Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tg5segofs Structured version   Visualization version   GIF version

Theorem tg5segofs 30879
Description: Rephrase axtg5seg 25409 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.)
Hypotheses
Ref Expression
tg5segofs.p 𝑃 = (Base‘𝐺)
tg5segofs.m = (dist‘𝐺)
tg5segofs.s 𝐼 = (Itv‘𝐺)
tg5segofs.g (𝜑𝐺 ∈ TarskiG)
tg5segofs.a (𝜑𝐴𝑃)
tg5segofs.b (𝜑𝐵𝑃)
tg5segofs.c (𝜑𝐶𝑃)
tg5segofs.d (𝜑𝐷𝑃)
tg5segofs.e (𝜑𝐸𝑃)
tg5segofs.f (𝜑𝐹𝑃)
tg5segofs.o 𝑂 = (AFS‘𝐺)
tg5segofs.h (𝜑𝐻𝑃)
tg5segofs.i (𝜑𝐼𝑃)
tg5segofs.1 (𝜑 → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝐸, 𝐹⟩, ⟨𝐻, 𝐼⟩⟩)
tg5segofs.2 (𝜑𝐴𝐵)
Assertion
Ref Expression
tg5segofs (𝜑 → (𝐶 𝐷) = (𝐻 𝐼))

Proof of Theorem tg5segofs
StepHypRef Expression
1 tg5segofs.p . 2 𝑃 = (Base‘𝐺)
2 tg5segofs.m . 2 = (dist‘𝐺)
3 tg5segofs.s . 2 𝐼 = (Itv‘𝐺)
4 tg5segofs.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tg5segofs.a . 2 (𝜑𝐴𝑃)
6 tg5segofs.b . 2 (𝜑𝐵𝑃)
7 tg5segofs.c . 2 (𝜑𝐶𝑃)
8 tg5segofs.e . 2 (𝜑𝐸𝑃)
9 tg5segofs.f . 2 (𝜑𝐹𝑃)
10 tg5segofs.h . 2 (𝜑𝐻𝑃)
11 tg5segofs.d . 2 (𝜑𝐷𝑃)
12 tg5segofs.i . 2 (𝜑𝐼𝑃)
13 tg5segofs.2 . 2 (𝜑𝐴𝐵)
14 tg5segofs.1 . . . . 5 (𝜑 → ⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝐸, 𝐹⟩, ⟨𝐻, 𝐼⟩⟩)
15 tg5segofs.o . . . . . 6 𝑂 = (AFS‘𝐺)
161, 2, 3, 4, 15, 5, 6, 7, 11, 8, 9, 10, 12brafs 30878 . . . . 5 (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑂⟨⟨𝐸, 𝐹⟩, ⟨𝐻, 𝐼⟩⟩ ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻)) ∧ ((𝐴 𝐵) = (𝐸 𝐹) ∧ (𝐵 𝐶) = (𝐹 𝐻)) ∧ ((𝐴 𝐷) = (𝐸 𝐼) ∧ (𝐵 𝐷) = (𝐹 𝐼)))))
1714, 16mpbid 222 . . . 4 (𝜑 → ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻)) ∧ ((𝐴 𝐵) = (𝐸 𝐹) ∧ (𝐵 𝐶) = (𝐹 𝐻)) ∧ ((𝐴 𝐷) = (𝐸 𝐼) ∧ (𝐵 𝐷) = (𝐹 𝐼))))
1817simp1d 1093 . . 3 (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻)))
1918simpld 474 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
2018simprd 478 . 2 (𝜑𝐹 ∈ (𝐸𝐼𝐻))
2117simp2d 1094 . . 3 (𝜑 → ((𝐴 𝐵) = (𝐸 𝐹) ∧ (𝐵 𝐶) = (𝐹 𝐻)))
2221simpld 474 . 2 (𝜑 → (𝐴 𝐵) = (𝐸 𝐹))
2321simprd 478 . 2 (𝜑 → (𝐵 𝐶) = (𝐹 𝐻))
2417simp3d 1095 . . 3 (𝜑 → ((𝐴 𝐷) = (𝐸 𝐼) ∧ (𝐵 𝐷) = (𝐹 𝐼)))
2524simpld 474 . 2 (𝜑 → (𝐴 𝐷) = (𝐸 𝐼))
2624simprd 478 . 2 (𝜑 → (𝐵 𝐷) = (𝐹 𝐼))
271, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 23, 25, 26axtg5seg 25409 1 (𝜑 → (𝐶 𝐷) = (𝐻 𝐼))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cop 4216   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  distcds 15997  TarskiGcstrkg 25374  Itvcitv 25380  AFScafs 30875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-trkgcb 25394  df-trkg 25397  df-afs 30876
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator