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Theorem pl42N 33548
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 2451 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
7 eqid 2451 . . 3  |-  ( +P `  K )  =  ( +P `  K )
81, 2, 3, 4, 5, 6, 7pl42lem4N 33547 . 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 1011 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  K  e.  HL )
10 hllat 32929 . . . . 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 1012 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  X  e.  B )
13 simpl3 1013 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Y  e.  B )
141, 3latjcl 16297 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1511, 12, 13, 14syl3anc 1268 . . . . . 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 1014 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Z  e.  B )
171, 4latmcl 16298 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
1811, 15, 16, 17syl3anc 1268 . . . . 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 1015 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  W  e.  B )
201, 3latjcl 16297 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  .\/  Y )  ./\  Z )  e.  B  /\  W  e.  B )  ->  (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  e.  B
)
2111, 18, 19, 20syl3anc 1268 . . . 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 1016 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  V  e.  B )
231, 4latmcl 16298 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( X 
.\/  Y )  ./\  Z )  .\/  W )  e.  B  /\  V  e.  B )  ->  (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  e.  B )
2411, 21, 22, 23syl3anc 1268 . . 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 16297 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  .\/  W
)  e.  B )
2611, 12, 19, 25syl3anc 1268 . . . . 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 16297 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
2811, 13, 22, 27syl3anc 1268 . . . . 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 16298 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  W )  e.  B  /\  ( Y  .\/  V )  e.  B )  ->  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )
3011, 26, 28, 29syl3anc 1268 . . . 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 16297 . . . 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 1268 . . 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 33326 . . 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 1268 . 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 238 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 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    C_ wss 3404   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   lecple 15197   occoc 15198   joincjn 16189   meetcmee 16190   Latclat 16291   HLchlt 32916   pmapcpmap 33062   +Pcpadd 33360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-riotaBAD 32525
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-undef 7020  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917  df-psubsp 33068  df-pmap 33069  df-padd 33361  df-polarityN 33468  df-psubclN 33500
This theorem is referenced by: (None)
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