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Theorem colperpexlem2 25844
Description: Lemma for colperpex 25846. 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 471 . . . . . . . . . 10 ((𝜑𝐴 = 𝑄) → 𝐴 = 𝑄)
32fveq2d 6336 . . . . . . . . 9 ((𝜑𝐴 = 𝑄) → (𝑆𝐴) = (𝑆𝑄))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (𝑆𝐴)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (𝑆𝑄)
63, 4, 53eqtr4g 2830 . . . . . . . 8 ((𝜑𝐴 = 𝑄) → 𝑀 = 𝐾)
76fveq1d 6334 . . . . . . 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 25778 . . . . . . . 8 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
1716adantr 466 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = 𝐶)
18 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
1918adantr 466 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
207, 17, 193eqtr3rd 2814 . . . . . 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 25782 . . . . . . 7 (𝜑 → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2423adantr 466 . . . . . 6 ((𝜑𝐴 = 𝑄) → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2520, 24mpbid 222 . . . . 5 ((𝜑𝐴 = 𝑄) → 𝐵 = 𝐶)
2625ex 397 . . . 4 (𝜑 → (𝐴 = 𝑄𝐵 = 𝐶))
2726necon3ad 2956 . . 3 (𝜑 → (𝐵𝐶 → ¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (𝜑 → ¬ 𝐴 = 𝑄)
2928neqned 2950 1 (𝜑𝐴𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  cfv 6031  ⟨“cs3 13796  Basecbs 16064  distcds 16158  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557  pInvGcmir 25768  ∟Gcrag 25809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573  df-mir 25769
This theorem is referenced by:  colperpexlem3  25845
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