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Theorem tgbtwnconn1lem2 25381
Description: Lemma for tgbtwnconn1 25383. (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 25274 . . . 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 477 . . . . . . . . 9 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
1817oveq1d 6625 . . . . . . . 8 ((𝜑𝐵 = 𝐶) → (𝐵 𝐶) = (𝐶 𝐶))
1916, 18eqtrd 2655 . . . . . . 7 ((𝜑𝐵 = 𝐶) → (𝐸 𝐻) = (𝐶 𝐶))
201, 2, 3, 9, 10, 12, 14, 19axtgcgrid 25275 . . . . . 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 25380 . . . . . . 7 (𝜑𝐻 = 𝐽)
3635adantr 481 . . . . . 6 ((𝜑𝐵 = 𝐶) → 𝐻 = 𝐽)
3720, 36eqtrd 2655 . . . . 5 ((𝜑𝐵 = 𝐶) → 𝐸 = 𝐽)
3837oveq2d 6626 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐹 𝐸) = (𝐹 𝐽))
3934adantr 481 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐹 𝐽) = (𝐵 𝐷))
4017oveq1d 6625 . . . 4 ((𝜑𝐵 = 𝐶) → (𝐵 𝐷) = (𝐶 𝐷))
4138, 39, 403eqtrd 2659 . . 3 ((𝜑𝐵 = 𝐶) → (𝐹 𝐸) = (𝐶 𝐷))
428, 41eqtrd 2655 . 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 477 . . . 4 ((𝜑𝐵𝐶) → 𝐵𝐶)
511, 2, 3, 4, 21, 22, 13, 6, 25, 29tgbtwnexch3 25302 . . . . 5 (𝜑𝐶 ∈ (𝐵𝐼𝐹))
5251adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐶 ∈ (𝐵𝐼𝐹))
5335oveq2d 6626 . . . . . . . 8 (𝜑 → (𝐴𝐼𝐻) = (𝐴𝐼𝐽))
5430, 53eleqtrd 2700 . . . . . . 7 (𝜑𝐸 ∈ (𝐴𝐼𝐽))
551, 2, 3, 4, 21, 23, 5, 27, 28, 54tgbtwnexch3 25302 . . . . . 6 (𝜑𝐸 ∈ (𝐷𝐼𝐽))
561, 2, 3, 4, 23, 5, 27, 55tgbtwncom 25296 . . . . 5 (𝜑𝐸 ∈ (𝐽𝐼𝐷))
5756adantr 481 . . . 4 ((𝜑𝐵𝐶) → 𝐸 ∈ (𝐽𝐼𝐷))
5835adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → 𝐻 = 𝐽)
5958oveq2d 6626 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐻) = (𝐸 𝐽))
6015adantr 481 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐻) = (𝐵 𝐶))
611, 2, 3, 43, 45, 49axtgcgrrflx 25274 . . . . 5 ((𝜑𝐵𝐶) → (𝐸 𝐽) = (𝐽 𝐸))
6259, 60, 613eqtr3d 2663 . . . 4 ((𝜑𝐵𝐶) → (𝐵 𝐶) = (𝐽 𝐸))
6333, 32eqtr4d 2658 . . . . 5 (𝜑 → (𝐶 𝐹) = (𝐸 𝐷))
6463adantr 481 . . . 4 ((𝜑𝐵𝐶) → (𝐶 𝐹) = (𝐸 𝐷))
651, 2, 3, 4, 21, 22, 23, 5, 26, 28tgbtwnexch3 25302 . . . . . 6 (𝜑𝐷 ∈ (𝐵𝐼𝐸))
6665adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐷 ∈ (𝐵𝐼𝐸))
671, 2, 3, 4, 21, 13, 6, 27, 29, 31tgbtwnexch3 25302 . . . . . . 7 (𝜑𝐹 ∈ (𝐶𝐼𝐽))
681, 2, 3, 4, 13, 6, 27, 67tgbtwncom 25296 . . . . . 6 (𝜑𝐹 ∈ (𝐽𝐼𝐶))
6968adantr 481 . . . . 5 ((𝜑𝐵𝐶) → 𝐹 ∈ (𝐽𝐼𝐶))
701, 2, 3, 4, 27, 6axtgcgrrflx 25274 . . . . . . 7 (𝜑 → (𝐽 𝐹) = (𝐹 𝐽))
7170, 34eqtr2d 2656 . . . . . 6 (𝜑 → (𝐵 𝐷) = (𝐽 𝐹))
7271adantr 481 . . . . 5 ((𝜑𝐵𝐶) → (𝐵 𝐷) = (𝐽 𝐹))
731, 2, 3, 4, 13, 6, 5, 23, 63tgcgrcomlr 25288 . . . . . . 7 (𝜑 → (𝐹 𝐶) = (𝐷 𝐸))
7473adantr 481 . . . . . 6 ((𝜑𝐵𝐶) → (𝐹 𝐶) = (𝐷 𝐸))
7574eqcomd 2627 . . . . 5 ((𝜑𝐵𝐶) → (𝐷 𝐸) = (𝐹 𝐶))
761, 2, 3, 43, 48, 46, 45, 49, 44, 47, 66, 69, 72, 75tgcgrextend 25293 . . . 4 ((𝜑𝐵𝐶) → (𝐵 𝐸) = (𝐽 𝐶))
771, 2, 3, 43, 47, 45axtgcgrrflx 25274 . . . 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 25277 . . 3 ((𝜑𝐵𝐶) → (𝐹 𝐸) = (𝐷 𝐶))
791, 2, 3, 43, 44, 45, 46, 47, 78tgcgrcomlr 25288 . 2 ((𝜑𝐵𝐶) → (𝐸 𝐹) = (𝐶 𝐷))
8042, 79pm2.61dane 2877 1 (𝜑 → (𝐸 𝐹) = (𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  cfv 5852  (class class class)co 6610  Basecbs 15788  distcds 15878  TarskiGcstrkg 25242  Itvcitv 25248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-trkgc 25260  df-trkgb 25261  df-trkgcb 25262  df-trkg 25265
This theorem is referenced by:  tgbtwnconn1lem3  25382
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