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Theorem cvlcvrp 34354
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 27067 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 1028 . . . . 5  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  CvLat )
2 cvllat 34340 . . . . 5  |-  ( K  e.  CvLat  ->  K  e.  Lat )
31, 2syl 16 . . . 4  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  Lat )
4 simp2 997 . . . 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 34303 . . . . 5  |-  ( P  e.  A  ->  P  e.  B )
873ad2ant3 1019 . . . 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 15565 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  P  e.  B )  ->  ( X  ./\  P
)  =  ( P 
./\  X ) )
113, 4, 8, 10syl3anc 1228 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  ( X  ./\  P )  =  ( P  ./\  X
) )
1211eqeq1d 2469 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  ( P  ./\ 
X )  =  .0.  ) )
13 cvlatl 34339 . . . 4  |-  ( K  e.  CvLat  ->  K  e.  AtLat
)
141, 13syl 16 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  K  e.  AtLat )
15 simp3 998 . . 3  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  P  e.  A )
16 eqid 2467 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
17 cvlcvrp.z . . . 4  |-  .0.  =  ( 0. `  K )
185, 16, 9, 17, 6atnle 34331 . . 3  |-  ( ( K  e.  AtLat  /\  P  e.  A  /\  X  e.  B )  ->  ( -.  P ( le `  K ) X  <->  ( P  ./\ 
X )  =  .0.  ) )
1914, 15, 4, 18syl3anc 1228 . 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 34353 . 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 281 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6285   Basecbs 14493   lecple 14565   joincjn 15434   meetcmee 15435   0.cp0 15527   Latclat 15535   CLatccla 15597   OMLcoml 34189    <o ccvr 34276   Atomscatm 34277   AtLatcal 34278   CvLatclc 34279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-poset 15436  df-plt 15448  df-lub 15464  df-glb 15465  df-join 15466  df-meet 15467  df-p0 15529  df-lat 15536  df-clat 15598  df-oposet 34190  df-ol 34192  df-oml 34193  df-covers 34280  df-ats 34281  df-atl 34312  df-cvlat 34336
This theorem is referenced by:  cvlatcvr1  34355  cvrp  34429
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