Theorem tgbtwncom 28415

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 28393	. . 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 28414	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶))
17	1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16	axtgpasch 28394	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴)))
18	12, 17	r19.29a 3141	1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  distcds 17229  TarskiGcstrkg 28354  Itvcitv 28360

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 5261

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 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-trkgc 28375  df-trkgb 28376  df-trkgcb 28377  df-trkg 28380

This theorem is referenced by:  tgbtwncomb  28416  tgbtwntriv1  28418  tgbtwnexch3  28421  tgbtwnexch2  28423  tgbtwnouttr  28424  tgbtwnexch  28425  tgtrisegint  28426  tgifscgr  28435  tgcgrxfr  28445  tgbtwnconn1lem1  28499  tgbtwnconn1lem2  28500  tgbtwnconn1lem3  28501  tgbtwnconn1  28502  tgbtwnconn3  28504  tgbtwnconn22  28506  tgbtwnconnln1  28507  tgbtwnconnln2  28508  legtri3  28517  legtrid  28518  legbtwn  28521  tgcgrsub2  28522  hlln  28534  btwnhl2  28540  btwnhl  28541  hlcgrex  28543  hlcgreulem  28544  tglineeltr  28558  mirreu3  28581  mirmir  28589  mireq  28592  miriso  28597  mirconn  28605  mirbtwnhl  28607  mirhl2  28608  mircgrextend  28609  miduniq  28612  colmid  28615  krippenlem  28617  krippen  28618  midexlem  28619  ragflat  28631  ragcgr  28634  footexALT  28645  footexlem1  28646  footexlem2  28647  colperpexlem1  28657  colperpexlem3  28659  mideulem2  28661  opphllem  28662  midex  28664  oppcom  28671  opphllem5  28678  opphllem6  28679  outpasch  28682  hlpasch  28683  lnopp2hpgb  28690  colhp  28697  midbtwn  28706  hypcgrlem1  28726  hypcgrlem2  28727  flatcgra  28751  cgrabtwn  28753  cgracol  28755  dfcgra2  28757  sacgr  28758  oacgr  28759  inagswap  28768  inaghl  28772