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Theorem lhpat3 33076
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 1055 . . . . . . 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 1062 . . . . . . . . . 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 32394 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . . . . . . . . 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 1026 . . . . . . . . . 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 2404 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
8 lhpat.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
97, 8atbase 32320 . . . . . . . . . 10  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
106, 9syl 17 . . . . . . . . 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 1064 . . . . . . . . . 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 1025 . . . . . . . . . 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 32397 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
153, 11, 12, 14syl3anc 1232 . . . . . . . . 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 1063 . . . . . . . . . 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 33028 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 17 . . . . . . . . 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 16034 . . . . . . . . 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 1234 . . . . . . . 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 924 . . . . . 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 4437 . . . . 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 32391 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3028, 29syl 17 . . . . . 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 1053 . . . . . 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 1050 . . . . . . 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 1051 . . . . . . 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 1052 . . . . . . 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 1054 . . . . . . 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 33075 . . . . . . 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 1236 . . . . . 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 32342 . . . . . 6  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .<_  R  <->  S  =  R ) )
3930, 31, 37, 38syl3anc 1232 . . . . 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 212 . . . 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 16033 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
435, 15, 19, 42syl3anc 1232 . . . . 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 4430 . . . 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 4400 . . . 4  |-  ( S  =  R  ->  ( S  .<_  W  <->  R  .<_  W ) )
4644, 45syl5ibrcom 224 . . 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 192 . 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 2652 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 186    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600   class class class wbr 4397   ` cfv 5571  (class class class)co 6280   Basecbs 14843   lecple 14918   joincjn 15899   meetcmee 15900   Latclat 16001   Atomscatm 32294   AtLatcal 32295   HLchlt 32381   LHypclh 33014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-lhyp 33018
This theorem is referenced by:  4atexlemntlpq  33098  4atexlemnclw  33100
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