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Theorem colperpexlem2 26523
Description: Lemma for colperpex 26525. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)
Hypotheses
Ref Expression
colperpex.p 𝑃 = (Base‘𝐺)
colperpex.d = (dist‘𝐺)
colperpex.i 𝐼 = (Itv‘𝐺)
colperpex.l 𝐿 = (LineG‘𝐺)
colperpex.g (𝜑𝐺 ∈ TarskiG)
colperpexlem.s 𝑆 = (pInvG‘𝐺)
colperpexlem.m 𝑀 = (𝑆𝐴)
colperpexlem.n 𝑁 = (𝑆𝐵)
colperpexlem.k 𝐾 = (𝑆𝑄)
colperpexlem.a (𝜑𝐴𝑃)
colperpexlem.b (𝜑𝐵𝑃)
colperpexlem.c (𝜑𝐶𝑃)
colperpexlem.q (𝜑𝑄𝑃)
colperpexlem.1 (𝜑 → ⟨“𝐴𝐵𝐶”⟩ ∈ (∟G‘𝐺))
colperpexlem.2 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
colperpexlem2.e (𝜑𝐵𝐶)
Assertion
Ref Expression
colperpexlem2 (𝜑𝐴𝑄)

Proof of Theorem colperpexlem2
StepHypRef Expression
1 colperpexlem2.e . . 3 (𝜑𝐵𝐶)
2 simpr 488 . . . . . . . . . 10 ((𝜑𝐴 = 𝑄) → 𝐴 = 𝑄)
32fveq2d 6656 . . . . . . . . 9 ((𝜑𝐴 = 𝑄) → (𝑆𝐴) = (𝑆𝑄))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (𝑆𝐴)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (𝑆𝑄)
63, 4, 53eqtr4g 2882 . . . . . . . 8 ((𝜑𝐴 = 𝑄) → 𝑀 = 𝐾)
76fveq1d 6654 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = (𝐾‘(𝑀𝐶)))
8 colperpex.p . . . . . . . . 9 𝑃 = (Base‘𝐺)
9 colperpex.d . . . . . . . . 9 = (dist‘𝐺)
10 colperpex.i . . . . . . . . 9 𝐼 = (Itv‘𝐺)
11 colperpex.l . . . . . . . . 9 𝐿 = (LineG‘𝐺)
12 colperpexlem.s . . . . . . . . 9 𝑆 = (pInvG‘𝐺)
13 colperpex.g . . . . . . . . 9 (𝜑𝐺 ∈ TarskiG)
14 colperpexlem.a . . . . . . . . 9 (𝜑𝐴𝑃)
15 colperpexlem.c . . . . . . . . 9 (𝜑𝐶𝑃)
168, 9, 10, 11, 12, 13, 14, 4, 15mirmir 26454 . . . . . . . 8 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
1716adantr 484 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = 𝐶)
18 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
1918adantr 484 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
207, 17, 193eqtr3rd 2866 . . . . . 6 ((𝜑𝐴 = 𝑄) → (𝑁𝐶) = 𝐶)
21 colperpexlem.b . . . . . . . 8 (𝜑𝐵𝑃)
22 colperpexlem.n . . . . . . . 8 𝑁 = (𝑆𝐵)
238, 9, 10, 11, 12, 13, 21, 22, 15mirinv 26458 . . . . . . 7 (𝜑 → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2423adantr 484 . . . . . 6 ((𝜑𝐴 = 𝑄) → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2520, 24mpbid 235 . . . . 5 ((𝜑𝐴 = 𝑄) → 𝐵 = 𝐶)
2625ex 416 . . . 4 (𝜑 → (𝐴 = 𝑄𝐵 = 𝐶))
2726necon3ad 3024 . . 3 (𝜑 → (𝐵𝐶 → ¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (𝜑 → ¬ 𝐴 = 𝑄)
2928neqned 3018 1 (𝜑𝐴𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wne 3011  cfv 6334  ⟨“cs3 14195  Basecbs 16474  distcds 16565  TarskiGcstrkg 26222  Itvcitv 26228  LineGclng 26229  pInvGcmir 26444  ∟Gcrag 26485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-trkgc 26240  df-trkgb 26241  df-trkgcb 26242  df-trkg 26245  df-mir 26445
This theorem is referenced by:  colperpexlem3  26524
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