Theorem polvalN 32902

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 5858	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2486	. . 3 |- A e. _V
4	3	elpw2 4557	. 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 32901	. . . 4 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
9	8	fveq1d 5850	. . 3 |- ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
10		iineq1 4285	. . . . 5 |- ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
11	10	ineq2d 3640	. . . 4 |- ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
12		eqid 2402	. . . 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 4534	. . . 4 |- ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) e. _V
14	11, 12, 13	fvmpt 5931	. . 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 2463	. 2 |- ( ( K e. B /\ X e. ~P A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
16	4, 15	sylan2br 474	1 |- ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1405   e. wcel 1842   _Vcvv 3058   i^i cin 3412   C_ wss 3413   ~Pcpw 3954   |^|_ciin 4271   |-> cmpt 4452   ` cfv 5568   occoc 14915   Atomscatm 32261   pmapcpmap 32494   _|_PcpolN 32899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-polarityN 32900

This theorem is referenced by:  polval2N  32903  pol0N  32906  polcon3N  32914  polatN  32928