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Metamath Proof Explorer

Theorem polvalN

Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011) (New usage is discouraged.)

Ref Expression
Hypotheses polfval.o 
|- ._|_ = ( oc ` K )
polfval.a 
|- A = ( Atoms ` K )
polfval.m 
|- M = ( pmap ` K )
polfval.p 
|- P = ( _|_P ` K )
Assertion polvalN 
|- ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )

Proof

Step Hyp Ref Expression
1 polfval.o 
 |-  ._|_ = ( oc ` K )
2 polfval.a 
 |-  A = ( Atoms ` K )
3 polfval.m 
 |-  M = ( pmap ` K )
4 polfval.p 
 |-  P = ( _|_P ` K )
5 2 fvexi 
 |-  A e. _V
6 5 elpw2 
 |-  ( X e. ~P A <-> X C_ A )
7 1 2 3 4 polfvalN 
 |-  ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
8 7 fveq1d 
 |-  ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
9 iineq1 
 |-  ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
10 9 ineq2d 
 |-  ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
11 eqid 
 |-  ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) )
12 5 inex1 
 |-  ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) e. _V
13 10 11 12 fvmpt 
 |-  ( 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 ) ) ) )
14 8 13 sylan9eq 
 |-  ( ( K e. B /\ X e. ~P A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
15 6 14 sylan2br 
 |-  ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )