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Theorem colperpexlem2 25540
Description: Lemma for colperpex 25542. 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 477 . . . . . . . . . 10 ((𝜑𝐴 = 𝑄) → 𝐴 = 𝑄)
32fveq2d 6157 . . . . . . . . 9 ((𝜑𝐴 = 𝑄) → (𝑆𝐴) = (𝑆𝑄))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (𝑆𝐴)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (𝑆𝑄)
63, 4, 53eqtr4g 2680 . . . . . . . 8 ((𝜑𝐴 = 𝑄) → 𝑀 = 𝐾)
76fveq1d 6155 . . . . . . 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 25474 . . . . . . . 8 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
1716adantr 481 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = 𝐶)
18 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
1918adantr 481 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
207, 17, 193eqtr3rd 2664 . . . . . 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 25478 . . . . . . 7 (𝜑 → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2423adantr 481 . . . . . 6 ((𝜑𝐴 = 𝑄) → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2520, 24mpbid 222 . . . . 5 ((𝜑𝐴 = 𝑄) → 𝐵 = 𝐶)
2625ex 450 . . . 4 (𝜑 → (𝐴 = 𝑄𝐵 = 𝐶))
2726necon3ad 2803 . . 3 (𝜑 → (𝐵𝐶 → ¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (𝜑 → ¬ 𝐴 = 𝑄)
2928neqned 2797 1 (𝜑𝐴𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  cfv 5852  ⟨“cs3 13532  Basecbs 15792  distcds 15882  TarskiGcstrkg 25246  Itvcitv 25252  LineGclng 25253  pInvGcmir 25464  ∟Gcrag 25505
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-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
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-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  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-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-trkgc 25264  df-trkgb 25265  df-trkgcb 25266  df-trkg 25269  df-mir 25465
This theorem is referenced by:  colperpexlem3  25541
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