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Theorem polfvalN 30702
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 2965 . 2  |-  ( K  e.  B  ->  K  e.  _V )
2 polfval.p . . 3  |-  P  =  ( _|_ P `  K )
3 fveq2 5729 . . . . . . 7  |-  ( h  =  K  ->  ( Atoms `  h )  =  ( Atoms `  K )
)
4 polfval.a . . . . . . 7  |-  A  =  ( Atoms `  K )
53, 4syl6eqr 2487 . . . . . 6  |-  ( h  =  K  ->  ( Atoms `  h )  =  A )
65pweqd 3805 . . . . 5  |-  ( h  =  K  ->  ~P ( Atoms `  h )  =  ~P A )
7 fveq2 5729 . . . . . . . . . 10  |-  ( h  =  K  ->  ( pmap `  h )  =  ( pmap `  K
) )
8 polfval.m . . . . . . . . . 10  |-  M  =  ( pmap `  K
)
97, 8syl6eqr 2487 . . . . . . . . 9  |-  ( h  =  K  ->  ( pmap `  h )  =  M )
10 fveq2 5729 . . . . . . . . . . 11  |-  ( h  =  K  ->  ( oc `  h )  =  ( oc `  K
) )
11 polfval.o . . . . . . . . . . 11  |-  ._|_  =  ( oc `  K )
1210, 11syl6eqr 2487 . . . . . . . . . 10  |-  ( h  =  K  ->  ( oc `  h )  = 
._|_  )
1312fveq1d 5731 . . . . . . . . 9  |-  ( h  =  K  ->  (
( oc `  h
) `  p )  =  (  ._|_  `  p
) )
149, 13fveq12d 5735 . . . . . . . 8  |-  ( h  =  K  ->  (
( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  ( M `  (  ._|_  `  p ) ) )
1514adantr 453 . . . . . . 7  |-  ( ( h  =  K  /\  p  e.  m )  ->  ( ( pmap `  h
) `  ( ( oc `  h ) `  p ) )  =  ( M `  (  ._|_  `  p ) ) )
1615iineq2dv 4116 . . . . . 6  |-  ( h  =  K  ->  |^|_ p  e.  m  ( ( pmap `  h ) `  ( ( oc `  h ) `  p
) )  =  |^|_ p  e.  m  ( M `
 (  ._|_  `  p
) ) )
175, 16ineq12d 3544 . . . . 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 4288 . . . 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 30701 . . . 4  |-  _|_ P  =  ( h  e. 
_V  |->  ( m  e. 
~P ( Atoms `  h
)  |->  ( ( Atoms `  h )  i^i  |^|_ p  e.  m  ( (
pmap `  h ) `  ( ( oc `  h ) `  p
) ) ) ) )
20 fvex 5743 . . . . . . 7  |-  ( Atoms `  K )  e.  _V
214, 20eqeltri 2507 . . . . . 6  |-  A  e. 
_V
2221pwex 4383 . . . . 5  |-  ~P A  e.  _V
2322mptex 5967 . . . 4  |-  ( m  e.  ~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p
) ) ) )  e.  _V
2418, 19, 23fvmpt 5807 . . 3  |-  ( K  e.  _V  ->  ( _|_ P `  K )  =  ( m  e. 
~P A  |->  ( A  i^i  |^|_ p  e.  m  ( M `  (  ._|_  `  p ) ) ) ) )
252, 24syl5eq 2481 . 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 set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2957    i^i cin 3320   ~Pcpw 3800   |^|_ciin 4095    e. cmpt 4267   ` cfv 5455   occoc 13538   Atomscatm 30062   pmapcpmap 30295   _|_ PcpolN 30700
This theorem is referenced by:  polvalN  30703
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-polarityN 30701
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