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

Theorem omlol 30052
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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2296 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2296 . . 3  |-  ( join `  K )  =  (
join `  K )
4 eqid 2296 . . 3  |-  ( meet `  K )  =  (
meet `  K )
5 eqid 2296 . . 3  |-  ( oc
`  K )  =  ( oc `  K
)
61, 2, 3, 4, 5isoml 30050 . 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 446 1  |-  ( K  e.  OML  ->  K  e.  OL )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   occoc 13232   joincjn 14094   meetcmee 14095   OLcol 29986   OMLcoml 29987
This theorem is referenced by:  omlop  30053  omllat  30054  omllaw3  30057  omllaw4  30058  cmtcomlemN  30060  cmtbr2N  30065  cmtbr3N  30066  omlfh1N  30070  omlfh3N  30071  omlspjN  30073  hlol  30173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-oml 29991
  Copyright terms: Public domain W3C validator