Theorem polvalN 33554

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 5706	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2513	. . 3 |- A e. _V
4	3	elpw2 4461	. 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 33553	. . . 4 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
9	8	fveq1d 5698	. . 3 |- ( K e. B -> ( P ` X ) = ( ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) ` X ) )
10		iineq1 4190	. . . . 5 |- ( m = X -> |^|_ p e. m ( M ` ( ._|_ ` p ) ) = |^|_ p e. X ( M ` ( ._|_ ` p ) ) )
11	10	ineq2d 3557	. . . 4 |- ( m = X -> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) = ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) )
12		eqid 2443	. . . 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 4438	. . . 4 |- ( A i^i |^|_ p e. X ( M ` ( ._|_ ` p ) ) ) e. _V
14	11, 12, 13	fvmpt 5779	. . 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 2495	. 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 1369   e. wcel 1756   _Vcvv 2977   i^i cin 3332   C_ wss 3333   ~Pcpw 3865   |^|_ciin 4177   e. cmpt 4355   ` cfv 5423   occoc 14251   Atomscatm 32913   pmapcpmap 33146   _|_PcpolN 33551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-polarityN 33552

This theorem is referenced by:  polval2N  33555  pol0N  33558  polcon3N  33566  polatN  33580