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Theorem islpoldN 31745
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 1637 . . 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 2711 . . 3  |-  ( ph  ->  A. x  e.  A  ( (  ._|_  `  x
)  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) )
102, 5, 93jca 1133 . 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 31744 . . 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 935   A.wal 1545    = wceq 1647    e. wcel 1715   A.wral 2628    C_ wss 3238   ~Pcpw 3714   {csn 3729   -->wf 5354   ` cfv 5358   Basecbs 13356   0gc0g 13610   LSubSpclss 15899  LSAtomsclsa 29235  LSHypclsh 29236  LPolclpoN 31741
This theorem is referenced by:  dochpolN  31751
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-map 6917  df-lpolN 31742
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