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Theorem 2atmat 32578
Description: The meet of two intersecting lines (expressed as joins of atoms) is an atom. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
2atmat.l  |-  .<_  =  ( le `  K )
2atmat.j  |-  .\/  =  ( join `  K )
2atmat.m  |-  ./\  =  ( meet `  K )
2atmat.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atmat  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A )

Proof of Theorem 2atmat
StepHypRef Expression
1 simp11 1027 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  K  e.  HL )
2 hllat 32381 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  K  e.  Lat )
4 eqid 2402 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 2atmat.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 2atmat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 32384 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
873ad2ant1 1018 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
9 simp21 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  R  e.  A )
104, 6atbase 32307 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
119, 10syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  R  e.  ( Base `  K
) )
12 simp22 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  S  e.  A )
134, 6atbase 32307 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1412, 13syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  S  e.  ( Base `  K
) )
154, 5latjass 16049 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  R  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) )
163, 8, 11, 14, 15syl13anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  ( R 
.\/  S ) ) )
17 simp33 1035 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  S  .<_  ( ( P  .\/  Q )  .\/  R ) )
184, 5latjcl 16005 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
193, 8, 11, 18syl3anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )
20 2atmat.l . . . . . . 7  |-  .<_  =  ( le `  K )
214, 20, 5latleeqj2 16018 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  R )  e.  ( Base `  K
) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  ( ( ( P  .\/  Q ) 
.\/  R )  .\/  S )  =  ( ( P  .\/  Q ) 
.\/  R ) ) )
223, 14, 19, 21syl3anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( S  .<_  ( ( P 
.\/  Q )  .\/  R )  <->  ( ( ( P  .\/  Q ) 
.\/  R )  .\/  S )  =  ( ( P  .\/  Q ) 
.\/  R ) ) )
2317, 22mpbid 210 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( ( P  .\/  Q )  .\/  R ) 
.\/  S )  =  ( ( P  .\/  Q )  .\/  R ) )
2416, 23eqtr3d 2445 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  .\/  R ) )
25 simp23 1032 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  P  =/=  Q )
26 simp32 1034 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  -.  R  .<_  ( P  .\/  Q ) )
27 simp12 1028 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  P  e.  A )
28 simp13 1029 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  Q  e.  A )
29 eqid 2402 . . . . . 6  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3020, 5, 6, 29islpln2a 32565 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
) )  ->  (
( ( P  .\/  Q )  .\/  R )  e.  ( LPlanes `  K
)  <->  ( P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )
311, 27, 28, 9, 30syl13anc 1232 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( ( P  .\/  Q )  .\/  R )  e.  ( LPlanes `  K
)  <->  ( P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )
3225, 26, 31mpbir2and 923 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  .\/  R )  e.  ( LPlanes `  K )
)
3324, 32eqeltrd 2490 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  .\/  ( R  .\/  S ) )  e.  ( LPlanes `  K )
)
34 eqid 2402 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
355, 6, 34llni2 32529 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
361, 27, 28, 25, 35syl31anc 1233 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( P  .\/  Q )  e.  ( LLines `  K )
)
37 simp31 1033 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  R  =/=  S )
385, 6, 34llni2 32529 . . . 4  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
391, 9, 12, 37, 38syl31anc 1233 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  ( R  .\/  S )  e.  ( LLines `  K )
)
40 2atmat.m . . . 4  |-  ./\  =  ( meet `  K )
415, 40, 6, 34, 292llnmj 32577 . . 3  |-  ( ( K  e.  HL  /\  ( P  .\/  Q )  e.  ( LLines `  K
)  /\  ( R  .\/  S )  e.  (
LLines `  K ) )  ->  ( ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  e.  A  <->  ( ( P  .\/  Q
)  .\/  ( R  .\/  S ) )  e.  ( LPlanes `  K )
) )
421, 36, 39, 41syl3anc 1230 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  <->  ( ( P 
.\/  Q )  .\/  ( R  .\/  S ) )  e.  ( LPlanes `  K ) ) )
4333, 42mpbird 232 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  P  =/=  Q
)  /\  ( R  =/=  S  /\  -.  R  .<_  ( P  .\/  Q
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  R )
) )  ->  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395   ` cfv 5569  (class class class)co 6278   Basecbs 14841   lecple 14916   joincjn 15897   meetcmee 15898   Latclat 15999   Atomscatm 32281   HLchlt 32368   LLinesclln 32508   LPlanesclpl 32509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516
This theorem is referenced by:  4atexlemc  33086
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