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Theorem 3at 38874
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 38861 for lines and 4at 38997 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 38873 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
543expia 1118 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
6 hllat 38746 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
7 simpl 482 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ Lat)
8 simpr1 1191 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
9 eqid 2726 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 38672 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃 ∈ (Base‘𝐾))
12 simpr2 1192 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
139, 3atbase 38672 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄 ∈ (Base‘𝐾))
159, 2latjcl 18404 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
167, 11, 14, 15syl3anc 1368 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 simpr3 1193 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
189, 3atbase 38672 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅 ∈ (Base‘𝐾))
209, 2latjcl 18404 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
217, 16, 19, 20syl3anc 1368 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
229, 1latref 18406 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
2321, 22syldan 590 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
24 breq2 5145 . . . . . 6 (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → (((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2523, 24syl5ibcom 244 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
266, 25sylan 579 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
27263adant3 1129 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2827adantr 480 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
295, 28impbid 211 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934   class class class wbr 5141  cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Latclat 18396  Atomscatm 38646  HLchlt 38733
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  llncvrlpln2  38941  2lplnja  39003
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