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Theorem hlomcmcv 30154
Description: A Hilbert lattice is orthomodular, complete, and has the covering (exchange) property. (Contributed by NM, 5-Nov-2012.)
Assertion
Ref Expression
hlomcmcv  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )

Proof of Theorem hlomcmcv
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2436 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2436 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
4 eqid 2436 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2436 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2436 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 eqid 2436 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat1 30150 . 2  |-  ( K  e.  HL  <->  ( ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
)  /\  ( A. x  e.  ( Atoms `  K ) A. y  e.  ( Atoms `  K )
( x  =/=  y  ->  E. z  e.  (
Atoms `  K ) ( z  =/=  x  /\  z  =/=  y  /\  z
( le `  K
) ( x (
join `  K )
y ) ) )  /\  E. x  e.  ( Base `  K
) E. y  e.  ( Base `  K
) E. z  e.  ( Base `  K
) ( ( ( 0. `  K ) ( lt `  K
) x  /\  x
( lt `  K
) y )  /\  ( y ( lt
`  K ) z  /\  z ( lt
`  K ) ( 1. `  K ) ) ) ) ) )
98simplbi 447 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   ltcplt 14398   joincjn 14401   0.cp0 14466   1.cp1 14467   CLatccla 14536   OMLcoml 29973   Atomscatm 30061   CvLatclc 30063   HLchlt 30148
This theorem is referenced by:  hloml  30155  hlclat  30156  hlcvl  30157  cvr1  30207  cvrp  30213  atcvr1  30214  atcvr2  30215
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-hlat 30149
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