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Theorem 2atmat 30295
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 987 . . . . . 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 30098 . . . . . 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 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
5 2atmat.j . . . . . . 7  |-  .\/  =  ( join `  K )
6 2atmat.a . . . . . . 7  |-  A  =  ( Atoms `  K )
74, 5, 6hlatjcl 30101 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
873ad2ant1 978 . . . . 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 990 . . . . . 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 30024 . . . . . 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 991 . . . . . 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 30024 . . . . . 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 14516 . . . . 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 1186 . . . 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 995 . . . . 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 14471 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  R )  e.  ( Base `  K ) )
193, 8, 11, 18syl3anc 1184 . . . . . 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 14485 . . . . . 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 1184 . . . . 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 202 . . . 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 2469 . . 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 992 . . . 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 994 . . . 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 988 . . . . 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 989 . . . . 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 2435 . . . . . 6  |-  ( LPlanes `  K )  =  (
LPlanes `  K )
3020, 5, 6, 29islpln2a 30282 . . . . 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 1186 . . . 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 889 . . 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 2509 . 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 2435 . . . . 5  |-  ( LLines `  K )  =  (
LLines `  K )
355, 6, 34llni2 30246 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
361, 27, 28, 25, 35syl31anc 1187 . . 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 993 . . . 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 30246 . . . 4  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
391, 9, 12, 37, 38syl31anc 1187 . . 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 30294 . . 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 1184 . 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 224 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   Latclat 14466   Atomscatm 29998   HLchlt 30085   LLinesclln 30225   LPlanesclpl 30226
This theorem is referenced by:  4atexlemc  30803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233
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