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Theorem 3at 29497
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 29484 for lines and 4at 29620 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 29496 . . 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 1153 . 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 29371 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
7 simpl 443 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
8 simpr1 961 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
9 eqid 2316 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 29297 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
118, 10syl 15 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  ( Base `  K
) )
12 simpr2 962 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
139, 3atbase 29297 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1412, 13syl 15 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  ( Base `  K
) )
159, 2latjcl 14205 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
167, 11, 14, 15syl3anc 1182 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
17 simpr3 963 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
189, 3atbase 29297 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
1917, 18syl 15 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  ( Base `  K
) )
209, 2latjcl 14205 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
217, 16, 19, 20syl3anc 1182 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
229, 1latref 14208 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
2321, 22syldan 456 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
24 breq2 4064 . . . . . 6  |-  ( ( ( P  .\/  Q
)  .\/  R )  =  ( ( S 
.\/  T )  .\/  U )  ->  ( (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R )  <-> 
( ( P  .\/  Q )  .\/  R ) 
.<_  ( ( S  .\/  T )  .\/  U ) ) )
2523, 24syl5ibcom 211 . . . . 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 457 . . . 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 975 . . 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 451 . 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 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Latclat 14200   Atomscatm 29271   HLchlt 29358
This theorem is referenced by:  llncvrlpln2  29564  2lplnja  29626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-join 14159  df-lat 14201  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359
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