ov 7352  df-trkgc 28397  df-trkgb 28398  df-trkgcb 28399  df-trkg 28402

This theorem is referenced by:  tgbtwncomb  28438  tgbtwntriv1  28440  tgbtwnexch3  28443  tgbtwnexch2  28445  tgbtwnouttr  28446  tgbtwnexch  28447  tgtrisegint  28448  tgifscgr  28457  tgcgrxfr  28467  tgbtwnconn1lem1  28521  tgbtwnconn1lem2  28522  tgbtwnconn1lem3  28523  tgbtwnconn1  28524  tgbtwnconn3  28526  tgbtwnconn22  28528  tgbtwnconnln1  28529  tgbtwnconnln2  28530  legtri3  28539  legtrid  28540  legbtwn  28543  tgcgrsub2  28544  hlln  28556  btwnhl2  28562  btwnhl  28563  hlcgrex  28565  hlcgreulem  28566  tglineeltr  28580  mirreu3  28603  mirmir  28611  mireq  28614  miriso  28619  mirconn  28627  mirbtwnhl  28629  mirhl2  28630  mircgrextend  28631  miduniq  28634  colmid  28637  krippenlem  28639  krippen  28640  midexlem  28641  ragflat  28653  ragcgr  28656  footexALT  28667  footexlem1  28668  footexlem2  28669  colperpexlem1  28679  colperpexlem3  28681  mideulem2  28683  opphllem  28684  midex  28686  oppcom  28693  opphllem5  28700  opphllem6  28701  outpasch  28704  hlpasch  28705  lnopp2hpgb  28712  colhp  28719  midbtwn  28728  hypcgrlem1  28748  hypcgrlem2  28749  flatcgra  28773  cgrabtwn  28775  cgracol  28777  dfcgra2  28779  sacgr  28780  oacgr  28781  inagswap  28790  inaghl  28794