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Theorem tglineneq 28330
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 28311 . . . 4 (πœ‘ β†’ 𝐴 β‰  𝐡)
101, 2, 3, 4, 5, 6, 9tglinerflx1 28319 . . 3 (πœ‘ β†’ 𝐴 ∈ (𝐴𝐿𝐡))
11 simplr 766 . . . 4 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 = 𝐷)
124adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐺 ∈ TarskiG)
137adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 ∈ 𝑃)
14 tglineinteq.d . . . . . . . 8 (πœ‘ β†’ 𝐷 ∈ 𝑃)
1514adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐷 ∈ 𝑃)
16 simpr 484 . . . . . . 7 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐴 ∈ (𝐢𝐿𝐷))
171, 3, 2, 12, 13, 15, 16tglngne 28236 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
1817adantlr 712 . . . . 5 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
1918neneqd 2937 . . . 4 (((πœ‘ ∧ 𝐢 = 𝐷) ∧ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ Β¬ 𝐢 = 𝐷)
2011, 19pm2.65da 814 . . 3 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ Β¬ 𝐴 ∈ (𝐢𝐿𝐷))
21 nelne1 3031 . . 3 ((𝐴 ∈ (𝐴𝐿𝐡) ∧ Β¬ 𝐴 ∈ (𝐢𝐿𝐷)) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
2210, 20, 21syl2an2r 682 . 2 ((πœ‘ ∧ 𝐢 = 𝐷) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
234ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐺 ∈ TarskiG)
246ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐡 ∈ 𝑃)
257ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ 𝑃)
265ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 ∈ 𝑃)
27 pm2.46 879 . . . . . . . . 9 (Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢) β†’ Β¬ 𝐡 = 𝐢)
288, 27syl 17 . . . . . . . 8 (πœ‘ β†’ Β¬ 𝐡 = 𝐢)
2928neqned 2939 . . . . . . 7 (πœ‘ β†’ 𝐡 β‰  𝐢)
3029ad2antrr 723 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐡 β‰  𝐢)
3114ad2antrr 723 . . . . . . . 8 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐷 ∈ 𝑃)
32 simplr 766 . . . . . . . 8 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 β‰  𝐷)
331, 2, 3, 23, 25, 31, 32tglinerflx1 28319 . . . . . . 7 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ (𝐢𝐿𝐷))
34 simpr 484 . . . . . . 7 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ (𝐴𝐿𝐡) = (𝐢𝐿𝐷))
3533, 34eleqtrrd 2828 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐢 ∈ (𝐴𝐿𝐡))
361, 3, 2, 23, 26, 24, 35tglngne 28236 . . . . . 6 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 β‰  𝐡)
371, 2, 3, 23, 24, 25, 26, 30, 35, 36lnrot1 28309 . . . . 5 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ 𝐴 ∈ (𝐡𝐿𝐢))
3837orcd 870 . . . 4 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
398ad2antrr 723 . . . 4 (((πœ‘ ∧ 𝐢 β‰  𝐷) ∧ (𝐴𝐿𝐡) = (𝐢𝐿𝐷)) β†’ Β¬ (𝐴 ∈ (𝐡𝐿𝐢) ∨ 𝐡 = 𝐢))
4038, 39pm2.65da 814 . . 3 ((πœ‘ ∧ 𝐢 β‰  𝐷) β†’ Β¬ (𝐴𝐿𝐡) = (𝐢𝐿𝐷))
4140neqned 2939 . 2 ((πœ‘ ∧ 𝐢 β‰  𝐷) β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
4222, 41pm2.61dane 3021 1 (πœ‘ β†’ (𝐴𝐿𝐡) β‰  (𝐢𝐿𝐷))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∨ wo 844   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  β€˜cfv 6533  (class class class)co 7401  Basecbs 17142  TarskiGcstrkg 28113  Itvcitv 28119  LineGclng 28120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-trkgc 28134  df-trkgb 28135  df-trkgcb 28136  df-trkg 28139
This theorem is referenced by:  tglineinteq  28331  perpneq  28400
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