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Theorem lpolpolsatN 32287
Description: Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolpolsat.a  |-  A  =  (LSAtoms `  W )
lpolpolsat.p  |-  P  =  (LPol `  W )
lpolpolsat.w  |-  ( ph  ->  W  e.  X )
lpolpolsat.o  |-  ( ph  -> 
._|_  e.  P )
lpolpolsat.q  |-  ( ph  ->  Q  e.  A )
Assertion
Ref Expression
lpolpolsatN  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )

Proof of Theorem lpolpolsatN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolpolsat.o . . 3  |-  ( ph  -> 
._|_  e.  P )
2 lpolpolsat.w . . . 4  |-  ( ph  ->  W  e.  X )
3 eqid 2436 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
4 eqid 2436 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
5 eqid 2436 . . . . 5  |-  ( 0g
`  W )  =  ( 0g `  W
)
6 lpolpolsat.a . . . . 5  |-  A  =  (LSAtoms `  W )
7 eqid 2436 . . . . 5  |-  (LSHyp `  W )  =  (LSHyp `  W )
8 lpolpolsat.p . . . . 5  |-  P  =  (LPol `  W )
93, 4, 5, 6, 7, 8islpolN 32281 . . . 4  |-  ( W  e.  X  ->  (  ._|_  e.  P  <->  (  ._|_  : ~P ( Base `  W
) --> ( LSubSp `  W
)  /\  ( (  ._|_  `  ( Base `  W
) )  =  {
( 0g `  W
) }  /\  A. x A. y ( ( x  C_  ( Base `  W )  /\  y  C_  ( Base `  W
)  /\  x  C_  y
)  ->  (  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) ) ) )
102, 9syl 16 . . 3  |-  ( ph  ->  (  ._|_  e.  P  <->  ( 
._|_  : ~P ( Base `  W ) --> ( LSubSp `  W )  /\  (
(  ._|_  `  ( Base `  W ) )  =  { ( 0g `  W ) }  /\  A. x A. y ( ( x  C_  ( Base `  W )  /\  y  C_  ( Base `  W
)  /\  x  C_  y
)  ->  (  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) ) ) )
111, 10mpbid 202 . 2  |-  ( ph  ->  (  ._|_  : ~P ( Base `  W ) --> ( LSubSp `  W )  /\  ( (  ._|_  `  ( Base `  W ) )  =  { ( 0g
`  W ) }  /\  A. x A. y ( ( x 
C_  ( Base `  W
)  /\  y  C_  ( Base `  W )  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W
)  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) ) )
12 simpr3 965 . . 3  |-  ( ( 
._|_  : ~P ( Base `  W ) --> ( LSubSp `  W )  /\  (
(  ._|_  `  ( Base `  W ) )  =  { ( 0g `  W ) }  /\  A. x A. y ( ( x  C_  ( Base `  W )  /\  y  C_  ( Base `  W
)  /\  x  C_  y
)  ->  (  ._|_  `  y )  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) ) )  ->  A. x  e.  A  ( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x ) )
13 lpolpolsat.q . . . 4  |-  ( ph  ->  Q  e.  A )
14 fveq2 5728 . . . . . . 7  |-  ( x  =  Q  ->  (  ._|_  `  x )  =  (  ._|_  `  Q ) )
1514eleq1d 2502 . . . . . 6  |-  ( x  =  Q  ->  (
(  ._|_  `  x )  e.  (LSHyp `  W )  <->  ( 
._|_  `  Q )  e.  (LSHyp `  W )
) )
1614fveq2d 5732 . . . . . . 7  |-  ( x  =  Q  ->  (  ._|_  `  (  ._|_  `  x
) )  =  ( 
._|_  `  (  ._|_  `  Q
) ) )
17 id 20 . . . . . . 7  |-  ( x  =  Q  ->  x  =  Q )
1816, 17eqeq12d 2450 . . . . . 6  |-  ( x  =  Q  ->  (
(  ._|_  `  (  ._|_  `  x ) )  =  x  <->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q ) )
1915, 18anbi12d 692 . . . . 5  |-  ( x  =  Q  ->  (
( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x )  <->  ( (  ._|_  `  Q )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  Q
) )  =  Q ) ) )
2019rspcv 3048 . . . 4  |-  ( Q  e.  A  ->  ( A. x  e.  A  ( (  ._|_  `  x
)  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x )  ->  ( (  ._|_  `  Q )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  Q ) )  =  Q ) ) )
2113, 20syl 16 . . 3  |-  ( ph  ->  ( A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  x
) )  =  x )  ->  ( (  ._|_  `  Q )  e.  (LSHyp `  W )  /\  (  ._|_  `  (  ._|_  `  Q ) )  =  Q ) ) )
22 simpr 448 . . 3  |-  ( ( (  ._|_  `  Q )  e.  (LSHyp `  W
)  /\  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )
2312, 21, 22syl56 32 . 2  |-  ( ph  ->  ( (  ._|_  : ~P ( Base `  W ) --> ( LSubSp `  W )  /\  ( (  ._|_  `  ( Base `  W ) )  =  { ( 0g
`  W ) }  /\  A. x A. y ( ( x 
C_  ( Base `  W
)  /\  y  C_  ( Base `  W )  /\  x  C_  y )  ->  (  ._|_  `  y
)  C_  (  ._|_  `  x ) )  /\  A. x  e.  A  ( (  ._|_  `  x )  e.  (LSHyp `  W
)  /\  (  ._|_  `  (  ._|_  `  x ) )  =  x ) ) )  ->  (  ._|_  `  (  ._|_  `  Q
) )  =  Q ) )
2411, 23mpd 15 1  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  Q ) )  =  Q )
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 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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