Theorem polvalN 30387

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 5701	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2474	. . 3 |- A e. _V
4	3	elpw2 4324	. 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 30386	. . . 4 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
9	8	fveq1d 5689	. . 3 |- ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
10		iineq1 4067	. . . . 5 |- ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
11	10	ineq2d 3502	. . . 4 |- ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
12		eqid 2404	. . . 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 4304	. . . 4 |- ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) e. _V
14	11, 12, 13	fvmpt 5765	. . 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 2456	. 2 |- ( ( K e. B /\ X e. ~P A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
16	4, 15	sylan2br 463	1 |- ( ( K e. B /\ X C_ A ) -> ( P ` X ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   i^i cin 3279   C_ wss 3280   ~Pcpw 3759   |^|_ciin 4054   e. cmpt 4226   ` cfv 5413   occoc 13492   Atomscatm 29746   pmapcpmap 29979   _|_ PcpolN 30384

This theorem is referenced by:  polval2N  30388  pol0N  30391  polcon3N  30399  polatN  30413

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-polarityN 30385