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Theorem cvrp 30052
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 23866 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 29993 . 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 29977 . 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 1217 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 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   joincjn 14389   meetcmee 14390   0.cp0 14454   CLatccla 14524   OMLcoml 29812    <o ccvr 29899   Atomscatm 29900   CvLatclc 29902   HLchlt 29987
This theorem is referenced by:  atcvrj1  30067
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988
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