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Theorem lpolsetN 37310
Description: The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolset.v  |-  V  =  ( Base `  W
)
lpolset.s  |-  S  =  ( LSubSp `  W )
lpolset.z  |-  .0.  =  ( 0g `  W )
lpolset.a  |-  A  =  (LSAtoms `  W )
lpolset.h  |-  H  =  (LSHyp `  W )
lpolset.p  |-  P  =  (LPol `  W )
Assertion
Ref Expression
lpolsetN  |-  ( W  e.  X  ->  P  =  { o  e.  ( S  ^m  ~P V
)  |  ( ( o `  V )  =  {  .0.  }  /\  A. x A. y
( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  A  ( (
o `  x )  e.  H  /\  (
o `  ( o `  x ) )  =  x ) ) } )
Distinct variable groups:    x, A    S, o    o, V    x, o, y, W
Allowed substitution hints:    A( y, o)    P( x, y, o)    S( x, y)    H( x, y, o)    V( x, y)    X( x, y, o)    .0. ( x, y, o)

Proof of Theorem lpolsetN
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lpolset.p . . 3  |-  P  =  (LPol `  W )
3 fveq2 5872 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
4 lpolset.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
53, 4syl6eqr 2516 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
6 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
7 lpolset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
86, 7syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
98pweqd 4020 . . . . . 6  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
105, 9oveq12d 6314 . . . . 5  |-  ( w  =  W  ->  (
( LSubSp `  w )  ^m  ~P ( Base `  w
) )  =  ( S  ^m  ~P V
) )
118fveq2d 5876 . . . . . . 7  |-  ( w  =  W  ->  (
o `  ( Base `  w ) )  =  ( o `  V
) )
12 fveq2 5872 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
13 lpolset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
1412, 13syl6eqr 2516 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1514sneqd 4044 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1611, 15eqeq12d 2479 . . . . . 6  |-  ( w  =  W  ->  (
( o `  ( Base `  w ) )  =  { ( 0g
`  w ) }  <-> 
( o `  V
)  =  {  .0.  } ) )
178sseq2d 3527 . . . . . . . . 9  |-  ( w  =  W  ->  (
x  C_  ( Base `  w )  <->  x  C_  V
) )
188sseq2d 3527 . . . . . . . . 9  |-  ( w  =  W  ->  (
y  C_  ( Base `  w )  <->  y  C_  V ) )
1917, 183anbi12d 1300 . . . . . . . 8  |-  ( w  =  W  ->  (
( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  <->  ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) ) )
2019imbi1d 317 . . . . . . 7  |-  ( w  =  W  ->  (
( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w )  /\  x  C_  y )  ->  (
o `  y )  C_  ( o `  x
) )  <->  ( (
x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (
o `  y )  C_  ( o `  x
) ) ) )
21202albidv 1716 . . . . . 6  |-  ( w  =  W  ->  ( A. x A. y ( ( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  ->  ( o `  y )  C_  (
o `  x )
)  <->  A. x A. y
( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
) ) )
22 fveq2 5872 . . . . . . . 8  |-  ( w  =  W  ->  (LSAtoms `  w )  =  (LSAtoms `  W ) )
23 lpolset.a . . . . . . . 8  |-  A  =  (LSAtoms `  W )
2422, 23syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  (LSAtoms `  w )  =  A )
25 fveq2 5872 . . . . . . . . . 10  |-  ( w  =  W  ->  (LSHyp `  w )  =  (LSHyp `  W ) )
26 lpolset.h . . . . . . . . . 10  |-  H  =  (LSHyp `  W )
2725, 26syl6eqr 2516 . . . . . . . . 9  |-  ( w  =  W  ->  (LSHyp `  w )  =  H )
2827eleq2d 2527 . . . . . . . 8  |-  ( w  =  W  ->  (
( o `  x
)  e.  (LSHyp `  w )  <->  ( o `  x )  e.  H
) )
2928anbi1d 704 . . . . . . 7  |-  ( w  =  W  ->  (
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x )  <->  ( (
o `  x )  e.  H  /\  (
o `  ( o `  x ) )  =  x ) ) )
3024, 29raleqbidv 3068 . . . . . 6  |-  ( w  =  W  ->  ( A. x  e.  (LSAtoms `  w ) ( ( o `  x )  e.  (LSHyp `  w
)  /\  ( o `  ( o `  x
) )  =  x )  <->  A. x  e.  A  ( ( o `  x )  e.  H  /\  ( o `  (
o `  x )
)  =  x ) ) )
3116, 21, 303anbi123d 1299 . . . . 5  |-  ( w  =  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 ) )  <->  ( (
o `  V )  =  {  .0.  }  /\  A. x A. y ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  -> 
( o `  y
)  C_  ( o `  x ) )  /\  A. x  e.  A  ( ( o `  x
)  e.  H  /\  ( o `  (
o `  x )
)  =  x ) ) ) )
3210, 31rabeqbidv 3104 . . . 4  |-  ( w  =  W  ->  { 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 ) ) }  =  { o  e.  ( S  ^m  ~P V )  |  ( ( o `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (
o `  y )  C_  ( o `  x
) )  /\  A. x  e.  A  (
( o `  x
)  e.  H  /\  ( o `  (
o `  x )
)  =  x ) ) } )
33 df-lpolN 37309 . . . 4  |- 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 ) ) } )
34 ovex 6324 . . . . 5  |-  ( S  ^m  ~P V )  e.  _V
3534rabex 4607 . . . 4  |-  { o  e.  ( S  ^m  ~P V )  |  ( ( o `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (
o `  y )  C_  ( o `  x
) )  /\  A. x  e.  A  (
( o `  x
)  e.  H  /\  ( o `  (
o `  x )
)  =  x ) ) }  e.  _V
3632, 33, 35fvmpt 5956 . . 3  |-  ( W  e.  _V  ->  (LPol `  W )  =  {
o  e.  ( S  ^m  ~P V )  |  ( ( o `
 V )  =  {  .0.  }  /\  A. x A. y ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  -> 
( o `  y
)  C_  ( o `  x ) )  /\  A. x  e.  A  ( ( o `  x
)  e.  H  /\  ( o `  (
o `  x )
)  =  x ) ) } )
372, 36syl5eq 2510 . 2  |-  ( W  e.  _V  ->  P  =  { o  e.  ( S  ^m  ~P V
)  |  ( ( o `  V )  =  {  .0.  }  /\  A. x A. y
( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  A  ( (
o `  x )  e.  H  /\  (
o `  ( o `  x ) )  =  x ) ) } )
381, 37syl 16 1  |-  ( W  e.  X  ->  P  =  { o  e.  ( S  ^m  ~P V
)  |  ( ( o `  V )  =  {  .0.  }  /\  A. x A. y
( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  ( o `  y )  C_  (
o `  x )
)  /\  A. x  e.  A  ( (
o `  x )  e.  H  /\  (
o `  ( o `  x ) )  =  x ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395    e. wcel 1819   A.wral 2807   {crab 2811   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   {csn 4032   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Basecbs 14643   0gc0g 14856   LSubSpclss 17704  LSAtomsclsa 34800  LSHypclsh 34801  LPolclpoN 37308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6299  df-lpolN 37309
This theorem is referenced by:  islpolN  37311
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