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Theorem 3at 32471
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 32458 for lines and 4at 32594 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 32470 . . 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 1197 . 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 32345 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
7 simpl 455 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
8 simpr1 1001 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
9 eqid 2400 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 32271 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
118, 10syl 17 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
12 simpr2 1002 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
139, 3atbase 32271 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1412, 13syl 17 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
159, 2latjcl 15895 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
167, 11, 14, 15syl3anc 1228 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
17 simpr3 1003 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
189, 3atbase 32271 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1917, 18syl 17 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
209, 2latjcl 15895 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
217, 16, 19, 20syl3anc 1228 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
229, 1latref 15897 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
2321, 22syldan 468 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
24 breq2 4396 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R )  <-> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
2523, 24syl5ibcom 220 . . . . 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 469 . . . 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 1015 . . 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 463 . 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 191 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840    =/= wne 2596   class class class wbr 4392   ` cfv 5523  (class class class)co 6232   Basecbs 14731   lecple 14806   joincjn 15787   Latclat 15889   Atomscatm 32245   HLchlt 32332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-preset 15771  df-poset 15789  df-plt 15802  df-lub 15818  df-glb 15819  df-join 15820  df-meet 15821  df-p0 15883  df-lat 15890  df-covers 32248  df-ats 32249  df-atl 32280  df-cvlat 32304  df-hlat 32333
This theorem is referenced by:  llncvrlpln2  32538  2lplnja  32600
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