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Theorem cvlcvrp 28697
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22915 analog.) (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
cvlcvrp.b  |-  B  =  ( Base `  K
)
cvlcvrp.j  |-  .\/  =  ( join `  K )
cvlcvrp.m  |-  ./\  =  ( meet `  K )
cvlcvrp.z  |-  .0.  =  ( 0. `  K )
cvlcvrp.c  |-  C  =  (  <o  `  K )
cvlcvrp.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cvlcvrp  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  X C
( X  .\/  P
) ) )

Proof of Theorem cvlcvrp
StepHypRef Expression
1 simp13 992 . . . . 5  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  CvLat )
2 cvllat 28683 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
31, 2syl 17 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
4 simp2 961 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  X  e.  B )
5 cvlcvrp.b . . . . . 6  |-  B  =  ( Base `  K
)
6 cvlcvrp.a . . . . . 6  |-  A  =  ( Atoms `  K )
75, 6atbase 28646 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
873ad2ant3 983 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  B )
9 cvlcvrp.m . . . . 5  |-  ./\  =  ( meet `  K )
105, 9latmcom 14143 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
)  =  ( P 
./\  X ) )
113, 4, 8, 10syl3anc 1187 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( X  ./\  P )  =  ( P  ./\  X
) )
1211eqeq1d 2266 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  ( P  ./\ 
X )  =  .0.  ) )
13 cvlatl 28682 . . . 4  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
141, 13syl 17 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  AtLat )
15 simp3 962 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  A )
16 eqid 2258 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
17 cvlcvrp.z . . . 4  |-  .0.  =  ( 0. `  K )
185, 16, 9, 17, 6atnle 28674 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P ( le `  K ) X  <->  ( P  ./\ 
X )  =  .0.  ) )
1914, 15, 4, 18syl3anc 1187 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P ( le `  K ) X  <->  ( P  ./\ 
X )  =  .0.  ) )
20 cvlcvrp.j . . 3  |-  .\/  =  ( join `  K )
21 cvlcvrp.c . . 3  |-  C  =  (  <o  `  K )
225, 16, 20, 21, 6cvlcvr1 28696 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( -.  P ( le `  K ) X  <->  X C
( X  .\/  P
) ) )
2312, 19, 223bitr2d 274 1  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  X C
( X  .\/  P
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   0.cp0 14105   Latclat 14113   CLatccla 14175   OMLcoml 28532    <o ccvr 28619   Atomscatm 28620   AtLatcal 28621   CvLatclc 28622
This theorem is referenced by:  cvlatcvr1  28698  cvrp  28772
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679
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