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Theorem 3at 35653
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 35640 for lines and 4at 35776 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 35652 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄) ∧ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈))
543expia 1111 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈)))
6 hllat 35526 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
7 simpl 476 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐾 ∈ Lat)
8 simpr1 1205 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃𝐴)
9 eqid 2778 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
109, 3atbase 35452 . . . . . . . . . 10 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑃 ∈ (Base‘𝐾))
12 simpr2 1207 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄𝐴)
139, 3atbase 35452 . . . . . . . . . 10 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑄 ∈ (Base‘𝐾))
159, 2latjcl 17448 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
167, 11, 14, 15syl3anc 1439 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 simpr3 1209 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅𝐴)
189, 3atbase 35452 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝑅 ∈ (Base‘𝐾))
209, 2latjcl 17448 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
217, 16, 19, 20syl3anc 1439 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾))
229, 1latref 17450 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 𝑄) 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
2321, 22syldan 585 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅))
24 breq2 4892 . . . . . 6 (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → (((𝑃 𝑄) 𝑅) ((𝑃 𝑄) 𝑅) ↔ ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2523, 24syl5ibcom 237 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
266, 25sylan 575 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
27263adant3 1123 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
2827adantr 474 . 2 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑃𝑄)) → (((𝑃 𝑄) 𝑅) = ((𝑆 𝑇) 𝑈) → ((𝑃 𝑄) 𝑅) ((𝑆 𝑇) 𝑈)))
295, 28impbid 204 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969   class class class wbr 4888  cfv 6137  (class class class)co 6924  Basecbs 16266  lecple 16356  joincjn 17341  Latclat 17442  Atomscatm 35426  HLchlt 35513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-proset 17325  df-poset 17343  df-plt 17355  df-lub 17371  df-glb 17372  df-join 17373  df-meet 17374  df-p0 17436  df-lat 17443  df-covers 35429  df-ats 35430  df-atl 35461  df-cvlat 35485  df-hlat 35514
This theorem is referenced by:  llncvrlpln2  35720  2lplnja  35782
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