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Theorem lhpat3 29608
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 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 )  ->  S  .<_  ( P  .\/  Q ) )
2 simpr 447 . . . . . . 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 1018 . . . . . . . . . 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 28926 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 15 . . . . . . . . 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 982 . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
8 lhpat.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
97, 8atbase 28852 . . . . . . . . . 10  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
106, 9syl 15 . . . . . . . . 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 1020 . . . . . . . . . 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 981 . . . . . . . . . 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 28929 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
153, 11, 12, 14syl3anc 1182 . . . . . . . . 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 1019 . . . . . . . . . 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 29560 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1916, 18syl 15 . . . . . . . . 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 14184 . . . . . . . . 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 1184 . . . . . . . 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 451 . . . . . . 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 887 . . . . . 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 4063 . . . . 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 451 . . . . . . 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 28923 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  AtLat )
3028, 29syl 15 . . . . . 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 1009 . . . . . 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 1006 . . . . . . 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 1007 . . . . . . 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 1008 . . . . . . 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 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  =/=  Q )
3620, 13, 21, 8, 17, 26lhpat2 29607 . . . . . . 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 1186 . . . . . 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 28874 . . . . . 6  |-  ( ( K  e.  AtLat  /\  S  e.  A  /\  R  e.  A )  ->  ( S  .<_  R  <->  S  =  R ) )
3930, 31, 37, 38syl3anc 1182 . . . . 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 201 . . . 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 423 . . 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 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  W )  .<_  W )
435, 15, 19, 42syl3anc 1182 . . . . 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 4056 . . . 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 4026 . . . 4  |-  ( S  =  R  ->  ( S  .<_  W  <->  R  .<_  W ) )
4644, 45syl5ibrcom 213 . . 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 183 . 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 2480 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 28826   AtLatcal 28827   HLchlt 28913   LHypclh 29546
This theorem is referenced by:  4atexlemntlpq  29630  4atexlemnclw  29632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-lhyp 29550
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