MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwncomb Structured version   Visualization version   GIF version

Theorem tgbtwncomb 25604
Description: Betweenness commutes, biconditional version. (Contributed by Thierry Arnoux, 3-Apr-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)
tgbtwncomb.3 (𝜑𝐶𝑃)
Assertion
Ref Expression
tgbtwncomb (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))

Proof of Theorem tgbtwncomb
StepHypRef Expression
1 tkgeom.p . . 3 𝑃 = (Base‘𝐺)
2 tkgeom.d . . 3 = (dist‘𝐺)
3 tkgeom.i . . 3 𝐼 = (Itv‘𝐺)
4 tkgeom.g . . . 4 (𝜑𝐺 ∈ TarskiG)
54adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG)
6 tgbtwntriv2.1 . . . 4 (𝜑𝐴𝑃)
76adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐴𝑃)
8 tgbtwntriv2.2 . . . 4 (𝜑𝐵𝑃)
98adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵𝑃)
10 tgbtwncomb.3 . . . 4 (𝜑𝐶𝑃)
1110adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐶𝑃)
12 simpr 479 . . 3 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐶))
131, 2, 3, 5, 7, 9, 11, 12tgbtwncom 25603 . 2 ((𝜑𝐵 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐴))
144adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐺 ∈ TarskiG)
1510adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐶𝑃)
168adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵𝑃)
176adantr 472 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐴𝑃)
18 simpr 479 . . 3 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐶𝐼𝐴))
191, 2, 3, 14, 15, 16, 17, 18tgbtwncom 25603 . 2 ((𝜑𝐵 ∈ (𝐶𝐼𝐴)) → 𝐵 ∈ (𝐴𝐼𝐶))
2013, 19impbida 913 1 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ↔ 𝐵 ∈ (𝐶𝐼𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  cfv 6049  (class class class)co 6814  Basecbs 16079  distcds 16172  TarskiGcstrkg 25549  Itvcitv 25555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6817  df-trkgc 25567  df-trkgb 25568  df-trkgcb 25569  df-trkg 25572
This theorem is referenced by:  colcom  25673  colrot1  25674  lnhl  25730  lncom  25737  lnrot1  25738  lnrot2  25739  mirreu3  25769
  Copyright terms: Public domain W3C validator