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Theorem colperpexlem2 28607
Description: Lemma for colperpex 28609. 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 483 . . . . . . . . . 10 ((𝜑𝐴 = 𝑄) → 𝐴 = 𝑄)
32fveq2d 6900 . . . . . . . . 9 ((𝜑𝐴 = 𝑄) → (𝑆𝐴) = (𝑆𝑄))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (𝑆𝐴)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (𝑆𝑄)
63, 4, 53eqtr4g 2790 . . . . . . . 8 ((𝜑𝐴 = 𝑄) → 𝑀 = 𝐾)
76fveq1d 6898 . . . . . . 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 28538 . . . . . . . 8 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
1716adantr 479 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = 𝐶)
18 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
1918adantr 479 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
207, 17, 193eqtr3rd 2774 . . . . . 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 28542 . . . . . . 7 (𝜑 → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2423adantr 479 . . . . . 6 ((𝜑𝐴 = 𝑄) → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2520, 24mpbid 231 . . . . 5 ((𝜑𝐴 = 𝑄) → 𝐵 = 𝐶)
2625ex 411 . . . 4 (𝜑 → (𝐴 = 𝑄𝐵 = 𝐶))
2726necon3ad 2942 . . 3 (𝜑 → (𝐵𝐶 → ¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (𝜑 → ¬ 𝐴 = 𝑄)
2928neqned 2936 1 (𝜑𝐴𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wne 2929  cfv 6549  ⟨“cs3 14829  Basecbs 17183  distcds 17245  TarskiGcstrkg 28303  Itvcitv 28309  LineGclng 28310  pInvGcmir 28528  ∟Gcrag 28569
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-trkgc 28324  df-trkgb 28325  df-trkgcb 28326  df-trkg 28329  df-mir 28529
This theorem is referenced by:  colperpexlem3  28608
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