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Theorem cvrp 28756
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22901 analog.) (Contributed by NM, 18-Nov-2011.)
Hypotheses
Ref Expression
cvrp.b  |-  B  =  ( Base `  K
)
cvrp.j  |-  .\/  =  ( join `  K )
cvrp.m  |-  ./\  =  ( meet `  K )
cvrp.z  |-  .0.  =  ( 0. `  K )
cvrp.c  |-  C  =  (  <o  `  K )
cvrp.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvrp  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  .0.  <->  X C
( X  .\/  P
) ) )

Proof of Theorem cvrp
StepHypRef Expression
1 hlomcmcv 28697 . 2  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
2 cvrp.b . . 3  |-  B  =  ( Base `  K
)
3 cvrp.j . . 3  |-  .\/  =  ( join `  K )
4 cvrp.m . . 3  |-  ./\  =  ( meet `  K )
5 cvrp.z . . 3  |-  .0.  =  ( 0. `  K )
6 cvrp.c . . 3  |-  C  =  (  <o  `  K )
7 cvrp.a . . 3  |-  A  =  ( Atoms `  K )
82, 3, 4, 5, 6, 7cvlcvrp 28681 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  X C
( X  .\/  P
) ) )
91, 8syl3an1 1220 1  |-  ( ( K  e.  HL  /\  X  e.  B  /\  P  e.  A )  ->  ( ( X  ./\  P )  =  .0.  <->  X C
( X  .\/  P
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   joincjn 14026   meetcmee 14027   0.cp0 14091   CLatccla 14161   OMLcoml 28516    <o ccvr 28603   Atomscatm 28604   CvLatclc 28606   HLchlt 28691
This theorem is referenced by:  atcvrj1  28771
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-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  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-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-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
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