Theorem polfvalN 35190

Description: 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
polfvalN	|- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )

Distinct variable groups:   A, m   m, p, K

Allowed substitution hints:   A( p)   B( m, p)   P( m, p)   M( m, p)   ._|_ ( m, p)

Proof of Theorem polfvalN

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. B -> K e. _V )
2		polfval.p	. . 3 |- P = ( _|_P ` K )
3		fveq2 6191	. . . . . . 7 |- ( h = K -> ( Atoms ` h ) = ( Atoms ` K ) )
4		polfval.a	. . . . . . 7 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( h = K -> ( Atoms ` h ) = A )
6	5	pweqd 4163	. . . . 5 |- ( h = K -> ~P ( Atoms ` h ) = ~P A )
7		fveq2 6191	. . . . . . . . . 10 |- ( h = K -> ( pmap ` h ) = ( pmap ` K ) )
8		polfval.m	. . . . . . . . . 10 |- M = ( pmap ` K )
9	7, 8	syl6eqr 2674	. . . . . . . . 9 |- ( h = K -> ( pmap ` h ) = M )
10		fveq2 6191	. . . . . . . . . . 11 |- ( h = K -> ( oc ` h ) = ( oc ` K ) )
11		polfval.o	. . . . . . . . . . 11 |- ._|_ = ( oc ` K )
12	10, 11	syl6eqr 2674	. . . . . . . . . 10 |- ( h = K -> ( oc ` h ) = ._|_ )
13	12	fveq1d 6193	. . . . . . . . 9 |- ( h = K -> ( ( oc ` h ) ` p ) = ( ._|_ ` p ) )
14	9, 13	fveq12d 6197	. . . . . . . 8 |- ( h = K -> ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = ( M ` ( ._|_ ` p ) ) )
15	14	adantr 481	. . . . . . 7 |- ( ( h = K /\ p e. m ) -> ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = ( M ` ( ._|_ ` p ) ) )
16	15	iineq2dv 4543	. . . . . 6 |- ( h = K -> |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) = |^|_ p e. m ( M ` ( ._|_ ` p ) ) )
17	5, 16	ineq12d 3815	. . . . 5 |- ( h = K -> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) = ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) )
18	6, 17	mpteq12dv 4733	. . . 4 |- ( h = K -> ( m e. ~P ( Atoms ` h ) |-> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) ) = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
19		df-polarityN 35189	. . . 4 |- _|_P = ( h e. _V |-> ( m e. ~P ( Atoms ` h ) |-> ( ( Atoms ` h ) i^i |^|_ p e. m ( ( pmap ` h ) ` ( ( oc ` h ) ` p ) ) ) ) )
20		fvex 6201	. . . . . . 7 |- ( Atoms ` K ) e. _V
21	4, 20	eqeltri 2697	. . . . . 6 |- A e. _V
22	21	pwex 4848	. . . . 5 |- ~P A e. _V
23	22	mptex 6486	. . . 4 |- ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) e. _V
24	18, 19, 23	fvmpt 6282	. . 3 |- ( K e. _V -> ( _|_P ` K ) = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
25	2, 24	syl5eq 2668	. 2 |- ( K e. _V -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )
26	1, 25	syl 17	1 |- ( K e. B -> P = ( m e. ~P A |-> ( A i^i |^|_ p e. m ( M ` ( ._|_ ` p ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   |^|_ciin 4521   |-> cmpt 4729   ` cfv 5888   occoc 15949   Atomscatm 34550   pmapcpmap 34783   _|_PcpolN 35188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-polarityN 35189

This theorem is referenced by:  polvalN  35191