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Theorem 2atm 30009
Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
Hypotheses
Ref Expression
2atm.l  |-  .<_  =  ( le `  K )
2atm.j  |-  .\/  =  ( join `  K )
2atm.m  |-  ./\  =  ( meet `  K )
2atm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atm  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )

Proof of Theorem 2atm
StepHypRef Expression
1 simp31 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( P  .\/  Q ) )
2 simp32 994 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( R  .\/  S ) )
3 simp11 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  HL )
4 hllat 29846 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  Lat )
6 simp23 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  A )
7 eqid 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
97, 8atbase 29772 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
106, 9syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  ( Base `  K )
)
11 simp12 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  A )
127, 8atbase 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  ( Base `  K )
)
14 simp13 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  A )
157, 8atbase 29772 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 15syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  ( Base `  K )
)
17 2atm.j . . . . . 6  |-  .\/  =  ( join `  K )
187, 17latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
195, 13, 16, 18syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
20 simp21 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
21 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  S  e.  A )
227, 17, 8hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
233, 20, 21, 22syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
24 2atm.l . . . . 5  |-  .<_  =  ( le `  K )
25 2atm.m . . . . 5  |-  ./\  =  ( meet `  K )
267, 24, 25latlem12 14462 . . . 4  |-  ( ( K  e.  Lat  /\  ( T  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
) )  ->  (
( T  .<_  ( P 
.\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
275, 10, 19, 23, 26syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( T  .<_  ( P  .\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) ) ) )
281, 2, 27mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) )
29 hlatl 29843 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
303, 29syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  AtLat
)
317, 25latmcl 14435 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
325, 19, 23, 31syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
33 eqid 2404 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
347, 24, 33, 8atlen0 29793 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K
)  /\  T  e.  A )  /\  T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )  ->  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3530, 32, 6, 28, 34syl31anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3635neneqd 2583 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  -.  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =  ( 0. `  K
) )
37 simp33 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  =/=  ( R  .\/  S ) )
3817, 25, 33, 82atmat0 30008 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
393, 11, 14, 20, 21, 37, 38syl33anc 1199 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
4039ord 367 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( -.  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( 0.
`  K ) ) )
4136, 40mt3d 119 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
4224, 8atcmp 29794 . . 3  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A )  ->  ( T  .<_  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
4330, 6, 41, 42syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) ) )
4428, 43mpbid 202 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   0.cp0 14421   Latclat 14429   Atomscatm 29746   AtLatcal 29747   HLchlt 29833
This theorem is referenced by:  cdlemk12  31332  cdlemk12u  31354  cdlemk47  31431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980
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