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Theorem pl42N 29323
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 2256 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
7 eqid 2256 . . 3  |-  ( + P `  K )  =  ( + P `  K )
81, 2, 3, 4, 5, 6, 7pl42lem4N 29322 . 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 963 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  K  e.  HL )
10 hllat 28704 . . . . 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 964 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  X  e.  B )
13 simpl3 965 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Y  e.  B )
141, 3latjcl 14104 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1511, 12, 13, 14syl3anc 1187 . . . . . 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 966 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Z  e.  B )
171, 4latmcl 14105 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
1811, 15, 16, 17syl3anc 1187 . . . . 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 967 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  W  e.  B )
201, 3latjcl 14104 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  .\/  Y )  ./\  Z )  e.  B  /\  W  e.  B )  ->  (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  e.  B
)
2111, 18, 19, 20syl3anc 1187 . . . 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 968 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  V  e.  B )
231, 4latmcl 14105 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( X 
.\/  Y )  ./\  Z )  .\/  W )  e.  B  /\  V  e.  B )  ->  (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  e.  B )
2411, 21, 22, 23syl3anc 1187 . . 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 14104 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  .\/  W
)  e.  B )
2611, 12, 19, 25syl3anc 1187 . . . . 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 14104 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
2811, 13, 22, 27syl3anc 1187 . . . . 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 14105 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  W )  e.  B  /\  ( Y  .\/  V )  e.  B )  ->  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )
3011, 26, 28, 29syl3anc 1187 . . . 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 14104 . . . 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 1187 . . 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 29101 . . 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 1187 . 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 227 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    C_ wss 3113   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   occoc 13164   joincjn 14026   meetcmee 14027   Latclat 14099   HLchlt 28691   pmapcpmap 28837   + Pcpadd 29135
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-polarityN 29243  df-psubclN 29275
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