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Theorem 3at 33593
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 33580 for lines and 4at 33716 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 33592 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
543expia 1258 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
6 hllat 33467 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
7 simpl 471 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ Lat)
8 simpr1 1059 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
9 eqid 2605 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 33393 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃 ∈ (Base‘𝐾))
12 simpr2 1060 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
139, 3atbase 33393 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄 ∈ (Base‘𝐾))
159, 2latjcl 16816 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
167, 11, 14, 15syl3anc 1317 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 simpr3 1061 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
189, 3atbase 33393 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅 ∈ (Base‘𝐾))
209, 2latjcl 16816 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
217, 16, 19, 20syl3anc 1317 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
229, 1latref 16818 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
2321, 22syldan 485 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
24 breq2 4577 . . . . . 6 (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → (((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2523, 24syl5ibcom 233 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
266, 25sylan 486 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
27263adant3 1073 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2827adantr 479 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
295, 28impbid 200 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wne 2775   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  Latclat 16810  Atomscatm 33367  HLchlt 33454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-lat 16811  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455
This theorem is referenced by:  llncvrlpln2  33660  2lplnja  33722
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