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Theorem polvalN 30387
Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
polfval.o  |-  ._|_  =  ( oc `  K )
polfval.a  |-  A  =  ( Atoms `  K )
polfval.m  |-  M  =  ( pmap `  K
)
polfval.p  |-  P  =  ( _|_ P `  K )
Assertion
Ref Expression
polvalN  |-  ( ( K  e.  B  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
Distinct variable groups:    K, p    X, p
Allowed substitution hints:    A( p)    B( p)    P( p)    M( p)    ._|_ (
p)

Proof of Theorem polvalN
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 polfval.a . . . 4  |-  A  =  ( Atoms `  K )
2 fvex 5701 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2474 . . 3  |-  A  e. 
_V
43elpw2 4324 . 2  |-  ( X  e.  ~P A  <->  X  C_  A
)
5 polfval.o . . . . 5  |-  ._|_  =  ( oc `  K )
6 polfval.m . . . . 5  |-  M  =  ( pmap `  K
)
7 polfval.p . . . . 5  |-  P  =  ( _|_ P `  K )
85, 1, 6, 7polfvalN 30386 . . . 4  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
98fveq1d 5689 . . 3  |-  ( K  e.  B  ->  ( P `  X )  =  ( ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) ) `
 X ) )
10 iineq1 4067 . . . . 5  |-  ( m  =  X  ->  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) )  =  |^|_ p  e.  X  ( M `
 (  ._|_  `  p
) ) )
1110ineq2d 3502 . . . 4  |-  ( m  =  X  ->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
12 eqid 2404 . . . 4  |-  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) )  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) )
133inex1 4304 . . . 4  |-  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) )  e.  _V
1411, 12, 13fvmpt 5765 . . 3  |-  ( X  e.  ~P A  -> 
( ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) `  X )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
159, 14sylan9eq 2456 . 2  |-  ( ( K  e.  B  /\  X  e.  ~P A
)  ->  ( P `  X )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p
) ) ) )
164, 15sylan2br 463 1  |-  ( ( K  e.  B  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   |^|_ciin 4054    e. cmpt 4226   ` cfv 5413   occoc 13492   Atomscatm 29746   pmapcpmap 29979   _|_ PcpolN 30384
This theorem is referenced by:  polval2N  30388  pol0N  30391  polcon3N  30399  polatN  30413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-polarityN 30385
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