Theorem polvalN 33470

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 5875	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2525	. . 3 |- A e. _V
4	3	elpw2 4567	. 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 33469	. . . 4 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
9	8	fveq1d 5867	. . 3 |- ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
10		iineq1 4293	. . . . 5 |- ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
11	10	ineq2d 3634	. . . 4 |- ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
12		eqid 2451	. . . 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 4544	. . . 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 2505	. 2 |- ( ( K e. B /\ X e. ~P A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
16	4, 15	sylan2br 479	1 |- ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   |^|_ciin 4279   |-> cmpt 4461   ` cfv 5582   occoc 15198   Atomscatm 32829   pmapcpmap 33062   _|_PcpolN 33467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-polarityN 33468

This theorem is referenced by:  polval2N  33471  pol0N  33474  polcon3N  33482  polatN  33496