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Theorem polvalN 29912
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 5577 . . . 4  |-  ( Atoms `  K )  e.  _V
31, 2eqeltri 2386 . . 3  |-  A  e. 
_V
43elpw2 4212 . 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 29911 . . . 4  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
98fveq1d 5565 . . 3  |-  ( K  e.  B  ->  ( P `  X )  =  ( ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) ) `
 X ) )
10 iineq1 3956 . . . . 5  |-  ( m  =  X  ->  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) )  =  |^|_ p  e.  X  ( M `
 (  ._|_  `  p
) ) )
1110ineq2d 3404 . . . 4  |-  ( m  =  X  ->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) )  =  ( A  i^i  |^|_
p  e.  X  ( M `  (  ._|_  `  p ) ) ) )
12 eqid 2316 . . . 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 4192 . . . 4  |-  ( A  i^i  |^|_ p  e.  X  ( M `  (  ._|_  `  p ) ) )  e.  _V
1411, 12, 13fvmpt 5640 . . 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 2368 . 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 1633    e. wcel 1701   _Vcvv 2822    i^i cin 3185    C_ wss 3186   ~Pcpw 3659   |^|_ciin 3943    e. cmpt 4114   ` cfv 5292   occoc 13263   Atomscatm 29271   pmapcpmap 29504   _|_ PcpolN 29909
This theorem is referenced by:  polval2N  29913  pol0N  29916  polcon3N  29924  polatN  29938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-polarityN 29910
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