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Theorem 2atmat 34357
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 1026 . . . . . 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 34160 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . . 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 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 2atmat.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 2atmat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 34163 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
873ad2ant1 1017 . . . . 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 1029 . . . . . 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 34086 . . . . . 6  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
119, 10syl 16 . . . . 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 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 )
) )  ->  S  e.  A )
134, 6atbase 34086 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1412, 13syl 16 . . . . 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 15575 . . . . 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 1230 . . . 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 1034 . . . . 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 15531 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
193, 8, 11, 18syl3anc 1228 . . . . . 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 15544 . . . . . 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 1228 . . . . 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 2510 . . 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 1031 . . . 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 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  .<_  ( P  .\/  Q ) )
27 simp12 1027 . . . . 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 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 )
) )  ->  Q  e.  A )
29 eqid 2467 . . . . . 6  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3020, 5, 6, 29islpln2a 34344 . . . . 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 1230 . . . 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 920 . . 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 2555 . 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 2467 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
355, 6, 34llni2 34308 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
361, 27, 28, 25, 35syl31anc 1231 . . 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 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 )
) )  ->  R  =/=  S )
385, 6, 34llni2 34308 . . . 4  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
391, 9, 12, 37, 38syl31anc 1231 . . 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 34356 . . 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 1228 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   joincjn 15424   meetcmee 15425   Latclat 15525   Atomscatm 34060   HLchlt 34147   LLinesclln 34287   LPlanesclpl 34288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295
This theorem is referenced by:  4atexlemc  34865
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