MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwnconn1lem1 Structured version   Visualization version   GIF version

Theorem tgbtwnconn1lem1 28247
Description: Lemma for tgbtwnconn1 28250. (Contributed by Thierry Arnoux, 30-Apr-2019.)
Hypotheses
Ref Expression
tgbtwnconn1.p 𝑃 = (Base‘𝐺)
tgbtwnconn1.i 𝐼 = (Itv‘𝐺)
tgbtwnconn1.g (𝜑𝐺 ∈ TarskiG)
tgbtwnconn1.a (𝜑𝐴𝑃)
tgbtwnconn1.b (𝜑𝐵𝑃)
tgbtwnconn1.c (𝜑𝐶𝑃)
tgbtwnconn1.d (𝜑𝐷𝑃)
tgbtwnconn1.1 (𝜑𝐴𝐵)
tgbtwnconn1.2 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
tgbtwnconn1.3 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
tgbtwnconn1.m = (dist‘𝐺)
tgbtwnconn1.e (𝜑𝐸𝑃)
tgbtwnconn1.f (𝜑𝐹𝑃)
tgbtwnconn1.h (𝜑𝐻𝑃)
tgbtwnconn1.j (𝜑𝐽𝑃)
tgbtwnconn1.4 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
tgbtwnconn1.5 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
tgbtwnconn1.6 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
tgbtwnconn1.7 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
tgbtwnconn1.8 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
tgbtwnconn1.9 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
tgbtwnconn1.10 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
tgbtwnconn1.11 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
Assertion
Ref Expression
tgbtwnconn1lem1 (𝜑𝐻 = 𝐽)

Proof of Theorem tgbtwnconn1lem1
StepHypRef Expression
1 tgbtwnconn1.p . 2 𝑃 = (Base‘𝐺)
2 tgbtwnconn1.m . 2 = (dist‘𝐺)
3 tgbtwnconn1.i . 2 𝐼 = (Itv‘𝐺)
4 tgbtwnconn1.g . 2 (𝜑𝐺 ∈ TarskiG)
5 tgbtwnconn1.b . 2 (𝜑𝐵𝑃)
6 tgbtwnconn1.j . 2 (𝜑𝐽𝑃)
7 tgbtwnconn1.a . 2 (𝜑𝐴𝑃)
8 tgbtwnconn1.h . 2 (𝜑𝐻𝑃)
9 tgbtwnconn1.1 . 2 (𝜑𝐴𝐵)
10 tgbtwnconn1.e . . 3 (𝜑𝐸𝑃)
11 tgbtwnconn1.d . . . 4 (𝜑𝐷𝑃)
12 tgbtwnconn1.3 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
13 tgbtwnconn1.4 . . . 4 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
141, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch 28173 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐸))
15 tgbtwnconn1.6 . . 3 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
161, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch 28173 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐻))
17 tgbtwnconn1.f . . 3 (𝜑𝐹𝑃)
18 tgbtwnconn1.c . . . 4 (𝜑𝐶𝑃)
19 tgbtwnconn1.2 . . . 4 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
20 tgbtwnconn1.5 . . . 4 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
211, 2, 3, 4, 7, 5, 18, 17, 19, 20tgbtwnexch 28173 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐹))
22 tgbtwnconn1.7 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
231, 2, 3, 4, 7, 5, 17, 6, 21, 22tgbtwnexch 28173 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐽))
241, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch3 28169 . . 3 (𝜑𝐸 ∈ (𝐵𝐼𝐻))
251, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch 28173 . . . . 5 (𝜑𝐶 ∈ (𝐴𝐼𝐽))
261, 2, 3, 4, 7, 5, 18, 6, 19, 25tgbtwnexch3 28169 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐽))
271, 2, 3, 4, 5, 18, 6, 26tgbtwncom 28163 . . 3 (𝜑𝐶 ∈ (𝐽𝐼𝐵))
281, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch3 28169 . . . 4 (𝜑𝐷 ∈ (𝐵𝐼𝐸))
291, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch3 28169 . . . . 5 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
301, 2, 3, 4, 18, 17, 6, 29tgbtwncom 28163 . . . 4 (𝜑𝐹 ∈ (𝐽𝐼𝐶))
311, 2, 3, 4, 6, 17axtgcgrrflx 28137 . . . . 5 (𝜑 → (𝐽 𝐹) = (𝐹 𝐽))
32 tgbtwnconn1.11 . . . . 5 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
3331, 32eqtr2d 2765 . . . 4 (𝜑 → (𝐵 𝐷) = (𝐽 𝐹))
34 tgbtwnconn1.8 . . . . . 6 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
35 tgbtwnconn1.9 . . . . . 6 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
3634, 35eqtr4d 2767 . . . . 5 (𝜑 → (𝐸 𝐷) = (𝐶 𝐹))
371, 2, 3, 4, 10, 11, 18, 17, 36tgcgrcomlr 28155 . . . 4 (𝜑 → (𝐷 𝐸) = (𝐹 𝐶))
381, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37tgcgrextend 28160 . . 3 (𝜑 → (𝐵 𝐸) = (𝐽 𝐶))
39 tgbtwnconn1.10 . . . 4 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
401, 2, 3, 4, 10, 8, 5, 18, 39tgcgrcomr 28153 . . 3 (𝜑 → (𝐸 𝐻) = (𝐶 𝐵))
411, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 40tgcgrextend 28160 . 2 (𝜑 → (𝐵 𝐻) = (𝐽 𝐵))
421, 2, 3, 4, 5, 6axtgcgrrflx 28137 . 2 (𝜑 → (𝐵 𝐽) = (𝐽 𝐵))
431, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 41, 42tgsegconeq 28161 1 (𝜑𝐻 = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wne 2932  cfv 6533  (class class class)co 7401  Basecbs 17140  distcds 17202  TarskiGcstrkg 28102  Itvcitv 28108
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5296
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-trkgc 28123  df-trkgb 28124  df-trkgcb 28125  df-trkg 28128
This theorem is referenced by:  tgbtwnconn1lem2  28248  tgbtwnconn1lem3  28249
  Copyright terms: Public domain W3C validator