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Theorem 3at 38349
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analogue of ps-1 38336 for lines and 4at 38472 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 ≀ = (leβ€˜πΎ)
3at.j ∨ = (joinβ€˜πΎ)
3at.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
3at (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
Proof of Theorem 3at
StepHypRef Expression
1 3at.l . . . 4 ≀ = (leβ€˜πΎ)
2 3at.j . . . 4 ∨ = (joinβ€˜πΎ)
3 3at.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
41, 2, 33atlem7 38348 . . 3 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄) ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ))
543expia 1121 . 2 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
6 hllat 38221 . . . . 5 (𝐾 ∈ HL β†’ 𝐾 ∈ Lat)
7 simpl 483 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐾 ∈ Lat)
8 simpr1 1194 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ 𝐴)
9 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
109, 3atbase 38147 . . . . . . . . . 10 (𝑃 ∈ 𝐴 β†’ 𝑃 ∈ (Baseβ€˜πΎ))
118, 10syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑃 ∈ (Baseβ€˜πΎ))
12 simpr2 1195 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ 𝐴)
139, 3atbase 38147 . . . . . . . . . 10 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
1412, 13syl 17 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
159, 2latjcl 18388 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
167, 11, 14, 15syl3anc 1371 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
17 simpr3 1196 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ 𝐴)
189, 3atbase 38147 . . . . . . . . 9 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ (Baseβ€˜πΎ))
1917, 18syl 17 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑅 ∈ (Baseβ€˜πΎ))
209, 2latjcl 18388 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Baseβ€˜πΎ) ∧ 𝑅 ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Baseβ€˜πΎ))
217, 16, 19, 20syl3anc 1371 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Baseβ€˜πΎ))
229, 1latref 18390 . . . . . . 7 ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ (Baseβ€˜πΎ)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))
2321, 22syldan 591 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅))
24 breq2 5151 . . . . . 6 (((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
2523, 24syl5ibcom 244 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
266, 25sylan 580 . . . 4 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
27263adant3 1132 . . 3 ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
2827adantr 481 . 2 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
295, 28impbid 211 1 (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑃 β‰  𝑄)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑅) ≀ ((𝑆 ∨ 𝑇) ∨ π‘ˆ) ↔ ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑆 ∨ 𝑇) ∨ π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  HLchlt 38208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209
This theorem is referenced by:  llncvrlpln2  38416  2lplnja  38478
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