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Theorem cvlcvrp 28797
Description: A Hilbert lattice satisfies the covering property of Definition 7.4 of [MaedaMaeda] p. 31 and its converse. (cvp 22947 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 28783 . . . . 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 28746 . . . . 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 14175 . . . 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 2292 . 2  |-  ( ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat )  /\  X  e.  B  /\  P  e.  A )  ->  (
( X  ./\  P
)  =  .0.  <->  ( P  ./\ 
X )  =  .0.  ) )
13 cvlatl 28782 . . . 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 2284 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
17 cvlcvrp.z . . . 4  |-  .0.  =  ( 0. `  K )
185, 16, 9, 17, 6atnle 28774 . . 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 28796 . 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 1628    e. wcel 1688   class class class wbr 4024   ` cfv 5221  (class class class)co 5819   Basecbs 13142   lecple 13209   joincjn 14072   meetcmee 14073   0.cp0 14137   Latclat 14145   CLatccla 14207   OMLcoml 28632    <o ccvr 28719   Atomscatm 28720   AtLatcal 28721   CvLatclc 28722
This theorem is referenced by:  cvlatcvr1  28798  cvrp  28872
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-undef 6291  df-riota 6299  df-poset 14074  df-plt 14086  df-lub 14102  df-glb 14103  df-join 14104  df-meet 14105  df-p0 14139  df-lat 14146  df-clat 14208  df-oposet 28633  df-ol 28635  df-oml 28636  df-covers 28723  df-ats 28724  df-atl 28755  df-cvlat 28779
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