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Theorem tgbtwnconn1lem2 26275
Description: Lemma for tgbtwnconn1 26277. (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
tgbtwnconn1lem2 (𝜑 → (𝐸 𝐹) = (𝐶 𝐷))

Proof of Theorem tgbtwnconn1lem2
StepHypRef Expression
1 tgbtwnconn1.p . . . . 5 𝑃 = (Base‘𝐺)
2 tgbtwnconn1.m . . . . 5 = (dist‘𝐺)
3 tgbtwnconn1.i . . . . 5 𝐼 = (Itv‘𝐺)
4 tgbtwnconn1.g . . . . 5 (𝜑𝐺 ∈ TarskiG)
5 tgbtwnconn1.e . . . . 5 (𝜑𝐸𝑃)
6 tgbtwnconn1.f . . . . 5 (𝜑𝐹𝑃)
71, 2, 3, 4, 5, 6axtgcgrrflx 26164 . . . 4 (𝜑 → (𝐸 𝐹) = (𝐹 𝐸))
87adantr 481 . . 3 ((𝜑𝐵 = 𝐶) → (𝐸 𝐹) = (𝐹 𝐸))
94adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐶) → 𝐺 ∈ TarskiG)
105adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐶) → 𝐸𝑃)
11 tgbtwnconn1.h . . . . . . . 8 (𝜑𝐻𝑃)
1211adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐶) → 𝐻𝑃)
13 tgbtwnconn1.c . . . . . . . 8 (𝜑𝐶𝑃)
1413adantr 481 . . . . . . 7 ((𝜑𝐵 = 𝐶) → 𝐶𝑃)
15 tgbtwnconn1.10 . . . . . . . . 9 (𝜑 → (𝐸 𝐻) = (𝐵 𝐶))
1615adantr 481 . . . . . . . 8 ((𝜑𝐵 = 𝐶) → (𝐸 𝐻) = (𝐵 𝐶))
17 simpr 485 . . . . . . . . 9 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
1817oveq1d 7166 . . . . . . . 8 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐶 𝐶))
1916, 18eqtrd 2860 . . . . . . 7 ((𝜑𝐵 = 𝐶) → (𝐸 𝐻) = (𝐶 𝐶))
201, 2, 3, 9, 10, 12, 14, 19axtgcgrid 26165 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐻)
21 tgbtwnconn1.a . . . . . . . 8 (𝜑𝐴𝑃)
22 tgbtwnconn1.b . . . . . . . 8 (𝜑𝐵𝑃)
23 tgbtwnconn1.d . . . . . . . 8 (𝜑𝐷𝑃)
24 tgbtwnconn1.1 . . . . . . . 8 (𝜑𝐴𝐵)
25 tgbtwnconn1.2 . . . . . . . 8 (𝜑𝐵 ∈ (𝐴𝐼𝐶))
26 tgbtwnconn1.3 . . . . . . . 8 (𝜑𝐵 ∈ (𝐴𝐼𝐷))
27 tgbtwnconn1.j . . . . . . . 8 (𝜑𝐽𝑃)
28 tgbtwnconn1.4 . . . . . . . 8 (𝜑𝐷 ∈ (𝐴𝐼𝐸))
29 tgbtwnconn1.5 . . . . . . . 8 (𝜑𝐶 ∈ (𝐴𝐼𝐹))
30 tgbtwnconn1.6 . . . . . . . 8 (𝜑𝐸 ∈ (𝐴𝐼𝐻))
31 tgbtwnconn1.7 . . . . . . . 8 (𝜑𝐹 ∈ (𝐴𝐼𝐽))
32 tgbtwnconn1.8 . . . . . . . 8 (𝜑 → (𝐸 𝐷) = (𝐶 𝐷))
33 tgbtwnconn1.9 . . . . . . . 8 (𝜑 → (𝐶 𝐹) = (𝐶 𝐷))
34 tgbtwnconn1.11 . . . . . . . 8 (𝜑 → (𝐹 𝐽) = (𝐵 𝐷))
351, 3, 4, 21, 22, 13, 23, 24, 25, 26, 2, 5, 6, 11, 27, 28, 29, 30, 31, 32, 33, 15, 34tgbtwnconn1lem1 26274 . . . . . . 7 (𝜑𝐻 = 𝐽)
3635adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐻 = 𝐽)
3720, 36eqtrd 2860 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐽)
3837oveq2d 7167 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐹 𝐸) = (𝐹 𝐽))
3934adantr 481 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐹 𝐽) = (𝐵 𝐷))
4017oveq1d 7166 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐵 𝐷) = (𝐶 𝐷))
4138, 39, 403eqtrd 2864 . . 3 ((𝜑𝐵 = 𝐶) → (𝐹 𝐸) = (𝐶 𝐷))
428, 41eqtrd 2860 . 2 ((𝜑𝐵 = 𝐶) → (𝐸 𝐹) = (𝐶 𝐷))
434adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐺 ∈ TarskiG)
446adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐹𝑃)
455adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐸𝑃)
