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Theorem 3at 29606
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 29593 for lines and 4at 29729 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  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q
) )  ->  (
( ( 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 29605 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q
)  /\  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) )  ->  (
( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) )
543expia 1155 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U )  ->  ( ( P 
.\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U ) ) )
6 hllat 29480 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
7 simpl 444 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
8 simpr1 963 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
9 eqid 2389 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 29406 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
118, 10syl 16 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
12 simpr2 964 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
139, 3atbase 29406 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1412, 13syl 16 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
159, 2latjcl 14408 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
167, 11, 14, 15syl3anc 1184 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
17 simpr3 965 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
189, 3atbase 29406 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1917, 18syl 16 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
209, 2latjcl 14408 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
217, 16, 19, 20syl3anc 1184 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
229, 1latref 14411 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
2321, 22syldan 457 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
24 breq2 4159 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R )  <-> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
2523, 24syl5ibcom 212 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) ) )
266, 25sylan 458 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) ) )
27263adant3 977 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( ( P 
.\/  Q )  .\/  R )  =  ( ( S  .\/  T ) 
.\/  U )  -> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
2827adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q
) )  ->  (
( ( P  .\/  Q )  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( ( P  .\/  Q )  .\/  R )  .<_  ( ( S  .\/  T )  .\/  U ) ) )
295, 28impbid 184 1  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  /\  ( -.  R  .<_  ( P  .\/  Q )  /\  P  =/=  Q
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U )  <-> 
( ( P  .\/  Q )  .\/  R )  =  ( ( S 
.\/  T )  .\/  U ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2552   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   Basecbs 13398   lecple 13465   joincjn 14330   Latclat 14403   Atomscatm 29380   HLchlt 29467
This theorem is referenced by:  llncvrlpln2  29673  2lplnja  29735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-undef 6481  df-riota 6487  df-poset 14332  df-plt 14344  df-lub 14360  df-join 14362  df-lat 14404  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468
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