Theorem tgbtwncom 25183

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 758	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐺 ∈ TarskiG)
6		tgbtwntriv2.2	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7	6	ad2antrr 758	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 ∈ 𝑃)
8		simplr 788	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ 𝑃)
9		simprl 790	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐵𝐼𝐵))
10	1, 2, 3, 5, 7, 8, 9	axtgbtwnid 25165	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝐵 = 𝑥)
11		simprr 792	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴))) → 𝑥 ∈ (𝐶𝐼𝐴))
12	10, 11	eqeltrd 2688	. 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 25182	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐵𝐼𝐶))
17	1, 2, 3, 4, 13, 6, 14, 6, 14, 15, 16	axtgpasch 25166	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥 ∈ (𝐵𝐼𝐵) ∧ 𝑥 ∈ (𝐶𝐼𝐴)))
18	12, 17	r19.29a 3060	1 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐴))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkgb 25148  df-trkgcb 25149  df-trkg 25152

This theorem is referenced by:  tgbtwncomb  25184  tgbtwntriv1  25186  tgbtwnexch3  25189  tgbtwnexch2  25191  tgbtwnouttr  25192  tgbtwnexch  25193  tgtrisegint  25194  tgifscgr  25203  tgcgrxfr  25213  tgbtwnconn1lem1  25267  tgbtwnconn1lem2  25268  tgbtwnconn1lem3  25269  tgbtwnconn1  25270  tgbtwnconn3  25272  tgbtwnconn22  25274  tgbtwnconnln1  25275  tgbtwnconnln2  25276  legtri3  25285  legtrid  25286  legbtwn  25289  tgcgrsub2  25290  hlln  25302  btwnhl2  25308  btwnhl  25309  hlcgrex  25311  hlcgreulem  25312  tglineeltr  25326  mirreu3  25349  mirmir  25357  mireq  25360  miriso  25365  mirconn  25373  mirbtwnhl  25375  mirhl2  25376  mircgrextend  25377  miduniq  25380  colmid  25383  krippenlem  25385  krippen  25386  midexlem  25387  ragflat  25399  ragcgr  25402  footex  25413  colperpexlem1  25422  colperpexlem3  25424  mideulem2  25426  opphllem  25427  midex  25429  oppcom  25436  opphllem5  25443  opphllem6  25444  outpasch  25447  hlpasch  25448  lnopp2hpgb  25455  colhp  25462  midbtwn  25471  hypcgrlem1  25491  hypcgrlem2  25492  cgrabtwn  25517  cgracol  25519  dfcgra2  25521  sacgr  25522  oacgr  25523  inagswap  25530  inaghl  25531