4623adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐷𝑃)
4713adantr 481 . . 3 ((𝜑𝐵𝐶) → 𝐶𝑃)
4822adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐵𝑃)
4927adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐽𝑃)
50 simpr 485 . . . 4 ((𝜑𝐵𝐶) → 𝐵𝐶)
511, 2, 3, 4, 21, 22, 13, 6, 25, 29tgbtwnexch3 26196 . . . . 5 (𝜑𝐶 ∈ (𝐵𝐼𝐹))
5251adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐹))
5335oveq2d 7167 . . . . . . . 8 (𝜑 → (𝐴𝐼𝐻) = (𝐴𝐼𝐽))
5430, 53eleqtrd 2919 . . . . . . 7 (𝜑𝐸 ∈ (𝐴𝐼𝐽))
551, 2, 3, 4, 21, 23, 5, 27, 28, 54tgbtwnexch3 26196 . . . . . 6 (𝜑𝐸 ∈ (𝐷𝐼𝐽))
561, 2, 3, 4, 23, 5, 27, 55tgbtwncom 26190 . . . . 5 (𝜑𝐸 ∈ (𝐽𝐼𝐷))
5756adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐽𝐼𝐷))
5835adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐻 = 𝐽)
5958oveq2d 7167 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐻) = (𝐸 𝐽))
6015adantr 481 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐻) = (𝐵 𝐶))
611, 2, 3, 43, 45, 49axtgcgrrflx 26164 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐽) = (𝐽 𝐸))
6259, 60, 613eqtr3d 2868 . . . 4 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐽 𝐸))
6333, 32eqtr4d 2863 . . . . 5 (𝜑 → (𝐶 𝐹) = (𝐸 𝐷))
6463adantr 481 . . . 4 ((𝜑𝐵𝐶) → (𝐶 𝐹) = (𝐸 𝐷))
651, 2, 3, 4, 21, 22, 23, 5, 26, 28tgbtwnexch3 26196 . . . . . 6 (𝜑𝐷 ∈ (𝐵𝐼𝐸))
6665adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐷 ∈ (𝐵𝐼𝐸))
671, 2, 3, 4, 21, 13, 6, 27, 29, 31tgbtwnexch3 26196 . . . . . . 7 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
681, 2, 3, 4, 13, 6, 27, 67tgbtwncom 26190 . . . . . 6 (𝜑𝐹 ∈ (𝐽𝐼𝐶))
6968adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐹 ∈ (𝐽𝐼𝐶))
701, 2, 3, 4, 27, 6axtgcgrrflx 26164 . . . . . . 7 (𝜑 → (𝐽 𝐹) = (𝐹 𝐽))
7170, 34eqtr2d 2861 . . . . . 6 (𝜑 → (𝐵 𝐷) = (𝐽 𝐹))
7271adantr 481 . . . . 5 ((𝜑𝐵𝐶) → (𝐵 𝐷) = (𝐽 𝐹))
731, 2, 3, 4, 13, 6, 5, 23, 63tgcgrcomlr 26182 . . . . . . 7 (𝜑 → (𝐹 𝐶) = (𝐷 𝐸))
7473adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → (𝐹 𝐶) = (𝐷 𝐸))
7574eqcomd 2831 . . . . 5 ((𝜑𝐵𝐶) → (𝐷 𝐸) = (𝐹 𝐶))
761, 2, 3, 43, 48, 46, 45, 49, 44, 47, 66, 69, 72, 75tgcgrextend 26187 . . . 4 ((𝜑𝐵𝐶) → (𝐵 𝐸) = (𝐽 𝐶))
771, 2, 3, 43, 47, 45axtgcgrrflx 26164 . . . 4 ((𝜑𝐵𝐶) → (𝐶 𝐸) = (𝐸 𝐶))
781, 2, 3, 43, 48, 47, 44, 49, 45, 46, 45, 47, 50, 52, 57, 62, 64, 76, 77axtg5seg 26167 . . 3 ((𝜑𝐵𝐶) → (𝐹 𝐸) = (𝐷 𝐶))
791, 2, 3, 43, 44, 45, 46, 47, 78tgcgrcomlr 26182 . 2 ((𝜑𝐵𝐶) → (𝐸 𝐹) = (𝐶 𝐷))
8042, 79pm2.61dane 3108 1 (𝜑 → (𝐸 𝐹) = (𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wne 3020  cfv 6351  (class class class)co 7151  Basecbs 16475  distcds 16566  TarskiGcstrkg 26132  Itvcitv 26138
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-ov 7154  df-trkgc 26150  df-trkgb 26151  df-trkgcb 26152  df-trkg 26155
This theorem is referenced by:  tgbtwnconn1lem3  26276
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