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Theorem lhpat3 30905
Description: There is only one atom under both  P  .\/  Q and co-atom  W. (Contributed by NM, 21-Nov-2012.)
Hypotheses
Ref Expression
lhpat.l  |-  .<_  =  ( le `  K )
lhpat.j  |-  .\/  =  ( join `  K )
lhpat.m  |-  ./\  =  ( meet `  K )
lhpat.a  |-  A  =  ( Atoms `  K )
lhpat.h  |-  H  =  ( LHyp `  K
)
lhpat2.r  |-  R  =  ( ( P  .\/  Q )  ./\  W )
Assertion
Ref Expression
lhpat3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( -.  S  .<_  W  <-> 
S  =/=  R ) )

Proof of Theorem lhpat3
StepHypRef Expression
1 simpl3r 1014 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  .<_  ( P  .\/  Q ) )
2 simpr 449 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  .<_  W )
3 simp1ll 1021 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  HL )
4 hllat 30223 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  K  e.  Lat )
6 simp2r 985 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  S  e.  A )
7 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
8 lhpat.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
97, 8atbase 30149 . . . . . . . . . 10  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
106, 9syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  S  e.  ( Base `  K ) )
11 simp1rl 1023 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  P  e.  A )
12 simp2l 984 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  Q  e.  A )
13 lhpat.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
147, 13, 8hlatjcl 30226 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
153, 11, 12, 14syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
16 simp1lr 1022 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  W  e.  H )
17 lhpat.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
187, 17lhpbase 30857 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  W  e.  ( Base `  K ) )
20 lhpat.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
21 lhpat.m . . . . . . . . . 10  |-  ./\  =  ( meet `  K )
227, 20, 21latlem12 14509 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( S  .<_  ( P 
.\/  Q )  /\  S  .<_  W )  <->  S  .<_  ( ( P  .\/  Q
)  ./\  W )
) )
235, 10, 15, 19, 22syl13anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( S  .<_  ( P  .\/  Q )  /\  S  .<_  W )  <-> 
S  .<_  ( ( P 
.\/  Q )  ./\  W ) ) )
2423adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  -> 
( ( S  .<_  ( P  .\/  Q )  /\  S  .<_  W )  <-> 
S  .<_  ( ( P 
.\/  Q )  ./\  W ) ) )
251, 2, 24mpbi2and 889 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  .<_  ( ( P 
.\/  Q )  ./\  W ) )
26 lhpat2.r . . . . . 6  |-  R  =  ( ( P  .\/  Q )  ./\  W )
2725, 26syl6breqr 4254 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  .<_  R )
283adantr 453 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  K  e.  HL )
29 hlatl 30220 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3028, 29syl 16 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  K  e.  AtLat )
31 simpl2r 1012 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  e.  A )
32 simpl1l 1009 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  -> 
( K  e.  HL  /\  W  e.  H ) )
33 simpl1r 1010 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
34 simpl2l 1011 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  Q  e.  A )
35 simpl3l 1013 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  P  =/=  Q )
3620, 13, 21, 8, 17, 26lhpat2 30904 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  R  e.  A
)
3732, 33, 34, 35, 36syl112anc 1189 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  R  e.  A )
3820, 8atcmp 30171 . . . . . 6  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .<_  R  <->  S  =  R ) )
3930, 31, 37, 38syl3anc 1185 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  -> 
( S  .<_  R  <->  S  =  R ) )
4027, 39mpbid 203 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  /\  S  .<_  W )  ->  S  =  R )
4140ex 425 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( S  .<_  W  ->  S  =  R )
)
427, 20, 21latmle2 14508 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
435, 15, 19, 42syl3anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( ( P  .\/  Q )  ./\  W )  .<_  W )
4426, 43syl5eqbr 4247 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  ->  R  .<_  W )
45 breq1 4217 . . . 4  |-  ( S  =  R  ->  ( S  .<_  W  <->  R  .<_  W ) )
4644, 45syl5ibrcom 215 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( S  =  R  ->  S  .<_  W ) )
4741, 46impbid 185 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( S  .<_  W  <->  S  =  R ) )
4847necon3bbid 2637 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  ( Q  e.  A  /\  S  e.  A )  /\  ( P  =/=  Q  /\  S  .<_  ( P  .\/  Q
) ) )  -> 
( -.  S  .<_  W  <-> 
S  =/=  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   AtLatcal 30124   HLchlt 30210   LHypclh 30843
This theorem is referenced by:  4atexlemntlpq  30927  4atexlemnclw  30929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-p1 14471  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847
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