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Theorem lpolvN 32298
Description: The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolv.v  |-  V  =  ( Base `  W
)
lpolv.z  |-  .0.  =  ( 0g `  W )
lpolv.p  |-  P  =  (LPol `  W )
lpolv.w  |-  ( ph  ->  W  e.  X )
lpolv.o  |-  ( ph  -> 
._|_  e.  P )
Assertion
Ref Expression
lpolvN  |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  }
)

Proof of Theorem lpolvN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolv.o . . 3  |-  ( ph  -> 
._|_  e.  P )
2 lpolv.w . . . 4  |-  ( ph  ->  W  e.  X )
3 lpolv.v . . . . 5  |-  V  =  ( Base `  W
)
4 eqid 2296 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 lpolv.z . . . . 5  |-  .0.  =  ( 0g `  W )
6 eqid 2296 . . . . 5  |-  (LSAtoms `  W
)  =  (LSAtoms `  W
)
7 eqid 2296 . . . . 5  |-  (LSHyp `  W )  =  (LSHyp `  W )
8 lpolv.p . . . . 5  |-  P  =  (LPol `  W )
93, 4, 5, 6, 7, 8islpolN 32295 . . . 4  |-  ( W  e.  X  ->  (  ._|_  e.  P  <->  (  ._|_  : ~P V --> ( LSubSp `  W )  /\  (
(  ._|_  `  V )  =  {  .0.  }  /\  A. x A. y ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  -> 
(  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  (LSAtoms `  W )
( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) ) ) )
102, 9syl 15 . . 3  |-  ( ph  ->  (  ._|_  e.  P  <->  ( 
._|_  : ~P V --> ( LSubSp `  W )  /\  (
(  ._|_  `  V )  =  {  .0.  }  /\  A. x A. y ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  -> 
(  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  (LSAtoms `  W )
( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) ) ) )
111, 10mpbid 201 . 2  |-  ( ph  ->  (  ._|_  : ~P V
--> ( LSubSp `  W )  /\  ( (  ._|_  `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)  /\  A. x  e.  (LSAtoms `  W )
( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) ) )
12 simpr1 961 . 2  |-  ( ( 
._|_  : ~P V --> ( LSubSp `  W )  /\  (
(  ._|_  `  V )  =  {  .0.  }  /\  A. x A. y ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  -> 
(  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  (LSAtoms `  W )
( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) )  -> 
(  ._|_  `  V )  =  {  .0.  } )
1311, 12syl 15 1  |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1530    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ~Pcpw 3638   {csn 3653   -->wf 5267   ` cfv 5271   Basecbs 13164   0gc0g 13416   LSubSpclss 15705  LSAtomsclsa 29786  LSHypclsh 29787  LPolclpoN 32292
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-lpolN 32293
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