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Theorem polfvalN 33553
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.
StepHypRef Expression
1 elex 2986 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 polfval.p . . 3  |-  P  =  ( _|_P `  K )
3 fveq2 5696 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 polfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2493 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3870 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 fveq2 5696 . . . . . . . . . 10  |-  ( h  =  K  ->  ( pmap `  h )  =  ( pmap `  K
) )
8 polfval.m . . . . . . . . . 10  |-  M  =  ( pmap `  K
)
97, 8syl6eqr 2493 . . . . . . . . 9  |-  ( h  =  K  ->  ( pmap `  h )  =  M )
10 fveq2 5696 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( oc `  h )  =  ( oc `  K
) )
11 polfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1210, 11syl6eqr 2493 . . . . . . . . . 10  |-  ( h  =  K  ->  ( oc `  h )  = 
._|_  )
1312fveq1d 5698 . . . . . . . . 9  |-  ( h  =  K  ->  (
( oc `  h
) `  p )  =  (  ._|_  `  p
) )
149, 13fveq12d 5702 . . . . . . . 8  |-  ( h  =  K  ->  (
( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  ( M `  (  ._|_  `  p ) ) )
1514adantr 465 . . . . . . 7  |-  ( ( h  =  K  /\  p  e.  m )  ->  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) )  =  ( M `  (  ._|_  `  p ) ) )
1615iineq2dv 4198 . . . . . 6  |-  ( h  =  K  ->  |^|_ p  e.  m  ( ( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  |^|_ p  e.  m  ( M `
 (  ._|_  `  p
) ) )
175, 16ineq12d 3558 . . . . 5  |-  ( h  =  K  ->  (
( Atoms `  h )  i^i  |^|_ p  e.  m  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) ) )  =  ( A  i^i  |^|_
p  e.  m  ( M `  (  ._|_  `  p ) ) ) )
186, 17mpteq12dv 4375 . . . 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 33552 . . . 4  |-  _|_P 
=  ( h  e. 
_V  |->  ( m  e. 
~P ( Atoms `  h
)  |->  ( ( Atoms `  h )  i^i  |^|_ p  e.  m  ( (
pmap `  h ) `  ( ( oc `  h ) `  p
) ) ) ) )
20 fvex 5706 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
214, 20eqeltri 2513 . . . . . 6  |-  A  e. 
_V
2221pwex 4480 . . . . 5  |-  ~P A  e.  _V
2322mptex 5953 . . . 4  |-  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) )  e.  _V
2418, 19, 23fvmpt 5779 . . 3  |-  ( K  e.  _V  ->  ( _|_P `  K )  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
252, 24syl5eq 2487 . 2  |-  ( K  e.  _V  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
261, 25syl 16 1  |-  ( K  e.  B  ->  P  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977    i^i cin 3332   ~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:  polvalN  33554
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