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Theorem 3at 35248
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 35235 for lines and 4at 35371 for volumes. I could not find this proof in the literature on projective geometry (where it is either given as an axiom or stated as an unproved fact), but it is similar to Theorem 15 of Veblen, "The Foundations of Geometry" (1911), p. 18, which uses different axioms. This proof was written before I became aware of Veblen's, and it is possible that a shorter proof could be obtained by using Veblen's proof for hints. (Contributed by NM, 23-Jun-2012.)
Hypotheses
Ref Expression
3at.l = (le‘𝐾)
3at.j = (join‘𝐾)
3at.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
3at (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))

Proof of Theorem 3at
StepHypRef Expression
1 3at.l . . . 4 = (le‘𝐾)
2 3at.j . . . 4 = (join‘𝐾)
3 3at.a . . . 4 𝐴 = (Atoms‘𝐾)
41, 2, 33atlem7 35247 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
543expia 1114 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
6 hllat 35122 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
7 simpl 474 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ Lat)
8 simpr1 1210 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
9 eqid 2748 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 35048 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃 ∈ (Base‘𝐾))
12 simpr2 1212 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
139, 3atbase 35048 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄 ∈ (Base‘𝐾))
159, 2latjcl 17223 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
167, 11, 14, 15syl3anc 1463 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 simpr3 1214 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
189, 3atbase 35048 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅 ∈ (Base‘𝐾))
209, 2latjcl 17223 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
217, 16, 19, 20syl3anc 1463 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
229, 1latref 17225 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
2321, 22syldan 488 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
24 breq2 4796 . . . . . 6 (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → (((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2523, 24syl5ibcom 235 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
266, 25sylan 489 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
27263adant3 1124 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2827adantr 472 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
295, 28impbid 202 1 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) ↔ ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wcel 2127  wne 2920   class class class wbr 4792  cfv 6037  (class class class)co 6801  Basecbs 16030  lecple 16121  joincjn 17116  Latclat 17217  Atomscatm 35022  HLchlt 35109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-preset 17100  df-poset 17118  df-plt 17130  df-lub 17146  df-glb 17147  df-join 17148  df-meet 17149  df-p0 17211  df-lat 17218  df-covers 35025  df-ats 35026  df-atl 35057  df-cvlat 35081  df-hlat 35110
This theorem is referenced by:  llncvrlpln2  35315  2lplnja  35377
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