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Theorem islpoldN 35064
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 436 . . . 4  |-  ( ph  ->  ( ( x  C_  V  /\  y  C_  V  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) ) )
54alrimivv 1776 . . 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 535 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  (
(  ._|_  `  x )  e.  H  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) )
98ralrimiva 2804 . . 3  |-  ( ph  ->  A. x  e.  A  ( (  ._|_  `  x
)  e.  H  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) )
102, 5, 93jca 1189 . 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 35063 . . 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 17 . 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 934 1  |-  ( ph  -> 
._|_  e.  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986   A.wal 1444    = wceq 1446    e. wcel 1889   A.wral 2739    C_ wss 3406   ~Pcpw 3953   {csn 3970   -->wf 5581   ` cfv 5585   Basecbs 15133   0gc0g 15350   LSubSpclss 18167  LSAtomsclsa 32552  LSHypclsh 32553  LPolclpoN 35060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7479  df-lpolN 35061
This theorem is referenced by:  dochpolN  35070
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