MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-odu Structured version   Unicode version

Definition df-odu 16326
Description: Define the dual of an ordered structure, which replaces the order component of the structure with its reverse. See odubas 16330, oduleval 16328, and oduleg 16329 for its principal properties.

EDITORIAL: likely usable to simplify many lattice proofs, as it allows for duality arguments to be formalized; for instance latmass 16385. (Contributed by Stefan O'Rear, 29-Jan-2015.)

Assertion
Ref Expression
df-odu  |- ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le `  ndx ) ,  `' ( le `  w )
>. ) )

Detailed syntax breakdown of Definition df-odu
StepHypRef Expression
1 codu 16325 . 2  class ODual
2 vw . . 3  setvar  w
3 cvv 3087 . . 3  class  _V
42cv 1436 . . . 4  class  w
5 cnx 15081 . . . . . 6  class  ndx
6 cple 15159 . . . . . 6  class  le
75, 6cfv 5601 . . . . 5  class  ( le
`  ndx )
84, 6cfv 5601 . . . . . 6  class  ( le
`  w )
98ccnv 4853 . . . . 5  class  `' ( le `  w )
107, 9cop 4008 . . . 4  class  <. ( le `  ndx ) ,  `' ( le `  w ) >.
11 csts 15082 . . . 4  class sSet
124, 10, 11co 6305 . . 3  class  ( w sSet  <. ( le `  ndx ) ,  `' ( le `  w ) >.
)
132, 3, 12cmpt 4484 . 2  class  ( w  e.  _V  |->  ( w sSet  <. ( le `  ndx ) ,  `' ( le `  w ) >.
) )
141, 13wceq 1437 1  wff ODual  =  ( w  e.  _V  |->  ( w sSet  <. ( le `  ndx ) ,  `' ( le `  w )
>. ) )
Colors of variables: wff setvar class
This definition is referenced by:  oduval  16327
  Copyright terms: Public domain W3C validator