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Theorem colperpexlem2 26039
Description: Lemma for colperpex 26041. 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 479 . . . . . . . . . 10 ((𝜑𝐴 = 𝑄) → 𝐴 = 𝑄)
32fveq2d 6436 . . . . . . . . 9 ((𝜑𝐴 = 𝑄) → (𝑆𝐴) = (𝑆𝑄))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (𝑆𝐴)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (𝑆𝑄)
63, 4, 53eqtr4g 2885 . . . . . . . 8 ((𝜑𝐴 = 𝑄) → 𝑀 = 𝐾)
76fveq1d 6434 . . . . . . 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 25973 . . . . . . . 8 (𝜑 → (𝑀‘(𝑀𝐶)) = 𝐶)
1716adantr 474 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝑀‘(𝑀𝐶)) = 𝐶)
18 colperpexlem.2 . . . . . . . 8 (𝜑 → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
1918adantr 474 . . . . . . 7 ((𝜑𝐴 = 𝑄) → (𝐾‘(𝑀𝐶)) = (𝑁𝐶))
207, 17, 193eqtr3rd 2869 . . . . . 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 25977 . . . . . . 7 (𝜑 → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2423adantr 474 . . . . . 6 ((𝜑𝐴 = 𝑄) → ((𝑁𝐶) = 𝐶𝐵 = 𝐶))
2520, 24mpbid 224 . . . . 5 ((𝜑𝐴 = 𝑄) → 𝐵 = 𝐶)
2625ex 403 . . . 4 (𝜑 → (𝐴 = 𝑄𝐵 = 𝐶))
2726necon3ad 3011 . . 3 (𝜑 → (𝐵𝐶 → ¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (𝜑 → ¬ 𝐴 = 𝑄)
2928neqned 3005 1 (𝜑𝐴𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wne 2998  cfv 6122  ⟨“cs3 13962  Basecbs 16221  distcds 16313  TarskiGcstrkg 25741  Itvcitv 25747  LineGclng 25748  pInvGcmir 25963  ∟Gcrag 26004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-rep 4993  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-reu 3123  df-rmo 3124  df-rab 3125  df-v 3415  df-sbc 3662  df-csb 3757  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-iun 4741  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-riota 6865  df-ov 6907  df-trkgc 25759  df-trkgb 25760  df-trkgcb 25761  df-trkg 25764  df-mir 25964
This theorem is referenced by:  colperpexlem3  26040
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