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Theorem polfvalN 34993
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 3127 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 polfval.p . . 3  |-  P  =  ( _|_P `  K )
3 fveq2 5871 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 polfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2526 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 4020 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 fveq2 5871 . . . . . . . . . 10  |-  ( h  =  K  ->  ( pmap `  h )  =  ( pmap `  K
) )
8 polfval.m . . . . . . . . . 10  |-  M  =  ( pmap `  K
)
97, 8syl6eqr 2526 . . . . . . . . 9  |-  ( h  =  K  ->  ( pmap `  h )  =  M )
10 fveq2 5871 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( oc `  h )  =  ( oc `  K
) )
11 polfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1210, 11syl6eqr 2526 . . . . . . . . . 10  |-  ( h  =  K  ->  ( oc `  h )  = 
._|_  )
1312fveq1d 5873 . . . . . . . . 9  |-  ( h  =  K  ->  (
( oc `  h
) `  p )  =  (  ._|_  `  p
) )
149, 13fveq12d 5877 . . . . . . . 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 4353 . . . . . 6  |-  ( h  =  K  ->  |^|_ p  e.  m  ( ( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  |^|_ p  e.  m  ( M `
 (  ._|_  `  p
) ) )
175, 16ineq12d 3706 . . . . 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 4530 . . . 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 34992 . . . 4  |-  _|_P 
=  ( h  e. 
_V  |->  ( m  e. 
~P ( Atoms `  h
)  |->  ( ( Atoms `  h )  i^i  |^|_ p  e.  m  ( (
pmap `  h ) `  ( ( oc `  h ) `  p
) ) ) ) )
20 fvex 5881 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
214, 20eqeltri 2551 . . . . . 6  |-  A  e. 
_V
2221pwex 4635 . . . . 5  |-  ~P A  e.  _V
2322mptex 6141 . . . 4  |-  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) )  e.  _V
2418, 19, 23fvmpt 5956 . . 3  |-  ( K  e.  _V  ->  ( _|_P `  K )  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
252, 24syl5eq 2520 . 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 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   ~Pcpw 4015   |^|_ciin 4331    |-> cmpt 4510   ` cfv 5593   occoc 14575   Atomscatm 34353   pmapcpmap 34586   _|_PcpolN 34991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-polarityN 34992
This theorem is referenced by:  polvalN  34994
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