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Theorem polvalN 30094
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 5539 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2353 . . 3  |-  A  e. 
_V
43elpw2 4175 . 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 30093 . . . 4  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
98fveq1d 5527 . . 3  |-  ( K  e.  B  ->  ( P `  X )  =  ( ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) ) `
 X ) )
10 iineq1 3919 . . . . 5  |-  ( m  =  X  ->  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) )  =  |^|_ p  e.  X  ( M `
 (  ._|_  `  p
) ) )
1110ineq2d 3370 . . . 4  |-  ( m  =  X  ->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
12 eqid 2283 . . . 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 4155 . . . 4  |-  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) )  e.  _V
1411, 12, 13fvmpt 5602 . . 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 2335 . 2  |-  ( ( K  e.  B  /\  X  e.  ~P A
)  ->  ( P `  X )  =  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p
) ) ) )
164, 15sylan2br 462 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 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   |^|_ciin 3906    e. cmpt 4077   ` cfv 5255   occoc 13216   Atomscatm 29453   pmapcpmap 29686   _|_ PcpolN 30091
This theorem is referenced by:  polval2N  30095  pol0N  30098  polcon3N  30106  polatN  30120
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-polarityN 30092
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