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Theorem hlomcmcv 32355
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 2402 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2402 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2402 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
4 eqid 2402 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2402 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2402 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 eqid 2402 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat1 32351 . 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 458 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    e. wcel 1842    =/= wne 2598   A.wral 2753   E.wrex 2754   class class class wbr 4394   ` cfv 5525  (class class class)co 6234   Basecbs 14733   lecple 14808   ltcplt 15786   joincjn 15789   0.cp0 15883   1.cp1 15884   CLatccla 15953   OMLcoml 32174   Atomscatm 32262   CvLatclc 32264   HLchlt 32349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-iota 5489  df-fv 5533  df-ov 6237  df-hlat 32350
This theorem is referenced by:  hloml  32356  hlclat  32357  hlcvl  32358  cvr1  32408  cvrp  32414  atcvr1  32415  atcvr2  32416
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