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Theorem pl42N 29440
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 2285 . . 3  |-  ( pmap `  K )  =  (
pmap `  K )
7 eqid 2285 . . 3  |-  ( + P `  K )  =  ( + P `  K )
81, 2, 3, 4, 5, 6, 7pl42lem4N 29439 . 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 960 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  K  e.  HL )
10 hllat 28821 . . . . 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 961 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  X  e.  B )
13 simpl3 962 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Y  e.  B )
141, 3latjcl 14151 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .\/  Y
)  e.  B )
1511, 12, 13, 14syl3anc 1184 . . . . . 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 963 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  Z  e.  B )
171, 4latmcl 14152 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( X  .\/  Y )  e.  B  /\  Z  e.  B )  ->  (
( X  .\/  Y
)  ./\  Z )  e.  B )
1811, 15, 16, 17syl3anc 1184 . . . . 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 964 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  W  e.  B )
201, 3latjcl 14151 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( X  .\/  Y )  ./\  Z )  e.  B  /\  W  e.  B )  ->  (
( ( X  .\/  Y )  ./\  Z )  .\/  W )  e.  B
)
2111, 18, 19, 20syl3anc 1184 . . . 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 965 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B  /\  V  e.  B
) )  ->  V  e.  B )
231, 4latmcl 14152 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( X 
.\/  Y )  ./\  Z )  .\/  W )  e.  B  /\  V  e.  B )  ->  (
( ( ( X 
.\/  Y )  ./\  Z )  .\/  W ) 
./\  V )  e.  B )
2411, 21, 22, 23syl3anc 1184 . . 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 14151 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  .\/  W
)  e.  B )
2611, 12, 19, 25syl3anc 1184 . . . . 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 14151 . . . . . 6  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  V  e.  B )  ->  ( Y  .\/  V
)  e.  B )
2811, 13, 22, 27syl3anc 1184 . . . . 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 14152 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  .\/  W )  e.  B  /\  ( Y  .\/  V )  e.  B )  ->  (
( X  .\/  W
)  ./\  ( Y  .\/  V ) )  e.  B )
3011, 26, 28, 29syl3anc 1184 . . . 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 14151 . . . 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 1184 . . 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 29218 . . 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 1184 . 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 936    = wceq 1624    e. wcel 1685    C_ wss 3154   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   occoc 13211   joincjn 14073   meetcmee 14074   Latclat 14146   HLchlt 28808   pmapcpmap 28954   + Pcpadd 29252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-polarityN 29360  df-psubclN 29392
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