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Theorem islpoldN 32282
Description: Properties that determine a polarity. (Contributed by NM, 26-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 )
islpold.w  |-  ( ph  ->  W  e.  X )
islpold.1  |-  ( ph  -> 
._|_  : ~P V --> S )
islpold.2  |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  }
)
islpold.3  |-  ( (
ph  /\  ( x  C_  V  /\  y  C_  V  /\  x  C_  y
) )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)
islpold.4  |-  ( (
ph  /\  x  e.  A )  ->  (  ._|_  `  x )  e.  H )
islpold.5  |-  ( (
ph  /\  x  e.  A )  ->  (  ._|_  `  (  ._|_  `  x
) )  =  x )
Assertion
Ref Expression
islpoldN  |-  ( ph  -> 
._|_  e.  P )
Distinct variable groups:    x, A    x, y, W    x,  ._|_ , y    ph, x, y
Allowed substitution hints:    A( y)    P( x, y)    S( x, y)    H( x, y)    V( x, y)    X( x, y)    .0. ( x, y)

Proof of Theorem islpoldN
StepHypRef Expression
1 islpold.1 . 2  |-  ( ph  -> 
._|_  : ~P V --> S )
2 islpold.2 . . 3  |-  ( ph  ->  (  ._|_  `  V )  =  {  .0.  }
)
3 islpold.3 . . . . 5  |-  ( (
ph  /\  ( x  C_  V  /\  y  C_  V  /\  x  C_  y
) )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)
43ex 424 . . . 4  |-  ( ph  ->  ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) ) )
54alrimivv 1642 . . 3  |-  ( ph  ->  A. x A. y
( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) ) )
6 islpold.4 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (  ._|_  `  x )  e.  H )
7 islpold.5 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  (  ._|_  `  (  ._|_  `  x
) )  =  x )
86, 7jca 519 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
(  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) )
98ralrimiva 2789 . . 3  |-  ( ph  ->  A. x  e.  A  ( (  ._|_  `  x
)  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) )
102, 5, 93jca 1134 . 2  |-  ( ph  ->  ( (  ._|_  `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) )
11 islpold.w . . 3  |-  ( ph  ->  W  e.  X )
12 lpolset.v . . . 4  |-  V  =  ( Base `  W
)
13 lpolset.s . . . 4  |-  S  =  ( LSubSp `  W )
14 lpolset.z . . . 4  |-  .0.  =  ( 0g `  W )
15 lpolset.a . . . 4  |-  A  =  (LSAtoms `  W )
16 lpolset.h . . . 4  |-  H  =  (LSHyp `  W )
17 lpolset.p . . . 4  |-  P  =  (LPol `  W )
1812, 13, 14, 15, 16, 17islpolN 32281 . . 3  |-  ( W  e.  X  ->  (  ._|_  e.  P  <->  (  ._|_  : ~P V --> S  /\  ( (  ._|_  `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) ) ) )
1911, 18syl 16 . 2  |-  ( ph  ->  (  ._|_  e.  P  <->  ( 
._|_  : ~P V --> S  /\  ( (  ._|_  `  V
)  =  {  .0.  }  /\  A. x A. y ( ( x 
C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y )  C_  (  ._|_  `  x )
)  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) ) ) )
201, 10, 19mpbir2and 889 1  |-  ( ph  -> 
._|_  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ~Pcpw 3799   {csn 3814   -->wf 5450   ` cfv 5454   Basecbs 13469   0gc0g 13723   LSubSpclss 16008  LSAtomsclsa 29772  LSHypclsh 29773  LPolclpoN 32278
This theorem is referenced by:  dochpolN  32288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-lpolN 32279
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