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Theorem lhpat3 34717
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 1047 . . . . . . 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 461 . . . . . . 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 1054 . . . . . . . . . 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 34035 . . . . . . . . . 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 1018 . . . . . . . . . 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 2460 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
8 lhpat.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
97, 8atbase 33961 . . . . . . . . . 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 1056 . . . . . . . . . 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 1017 . . . . . . . . . 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 34038 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
153, 11, 12, 14syl3anc 1223 . . . . . . . . 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 1055 . . . . . . . . . 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 34669 . . . . . . . . . 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 15554 . . . . . . . . 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 1225 . . . . . . . 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 465 . . . . . . 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 914 . . . . . 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 4480 . . . . 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 465 . . . . . . 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 34032 . . . . . . 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 1045 . . . . . 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 1042 . . . . . . 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 1043 . . . . . . 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 1044 . . . . . . 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 1046 . . . . . . 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 34716 . . . . . . 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 1227 . . . . . 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 33983 . . . . . 6  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .<_  R  <->  S  =  R ) )
3930, 31, 37, 38syl3anc 1223 . . . . 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 210 . . . 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 434 . . 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 15553 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
435, 15, 19, 42syl3anc 1223 . . . . 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 4473 . . . 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 4443 . . . 4  |-  ( S  =  R  ->  ( S  .<_  W  <->  R  .<_  W ) )
4644, 45syl5ibrcom 222 . . 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 191 . 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 2707 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   meetcmee 15421   Latclat 15521   Atomscatm 33935   AtLatcal 33936   HLchlt 34022   LHypclh 34655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659
This theorem is referenced by:  4atexlemntlpq  34739  4atexlemnclw  34741
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