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Definition df-lpolN 30360
Description: Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.)
Assertion
Ref Expression
df-lpolN  |- LPol  =  ( w  e.  _V  |->  { o  e.  ( (
LSubSp `  w )  ^m  ~P ( Base `  w
) )  |  ( ( o `  ( Base `  w ) )  =  { ( 0g
`  w ) }  /\  A. x A. y ( ( x 
C_  ( Base `  w
)  /\  y  C_  ( Base `  w )  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x ) ) } )
Distinct variable group:    w, o, x, y

Detailed syntax breakdown of Definition df-lpolN
StepHypRef Expression
1 clpoN 30359 . 2  class LPol
2 vw . . 3  set  w
3 cvv 2727 . . 3  class  _V
42cv 1618 . . . . . . . 8  class  w
5 cbs 13022 . . . . . . . 8  class  Base
64, 5cfv 4592 . . . . . . 7  class  ( Base `  w )
7 vo . . . . . . . 8  set  o
87cv 1618 . . . . . . 7  class  o
96, 8cfv 4592 . . . . . 6  class  ( o `
 ( Base `  w
) )
10 c0g 13274 . . . . . . . 8  class  0g
114, 10cfv 4592 . . . . . . 7  class  ( 0g
`  w )
1211csn 3544 . . . . . 6  class  { ( 0g `  w ) }
139, 12wceq 1619 . . . . 5  wff  ( o `
 ( Base `  w
) )  =  {
( 0g `  w
) }
14 vx . . . . . . . . . . 11  set  x
1514cv 1618 . . . . . . . . . 10  class  x
1615, 6wss 3078 . . . . . . . . 9  wff  x  C_  ( Base `  w )
17 vy . . . . . . . . . . 11  set  y
1817cv 1618 . . . . . . . . . 10  class  y
1918, 6wss 3078 . . . . . . . . 9  wff  y  C_  ( Base `  w )
2015, 18wss 3078 . . . . . . . . 9  wff  x  C_  y
2116, 19, 20w3a 939 . . . . . . . 8  wff  ( x 
C_  ( Base `  w
)  /\  y  C_  ( Base `  w )  /\  x  C_  y )
2218, 8cfv 4592 . . . . . . . . 9  class  ( o `
 y )
2315, 8cfv 4592 . . . . . . . . 9  class  ( o `
 x )
2422, 23wss 3078 . . . . . . . 8  wff  ( o `
 y )  C_  ( o `  x
)
2521, 24wi 6 . . . . . . 7  wff  ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  ->  ( o `  y )  C_  (
o `  x )
)
2625, 17wal 1532 . . . . . 6  wff  A. y
( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w )  /\  x  C_  y )  ->  (
o `  y )  C_  ( o `  x
) )
2726, 14wal 1532 . . . . 5  wff  A. x A. y ( ( x 
C_  ( Base `  w
)  /\  y  C_  ( Base `  w )  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)
28 clsh 27854 . . . . . . . . 9  class LSHyp
294, 28cfv 4592 . . . . . . . 8  class  (LSHyp `  w )
3023, 29wcel 1621 . . . . . . 7  wff  ( o `
 x )  e.  (LSHyp `  w )
3123, 8cfv 4592 . . . . . . . 8  class  ( o `
 ( o `  x ) )
3231, 15wceq 1619 . . . . . . 7  wff  ( o `
 ( o `  x ) )  =  x
3330, 32wa 360 . . . . . 6  wff  ( ( o `  x )  e.  (LSHyp `  w
)  /\  ( o `  ( o `  x
) )  =  x )
34 clsa 27853 . . . . . . 7  class LSAtoms
354, 34cfv 4592 . . . . . 6  class  (LSAtoms `  w
)
3633, 14, 35wral 2509 . . . . 5  wff  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x )
3713, 27, 36w3a 939 . . . 4  wff  ( ( o `  ( Base `  w ) )  =  { ( 0g `  w ) }  /\  A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x ) )
38 clss 15524 . . . . . 6  class  LSubSp
394, 38cfv 4592 . . . . 5  class  ( LSubSp `  w )
406cpw 3530 . . . . 5  class  ~P ( Base `  w )
41 cmap 6658 . . . . 5  class  ^m
4239, 40, 41co 5710 . . . 4  class  ( (
LSubSp `  w )  ^m  ~P ( Base `  w
) )
4337, 7, 42crab 2512 . . 3  class  { o  e.  ( ( LSubSp `  w )  ^m  ~P ( Base `  w )
)  |  ( ( o `  ( Base `  w ) )  =  { ( 0g `  w ) }  /\  A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x ) ) }
442, 3, 43cmpt 3974 . 2  class  ( w  e.  _V  |->  { o  e.  ( ( LSubSp `  w )  ^m  ~P ( Base `  w )
)  |  ( ( o `  ( Base `  w ) )  =  { ( 0g `  w ) }  /\  A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x ) ) } )
451, 44wceq 1619 1  wff LPol  =  ( w  e.  _V  |->  { o  e.  ( (
LSubSp `  w )  ^m  ~P ( Base `  w
) )  |  ( ( o `  ( Base `  w ) )  =  { ( 0g
`  w ) }  /\  A. x A. y ( ( x 
C_  ( Base `  w
)  /\  y  C_  ( Base `  w )  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  (LSAtoms `  w )
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x ) ) } )
Colors of variables: wff set class
This definition is referenced by:  lpolsetN  30361
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