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Theorem 3at 36662
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 36649 for lines and 4at 36785 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 36661 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
543expia 1117 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
6 hllat 36535 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
7 simpl 485 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ Lat)
8 simpr1 1190 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
9 eqid 2820 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 36461 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃 ∈ (Base‘𝐾))
12 simpr2 1191 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
139, 3atbase 36461 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄 ∈ (Base‘𝐾))
159, 2latjcl 17640 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
167, 11, 14, 15syl3anc 1367 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 simpr3 1192 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
189, 3atbase 36461 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅 ∈ (Base‘𝐾))
209, 2latjcl 17640 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
217, 16, 19, 20syl3anc 1367 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
229, 1latref 17642 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
2321, 22syldan 593 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
24 breq2 5046 . . . . . 6 (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → (((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2523, 24syl5ibcom 247 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
266, 25sylan 582 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
27263adant3 1128 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2827adantr 483 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
295, 28impbid 214 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 208  wa 398  w3a 1083   = wceq 1537  wcel 2114  wne 3006   class class class wbr 5042  cfv 6331  (class class class)co 7133  Basecbs 16462  lecple 16551  joincjn 17533  Latclat 17634  Atomscatm 36435  HLchlt 36522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-proset 17517  df-poset 17535  df-plt 17547  df-lub 17563  df-glb 17564  df-join 17565  df-meet 17566  df-p0 17628  df-lat 17635  df-covers 36438  df-ats 36439  df-atl 36470  df-cvlat 36494  df-hlat 36523
This theorem is referenced by:  llncvrlpln2  36729  2lplnja  36791
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