Theorem tgbtwncom 28467

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 28445	. . 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 2834	. 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 28466	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶))
17	1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16	axtgpasch 28446	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴)))
18	12, 17	r19.29a 3148	1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  distcds 17280  TarskiGcstrkg 28406  Itvcitv 28412

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408  df-trkgc 28427  df-trkgb 28428  df-trkgcb 28429  df-trkg 28432

This theorem is referenced by:  tgbtwncomb  28468  tgbtwntriv1  28470  tgbtwnexch3  28473  tgbtwnexch2  28475  tgbtwnouttr  28476  tgbtwnexch  28477  tgtrisegint  28478  tgifscgr  28487  tgcgrxfr  28497  tgbtwnconn1lem1  28551  tgbtwnconn1lem2  28552  tgbtwnconn1lem3  28553  tgbtwnconn1  28554  tgbtwnconn3  28556  tgbtwnconn22  28558  tgbtwnconnln1  28559  tgbtwnconnln2  28560  legtri3  28569  legtrid  28570  legbtwn  28573  tgcgrsub2  28574  hlln  28586  btwnhl2  28592  btwnhl  28593  hlcgrex  28595  hlcgreulem  28596  tglineeltr  28610  mirreu3  28633  mirmir  28641  mireq  28644  miriso  28649  mirconn  28657  mirbtwnhl  28659  mirhl2  28660  mircgrextend  28661  miduniq  28664  colmid  28667  krippenlem  28669  krippen  28670  midexlem  28671  ragflat  28683  ragcgr  28686  footexALT  28697  footexlem1  28698  footexlem2  28699  colperpexlem1  28709  colperpexlem3  28711  mideulem2  28713  opphllem  28714  midex  28716  oppcom  28723  opphllem5  28730  opphllem6  28731  outpasch  28734  hlpasch  28735  lnopp2hpgb  28742  colhp  28749  midbtwn  28758  hypcgrlem1  28778  hypcgrlem2  28779  flatcgra  28803  cgrabtwn  28805  cgracol  28807  dfcgra2  28809  sacgr  28810  oacgr  28811  inagswap  28820  inaghl  28824