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Theorem pl42N 32980
Description: Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pl42.b  |-  B  =  ( Base `  K
)
pl42.l  |-  .<_  =  ( le `  K )
pl42.j  |-  .\/  =  ( join `  K )
pl42.m  |-  ./\  =  ( meet `  K )
pl42.o  |-  ._|_  =  ( oc `  K )
Assertion
Ref Expression
pl42N  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .<_  (  ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  ./\  V
)  .<_  ( ( X 
.\/  Y )  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) ) ) )

Proof of Theorem pl42N
StepHypRef Expression
1 pl42.b . . 3  |-  B  =  ( Base `  K
)
2 pl42.l . . 3  |-  .<_  =  ( le `  K )
3 pl42.j . . 3  |-  .\/  =  ( join `  K )
4 pl42.m . . 3  |-  ./\  =  ( meet `  K )
5 pl42.o . . 3  |-  ._|_  =  ( oc `  K )
6 eqid 2402 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
7 eqid 2402 . . 3  |-  ( +P `  K )  =  ( +P `  K )
81, 2, 3, 4, 5, 6, 7pl42lem4N 32979 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .<_  (  ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( ( pmap `  K ) `  ( ( ( ( X  .\/  Y ) 
./\  Z )  .\/  W )  ./\  V )
)  C_  ( ( pmap `  K ) `  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W ) 
./\  ( Y  .\/  V ) ) ) ) ) )
9 simpl1 1000 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  K  e.  HL )
10 hllat 32361 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
119, 10syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  K  e.  Lat )
12 simpl2 1001 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  X  e.  B )
13 simpl3 1002 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Y  e.  B )
141, 3latjcl 16003 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1511, 12, 13, 14syl3anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  ( X  .\/  Y )  e.  B )
16 simpr1 1003 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Z  e.  B )
171, 4latmcl 16004 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
1811, 15, 16, 17syl3anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
19 simpr2 1004 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  W  e.  B )
201, 3latjcl 16003 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  .\/  Y )  ./\  Z )  e.  B  /\  W  e.  B )  ->  (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  e.  B
)
2111, 18, 19, 20syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  e.  B
)
22 simpr3 1005 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  V  e.  B )
231, 4latmcl 16004 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( X 
.\/  Y )  ./\  Z )  .\/  W )  e.  B  /\  V  e.  B )  ->  (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  e.  B )
2411, 21, 22, 23syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  e.  B )
251, 3latjcl 16003 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  .\/  W
)  e.  B )
2611, 12, 19, 25syl3anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  ( X  .\/  W )  e.  B )
271, 3latjcl 16003 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
2811, 13, 22, 27syl3anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  ( Y  .\/  V )  e.  B )
291, 4latmcl 16004 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  W )  e.  B  /\  ( Y  .\/  V )  e.  B )  ->  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )
3011, 26, 28, 29syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )
311, 3latjcl 16003 . . . 4  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )  ->  (
( X  .\/  Y
)  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) )  e.  B
)
3211, 15, 30, 31syl3anc 1230 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .\/  Y
)  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) )  e.  B
)
331, 2, 6pmaple 32758 . . 3  |-  ( ( K  e.  HL  /\  ( ( ( ( X  .\/  Y ) 
./\  Z )  .\/  W )  ./\  V )  e.  B  /\  (
( X  .\/  Y
)  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) )  e.  B
)  ->  ( (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  .<_  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W ) 
./\  ( Y  .\/  V ) ) )  <->  ( ( pmap `  K ) `  ( ( ( ( X  .\/  Y ) 
./\  Z )  .\/  W )  ./\  V )
)  C_  ( ( pmap `  K ) `  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W ) 
./\  ( Y  .\/  V ) ) ) ) ) )
349, 24, 32, 33syl3anc 1230 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( ( ( ( X  .\/  Y ) 
./\  Z )  .\/  W )  ./\  V )  .<_  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W ) 
./\  ( Y  .\/  V ) ) )  <->  ( ( pmap `  K ) `  ( ( ( ( X  .\/  Y ) 
./\  Z )  .\/  W )  ./\  V )
)  C_  ( ( pmap `  K ) `  ( ( X  .\/  Y )  .\/  ( ( X  .\/  W ) 
./\  ( Y  .\/  V ) ) ) ) ) )
358, 34sylibrd 234 1  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  (
( X  .<_  (  ._|_  `  Y )  /\  Z  .<_  (  ._|_  `  W ) )  ->  ( (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  ./\  V
)  .<_  ( ( X 
.\/  Y )  .\/  ( ( X  .\/  W )  ./\  ( Y  .\/  V ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3413   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   occoc 14915   joincjn 15895   meetcmee 15896   Latclat 15997   HLchlt 32348   pmapcpmap 32494   +Pcpadd 32792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-polarityN 32900  df-psubclN 32932
This theorem is referenced by: (None)
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