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Theorem colperpexlem2 28522
Description: Lemma for colperpex 28524. Second part of lemma 8.20 of [Schwabhauser] p. 62. (Contributed by Thierry Arnoux, 10-Nov-2019.)
Hypotheses
Ref Expression
colperpex.p 𝑃 = (Baseβ€˜πΊ)
colperpex.d βˆ’ = (distβ€˜πΊ)
colperpex.i 𝐼 = (Itvβ€˜πΊ)
colperpex.l 𝐿 = (LineGβ€˜πΊ)
colperpex.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
colperpexlem.s 𝑆 = (pInvGβ€˜πΊ)
colperpexlem.m 𝑀 = (π‘†β€˜π΄)
colperpexlem.n 𝑁 = (π‘†β€˜π΅)
colperpexlem.k 𝐾 = (π‘†β€˜π‘„)
colperpexlem.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
colperpexlem.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
colperpexlem.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
colperpexlem.q (πœ‘ β†’ 𝑄 ∈ 𝑃)
colperpexlem.1 (πœ‘ β†’ βŸ¨β€œπ΄π΅πΆβ€βŸ© ∈ (∟Gβ€˜πΊ))
colperpexlem.2 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) = (π‘β€˜πΆ))
colperpexlem2.e (πœ‘ β†’ 𝐡 β‰  𝐢)
Assertion
Ref Expression
colperpexlem2 (πœ‘ β†’ 𝐴 β‰  𝑄)

Proof of Theorem colperpexlem2
StepHypRef Expression
1 colperpexlem2.e . . 3 (πœ‘ β†’ 𝐡 β‰  𝐢)
2 simpr 484 . . . . . . . . . 10 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ 𝐴 = 𝑄)
32fveq2d 6895 . . . . . . . . 9 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ (π‘†β€˜π΄) = (π‘†β€˜π‘„))
4 colperpexlem.m . . . . . . . . 9 𝑀 = (π‘†β€˜π΄)
5 colperpexlem.k . . . . . . . . 9 𝐾 = (π‘†β€˜π‘„)
63, 4, 53eqtr4g 2792 . . . . . . . 8 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ 𝑀 = 𝐾)
76fveq1d 6893 . . . . . . 7 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ (π‘€β€˜(π‘€β€˜πΆ)) = (πΎβ€˜(π‘€β€˜πΆ)))
8 colperpex.p . . . . . . . . 9 𝑃 = (Baseβ€˜πΊ)
9 colperpex.d . . . . . . . . 9 βˆ’ = (distβ€˜πΊ)
10 colperpex.i . . . . . . . . 9 𝐼 = (Itvβ€˜πΊ)
11 colperpex.l . . . . . . . . 9 𝐿 = (LineGβ€˜πΊ)
12 colperpexlem.s . . . . . . . . 9 𝑆 = (pInvGβ€˜πΊ)
13 colperpex.g . . . . . . . . 9 (πœ‘ β†’ 𝐺 ∈ TarskiG)
14 colperpexlem.a . . . . . . . . 9 (πœ‘ β†’ 𝐴 ∈ 𝑃)
15 colperpexlem.c . . . . . . . . 9 (πœ‘ β†’ 𝐢 ∈ 𝑃)
168, 9, 10, 11, 12, 13, 14, 4, 15mirmir 28453 . . . . . . . 8 (πœ‘ β†’ (π‘€β€˜(π‘€β€˜πΆ)) = 𝐢)
1716adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ (π‘€β€˜(π‘€β€˜πΆ)) = 𝐢)
18 colperpexlem.2 . . . . . . . 8 (πœ‘ β†’ (πΎβ€˜(π‘€β€˜πΆ)) = (π‘β€˜πΆ))
1918adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ (πΎβ€˜(π‘€β€˜πΆ)) = (π‘β€˜πΆ))
207, 17, 193eqtr3rd 2776 . . . . . 6 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ (π‘β€˜πΆ) = 𝐢)
21 colperpexlem.b . . . . . . . 8 (πœ‘ β†’ 𝐡 ∈ 𝑃)
22 colperpexlem.n . . . . . . . 8 𝑁 = (π‘†β€˜π΅)
238, 9, 10, 11, 12, 13, 21, 22, 15mirinv 28457 . . . . . . 7 (πœ‘ β†’ ((π‘β€˜πΆ) = 𝐢 ↔ 𝐡 = 𝐢))
2423adantr 480 . . . . . 6 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ ((π‘β€˜πΆ) = 𝐢 ↔ 𝐡 = 𝐢))
2520, 24mpbid 231 . . . . 5 ((πœ‘ ∧ 𝐴 = 𝑄) β†’ 𝐡 = 𝐢)
2625ex 412 . . . 4 (πœ‘ β†’ (𝐴 = 𝑄 β†’ 𝐡 = 𝐢))
2726necon3ad 2948 . . 3 (πœ‘ β†’ (𝐡 β‰  𝐢 β†’ Β¬ 𝐴 = 𝑄))
281, 27mpd 15 . 2 (πœ‘ β†’ Β¬ 𝐴 = 𝑄)
2928neqned 2942 1 (πœ‘ β†’ 𝐴 β‰  𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  β€˜cfv 6542  βŸ¨β€œcs3 14817  Basecbs 17171  distcds 17233  TarskiGcstrkg 28218  Itvcitv 28224  LineGclng 28225  pInvGcmir 28443  βˆŸGcrag 28484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-trkgc 28239  df-trkgb 28240  df-trkgcb 28241  df-trkg 28244  df-mir 28444
This theorem is referenced by:  colperpexlem3  28523
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