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Theorem lpolsetN 31741
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 2872 . 2  |-  ( W  e.  X  ->  W  e.  _V )
2 lpolset.p . . 3  |-  P  =  (LPol `  W )
3 fveq2 5608 . . . . . . 7  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  ( LSubSp `  W )
)
4 lpolset.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
53, 4syl6eqr 2408 . . . . . 6  |-  ( w  =  W  ->  ( LSubSp `
 w )  =  S )
6 fveq2 5608 . . . . . . . 8  |-  ( w  =  W  ->  ( Base `  w )  =  ( Base `  W
) )
7 lpolset.v . . . . . . . 8  |-  V  =  ( Base `  W
)
86, 7syl6eqr 2408 . . . . . . 7  |-  ( w  =  W  ->  ( Base `  w )  =  V )
98pweqd 3706 . . . . . 6  |-  ( w  =  W  ->  ~P ( Base `  w )  =  ~P V )
105, 9oveq12d 5963 . . . . 5  |-  ( w  =  W  ->  (
( LSubSp `  w )  ^m  ~P ( Base `  w
) )  =  ( S  ^m  ~P V
) )
118fveq2d 5612 . . . . . . 7  |-  ( w  =  W  ->  (
o `  ( Base `  w ) )  =  ( o `  V
) )
12 fveq2 5608 . . . . . . . . 9  |-  ( w  =  W  ->  ( 0g `  w )  =  ( 0g `  W
) )
13 lpolset.z . . . . . . . . 9  |-  .0.  =  ( 0g `  W )
1412, 13syl6eqr 2408 . . . . . . . 8  |-  ( w  =  W  ->  ( 0g `  w )  =  .0.  )
1514sneqd 3729 . . . . . . 7  |-  ( w  =  W  ->  { ( 0g `  w ) }  =  {  .0.  } )
1611, 15eqeq12d 2372 . . . . . 6  |-  ( w  =  W  ->  (
( o `  ( Base `  w ) )  =  { ( 0g
`  w ) }  <-> 
( o `  V
)  =  {  .0.  } ) )
178sseq2d 3282 . . . . . . . . 9  |-  ( w  =  W  ->  (
x  C_  ( Base `  w )  <->  x  C_  V
) )
188sseq2d 3282 . . . . . . . . 9  |-  ( w  =  W  ->  (
y  C_  ( Base `  w )  <->  y  C_  V ) )
1917, 183anbi12d 1253 . . . . . . . 8  |-  ( w  =  W  ->  (
( x  C_  ( Base `  w )  /\  y  C_  ( Base `  w
)  /\  x  C_  y
)  <->  ( x  C_  V  /\  y  C_  V  /\  x  C_  y ) ) )
2019imbi1d 308 . . . . . . 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 1627 . . . . . 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 5608 . . . . . . . 8  |-  ( w  =  W  ->  (LSAtoms `  w )  =  (LSAtoms `  W ) )
23 lpolset.a . . . . . . . 8  |-  A  =  (LSAtoms `  W )
2422, 23syl6eqr 2408 . . . . . . 7  |-  ( w  =  W  ->  (LSAtoms `  w )  =  A )
25 fveq2 5608 . . . . . . . . . 10  |-  ( w  =  W  ->  (LSHyp `  w )  =  (LSHyp `  W ) )
26 lpolset.h . . . . . . . . . 10  |-  H  =  (LSHyp `  W )
2725, 26syl6eqr 2408 . . . . . . . . 9  |-  ( w  =  W  ->  (LSHyp `  w )  =  H )
2827eleq2d 2425 . . . . . . . 8  |-  ( w  =  W  ->  (
( o `  x
)  e.  (LSHyp `  w )  <->  ( o `  x )  e.  H
) )
2928anbi1d 685 . . . . . . 7  |-  ( w  =  W  ->  (
( ( o `  x )  e.  (LSHyp `  w )  /\  (
o `  ( o `  x ) )  =  x )  <->  ( (
o `  x )  e.  H  /\  (
o `  ( o `  x ) )  =  x ) ) )
3024, 29raleqbidv 2824 . . . . . 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 1252 . . . . 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 2859 . . . 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 31740 . . . 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 5970 . . . . 5  |-  ( S  ^m  ~P V )  e.  _V
3534rabex 4246 . . . 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 5685 . . 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 2402 . 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 15 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1540    = wceq 1642    e. wcel 1710   A.wral 2619   {crab 2623   _Vcvv 2864    C_ wss 3228   ~Pcpw 3701   {csn 3716   ` cfv 5337  (class class class)co 5945    ^m cmap 6860   Basecbs 13245   0gc0g 13499   LSubSpclss 15788  LSAtomsclsa 29233  LSHypclsh 29234  LPolclpoN 31739
This theorem is referenced by:  islpolN  31742
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-lpolN 31740
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