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Theorem tgbtwnconn1lem1 25687
Description: Lemma for tgbtwnconn1 25690. (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 25613 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐸))
15 tgbtwnconn1.6 . . 3 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
161, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch 25613 . 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 25613 . . 3 (𝜑𝐵 ∈ (𝐴𝐼𝐹))
22 tgbtwnconn1.7 . . 3 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
231, 2, 3, 4, 7, 5, 17, 6, 21, 22tgbtwnexch 25613 . 2 (𝜑𝐵 ∈ (𝐴𝐼𝐽))
241, 2, 3, 4, 7, 5, 10, 8, 14, 15tgbtwnexch3 25609 . . 3 (𝜑𝐸 ∈ (𝐵𝐼𝐻))
251, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch 25613 . . . . 5 (𝜑𝐶 ∈ (𝐴𝐼𝐽))
261, 2, 3, 4, 7, 5, 18, 6, 19, 25tgbtwnexch3 25609 . . . 4 (𝜑𝐶 ∈ (𝐵𝐼𝐽))
271, 2, 3, 4, 5, 18, 6, 26tgbtwncom 25603 . . 3 (𝜑𝐶 ∈ (𝐽𝐼𝐵))
281, 2, 3, 4, 7, 5, 11, 10, 12, 13tgbtwnexch3 25609 . . . 4 (𝜑𝐷 ∈ (𝐵𝐼𝐸))
291, 2, 3, 4, 7, 18, 17, 6, 20, 22tgbtwnexch3 25609 . . . . 5 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
301, 2, 3, 4, 18, 17, 6, 29tgbtwncom 25603 . . . 4 (𝜑𝐹 ∈ (𝐽𝐼𝐶))
311, 2, 3, 4, 6, 17axtgcgrrflx 25581 . . . . 5 (𝜑 → (𝐽 𝐹) = (𝐹 𝐽))
32 tgbtwnconn1.11 . . . . 5 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
3331, 32eqtr2d 2795 . . . 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 2797 . . . . 5 (𝜑 → (𝐸 𝐷) = (𝐶 𝐹))
371, 2, 3, 4, 10, 11, 18, 17, 36tgcgrcomlr 25595 . . . 4 (𝜑 → (𝐷 𝐸) = (𝐹 𝐶))
381, 2, 3, 4, 5, 11, 10, 6, 17, 18, 28, 30, 33, 37tgcgrextend 25600 . . 3 (𝜑 → (𝐵 𝐸) = (𝐽 𝐶))
39 tgbtwnconn1.10 . . . 4 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
401, 2, 3, 4, 10, 8, 5, 18, 39tgcgrcomr 25593 . . 3 (𝜑 → (𝐸 𝐻) = (𝐶 𝐵))
411, 2, 3, 4, 5, 10, 8, 6, 18, 5, 24, 27, 38, 40tgcgrextend 25600 . 2 (𝜑 → (𝐵 𝐻) = (𝐽 𝐵))
421, 2, 3, 4, 5, 6axtgcgrrflx 25581 . 2 (𝜑 → (𝐵 𝐽) = (𝐽 𝐵))
431, 2, 3, 4, 5, 6, 5, 7, 8, 6, 9, 16, 23, 41, 42tgsegconeq 25601 1 (𝜑𝐻 = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  wne 2932  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-ne 2933  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:  tgbtwnconn1lem2  25688  tgbtwnconn1lem3  25689
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