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Theorem tglineneq 27885
Description: Given three non-colinear points, build two different lines. (Contributed by Thierry Arnoux, 6-Aug-2019.)
Hypotheses
Ref Expression
tglineintmo.p 𝑃 = (Baseβ€˜πΊ)
tglineintmo.i 𝐼 = (Itvβ€˜πΊ)
tglineintmo.l 𝐿 = (LineGβ€˜πΊ)
tglineintmo.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tglineinteq.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
tglineinteq.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
tglineinteq.c (πœ‘ β†’ 𝐢 ∈ 𝑃)
tglineinteq.d (πœ‘ β†’ 𝐷 ∈ 𝑃)
tglineinteq.e (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
Assertion
Ref Expression
tglineneq (πœ‘ β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
Proof of Theorem tglineneq
StepHypRef Expression
1 tglineintmo.p . . . 4 𝑃 = (Baseβ€˜πΊ)
2 tglineintmo.i . . . 4 𝐼 = (Itvβ€˜πΊ)
3 tglineintmo.l . . . 4 𝐿 = (LineGβ€˜πΊ)
4 tglineintmo.g . . . 4 (πœ‘ β†’ 𝐺 ∈ TarskiG)
5 tglineinteq.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
6 tglineinteq.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
7 tglineinteq.c . . . . 5 (πœ‘ β†’ 𝐢 ∈ 𝑃)
8 tglineinteq.e . . . . 5 (πœ‘ β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
91, 2, 3, 4, 5, 6, 7, 8ncolne1 27866 . . . 4 (πœ‘ β†’ 𝐴 β‰  𝐡)
101, 2, 3, 4, 5, 6, 9tglinerflx1 27874 . . 3 (πœ‘ β†’ 𝐴 ∈ (𝐴𝐿𝐡))
11 simplr 768 . . . 4 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 = 𝐷)
124adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐺 ∈ TarskiG)
137adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 ∈ 𝑃)
14 tglineinteq.d . . . . . . . 8 (πœ‘ β†’ 𝐷 ∈ 𝑃)
1514adantr 482 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐷 ∈ 𝑃)
16 simpr 486 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐴 ∈ (𝐢𝐿𝐷))
171, 3, 2, 12, 13, 15, 16tglngne 27791 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
1817adantlr 714 . . . . 5 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
1918neneqd 2946 . . . 4 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ Β¬ 𝐢 = 𝐷)
2011, 19pm2.65da 816 . . 3 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ Β¬ 𝐴 ∈ (𝐢𝐿𝐷))
21 nelne1 3040 . . 3 ((𝐴 ∈ (𝐴𝐿𝐡) ∧ Β¬ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
2210, 20, 21syl2an2r 684 . 2 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
234ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐺 ∈ TarskiG)
246ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐡 ∈ 𝑃)
257ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ 𝑃)
265ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 ∈ 𝑃)
27 pm2.46 882 . . . . . . . . 9 (Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢) β†’ Β¬ 𝐡 = 𝐢)
288, 27syl 17 . . . . . . . 8 (πœ‘ β†’ Β¬ 𝐡 = 𝐢)
2928neqned 2948 . . . . . . 7 (πœ‘ β†’ 𝐡 β‰  𝐢)
3029ad2antrr 725 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐡 β‰  𝐢)
3114ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐷 ∈ 𝑃)
32 simplr 768 . . . . . . . 8 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
331, 2, 3, 23, 25, 31, 32tglinerflx1 27874 . . . . . . 7 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ (𝐢𝐿𝐷))
34 simpr 486 . . . . . . 7 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ (𝐴𝐿𝐡) = (𝐢𝐿𝐷))
3533, 34eleqtrrd 2837 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ (𝐴𝐿𝐡))
361, 3, 2, 23, 26, 24, 35tglngne 27791 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 β‰  𝐡)
371, 2, 3, 23, 24, 25, 26, 30, 35, 36lnrot1 27864 . . . . 5 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 ∈ (𝐡𝐿𝐢))
3837orcd 872 . . . 4 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
398ad2antrr 725 . . . 4 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
4038, 39pm2.65da 816 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝐷) β†’ Β¬ (𝐴𝐿𝐡) = (𝐢𝐿𝐷))
4140neqned 2948 . 2 ((πœ‘ ∧ 𝐢 β‰  𝐷) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
4222, 41pm2.61dane 3030 1 (πœ‘ β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  β€˜cfv 6541  (class class class)co 7406  Basecbs 17141  TarskiGcstrkg 27668  Itvcitv 27674  LineGclng 27675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-1st 7972  df-2nd 7973  df-trkgc 27689  df-trkgb 27690  df-trkgcb 27691  df-trkg 27694
This theorem is referenced by:  tglineinteq  27886  perpneq  27955
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