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 ) ) ) )