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Theorem islpoldN 31674
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 423 . . . 4  |-  ( ph  ->  ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) ) )
54alrimivv 1618 . . 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 518 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
(  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) )
98ralrimiva 2626 . . 3  |-  ( ph  ->  A. x  e.  A  ( (  ._|_  `  x
)  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) )
102, 5, 93jca 1132 . 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 31673 . . 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 15 . 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 888 1  |-  ( ph  -> 
._|_  e.  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   A.wal 1527    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   {csn 3640   -->wf 5251   ` cfv 5255   Basecbs 13148   0gc0g 13400   LSubSpclss 15689  LSAtomsclsa 29164  LSHypclsh 29165  LPolclpoN 31670
This theorem is referenced by:  dochpolN  31680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-lpolN 31671
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