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Theorem hlomcmcv 33307
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 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2451 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2451 . . 3  |-  ( lt
`  K )  =  ( lt `  K
)
4 eqid 2451 . . 3  |-  ( join `  K )  =  (
join `  K )
5 eqid 2451 . . 3  |-  ( 0.
`  K )  =  ( 0. `  K
)
6 eqid 2451 . . 3  |-  ( 1.
`  K )  =  ( 1. `  K
)
7 eqid 2451 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
81, 2, 3, 4, 5, 6, 7ishlat1 33303 . 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 460 1  |-  ( K  e.  HL  ->  ( K  e.  OML  /\  K  e.  CLat  /\  K  e.  CvLat
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   ltcplt 15213   joincjn 15216   0.cp0 15309   1.cp1 15310   CLatccla 15379   OMLcoml 33126   Atomscatm 33214   CvLatclc 33216   HLchlt 33301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-iota 5479  df-fv 5524  df-ov 6193  df-hlat 33302
This theorem is referenced by:  hloml  33308  hlclat  33309  hlcvl  33310  cvr1  33360  cvrp  33366  atcvr1  33367  atcvr2  33368
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