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Theorem 3at 30289
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 30276 for lines and 4at 30412 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 30288 . . 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 1156 . 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 30163 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
7 simpl 445 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
8 simpr1 964 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
9 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 30089 . . . . . . . . . 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 965 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
139, 3atbase 30089 . . . . . . . . . 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 14481 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
167, 11, 14, 15syl3anc 1185 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
17 simpr3 966 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
189, 3atbase 30089 . . . . . . . . 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 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
217, 16, 19, 20syl3anc 1185 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
229, 1latref 14484 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
2321, 22syldan 458 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
24 breq2 4218 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R )  <-> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
2523, 24syl5ibcom 213 . . . . 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 459 . . . 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 978 . . 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 453 . 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 185 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   Latclat 14476   Atomscatm 30063   HLchlt 30150
This theorem is referenced by:  llncvrlpln2  30356  2lplnja  30418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-join 14435  df-lat 14477  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151
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