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Theorem polvalN 33470
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 5875 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2525 . . 3  |-  A  e. 
_V
43elpw2 4567 . 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 33469 . . . 4  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
98fveq1d 5867 . . 3  |-  ( K  e.  B  ->  ( P `  X )  =  ( ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) ) `
 X ) )
10 iineq1 4293 . . . . 5  |-  ( m  =  X  ->  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) )  =  |^|_ p  e.  X  ( M `
 (  ._|_  `  p
) ) )
1110ineq2d 3634 . . . 4  |-  ( m  =  X  ->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
12 eqid 2451 . . . 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 4544 . . . 4  |-  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) )  e.  _V
1411, 12, 13fvmpt 5948 . . 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 2505 . 2  |-  ( ( K  e.  B  /\  X  e.  ~P A
)  ->  ( P `  X )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p
) ) ) )
164, 15sylan2br 479 1  |-  ( ( K  e.  B  /\  X  C_  A )  -> 
( P `  X
)  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   _Vcvv 3045    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   |^|_ciin 4279    |-> cmpt 4461   ` cfv 5582   occoc 15198   Atomscatm 32829   pmapcpmap 33062   _|_PcpolN 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-polarityN 33468
This theorem is referenced by:  polval2N  33471  pol0N  33474  polcon3N  33482  polatN  33496
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