Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omlol Structured version   Unicode version

Theorem omlol 30040
Description: An orthomodular lattice is an ortholattice. (Contributed by NM, 18-Sep-2011.)
Assertion
Ref Expression
omlol  |-  ( K  e.  OML  ->  K  e.  OL )

Proof of Theorem omlol
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2438 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2438 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2438 . . 3  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2438 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
61, 2, 3, 4, 5isoml 30038 . 2  |-  ( K  e.  OML  <->  ( K  e.  OL  /\  A. x  e.  ( Base `  K
) A. y  e.  ( Base `  K
) ( x ( le `  K ) y  ->  y  =  ( x ( join `  K ) ( y ( meet `  K
) ( ( oc
`  K ) `  x ) ) ) ) ) )
76simplbi 448 1  |-  ( K  e.  OML  ->  K  e.  OL )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   A.wral 2707   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   occoc 13539   joincjn 14403   meetcmee 14404   OLcol 29974   OMLcoml 29975
This theorem is referenced by:  omlop  30041  omllat  30042  omllaw3  30045  omllaw4  30046  cmtcomlemN  30048  cmtbr2N  30053  cmtbr3N  30054  omlfh1N  30058  omlfh3N  30059  omlspjN  30061  hlol  30161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-ov 6086  df-oml 29979
  Copyright terms: Public domain W3C validator