Theorem polvalN 34701

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.

Step	Hyp	Ref	Expression
1		polfval.a	. . . 4 |- A = ( Atoms ` K )
2		fvex 5874	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2551	. . 3 |- A e. _V
4	3	elpw2 4611	. 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 )
8	5, 1, 6, 7	polfvalN 34700	. . . 4 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
9	8	fveq1d 5866	. . 3 |- ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
10		iineq1 4340	. . . . 5 |- ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
11	10	ineq2d 3700	. . . 4 |- ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
12		eqid 2467	. . . 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 ) ) ) )
13	3	inex1 4588	. . . 4 |- ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) e. _V
14	11, 12, 13	fvmpt 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 ) ) ) )
15	9, 14	sylan9eq 2528	. 2 |- ( ( K e. B /\ X e. ~P A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
16	4, 15	sylan2br 476	1 |- ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   i^i cin 3475   C_ wss 3476   ~Pcpw 4010   |^|_ciin 4326   |-> cmpt 4505   ` cfv 5586   occoc 14559   Atomscatm 34060   pmapcpmap 34293   _|_PcpolN 34698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-polarityN 34699

This theorem is referenced by:  polval2N  34702  pol0N  34705  polcon3N  34713  polatN  34727