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Theorem 3at 33492
Description: Any three non-colinear atoms in a (lattice) plane determine the plane uniquely. This is the 2-dimensional analog of ps-1 33479 for lines and 4at 33615 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 33491 . . 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 1190 . 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 33366 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
7 simpl 457 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  K  e.  Lat )
8 simpr1 994 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  P  e.  A )
9 eqid 2454 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
109, 3atbase 33292 . . . . . . . . . 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 995 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  Q  e.  A )
139, 3atbase 33292 . . . . . . . . . 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 15343 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
167, 11, 14, 15syl3anc 1219 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
17 simpr3 996 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  R  e.  A )
189, 3atbase 33292 . . . . . . . . 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 15343 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
217, 16, 19, 20syl3anc 1219 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
229, 1latref 15345 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
2321, 22syldan 470 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  R )  .<_  ( ( P  .\/  Q )  .\/  R ) )
24 breq2 4407 . . . . . 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 471 . . . 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 1008 . . 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 465 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   HLchlt 33353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354
This theorem is referenced by:  llncvrlpln2  33559  2lplnja  33621
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